{"text":"\\section{Basic Results} \\label{sec:basic_results}\n\nIn this section, we prove some basic results concerning the complexity of \\textsc{ListIso}\\xspace and \\textsc{ListAut}\\xspace.\n\n\\begin{lemma} \\label{lem:lautlgi_equiv}\nBoth problems \\textsc{ListAut}\\xspace and \\textsc{ListIso}\\xspace are polynomially equivalent.\n\\end{lemma}\n\n\\begin{proof}\nTo see that \\textsc{ListAut}\\xspace is polynomially reducible to \\textsc{ListIso}\\xspace just set $H$ to be a copy of $G$ and keep the\nlists for all vertices of $G$. It is straightforward to check that these two instances are\nequivalent. For the other direction, we build an instance $G'$ and $\\frakL'$ of \\textsc{ListAut}\\xspace as follows. Let\n$G'$ be a disjoint union of $G$ and $H$. And let $\\frakL'(v) = \\frakL(v)$ for all $v\\in V(G)$ and set\n$\\frakL'(w) = V(G)$ for all $w \\in V(H)$. It is easy to see that there exists list-compatible\nisomorphism from $G$ to $H$, if and only if there exists a list-compatible automorphism of $G'$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:list_size_two}\nThe problem \\textsc{ListIso}\\xspace can be solved in time $\\calO(n+m)$ when all lists are of size at\nmost two.\n\\end{lemma}\n\n\\begin{proof}\nWe construct a list-compatible isomorphism $\\pi : G \\to H$ by solving a 2-SAT formula which can be\ndone in linear time~\\cite{even_2sat,aspvall_2sat}. When $w \\in \\frakL(v)$, we assume that $\\deg(v) =\n\\deg(w)$, otherwise we remove $w$ from $\\frakL(v)$. Notice that if $\\frakL(u) = \\{w\\}$, we can set $\\pi(u)\n= w$ and for every $v \\in N(u)$, we modify $\\frakL(v) := L(v) \\cap N(w)$. Now, for every vertex $u_i$\nwith $\\frakL(u_i) = \\{w^0_i, w^1_i\\}$, we introduce a variable $x_i$ such that $\\pi(u_i) = w^{x_i}_i$.\nClearly, the mapping $\\pi$ is compatible with the lists.\n\nWe construct a 2-SAT formula such that there exists a list-compatible isomorphism if and only if it\nis satisfiable. First, if $\\frakL(u_i) \\cap \\frakL(u_j) \\ne \\emptyset$, we add implications for $x_i$ and\n$x_j$ such that $\\pi(u_i) \\ne \\pi(u_j)$. Next, when $\\pi(u_i) = w^j_i$, we add implications that\nevery $u_j \\in N(u_i)$ is mapped to $N(w^j_i)$. If $\\frakL(u_j) \\cap N(w^j_i) \\ne \\emptyset$, otherwise\n$u_i$ cannot be mapped to $w^j_i$ and $x_i \\ne j$. Therefore, $\\pi$ obtained from a satisfiable\nassignment maps $N[u]$ bijectively to $N[\\pi(u)]$ and it is an isomorphism. The total number of\nvariables in $n$, and the total number of clauses is $\\calO(n+m)$, so the running time is\n$\\calO(n+m)$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:disconnected}\nLet $G_1,\\dots,G_k$ be the components of $G$ and $H_1,\\dots,H_k$ be the components of $H$.\nIf we can decide \\textsc{ListIso}\\xspace in polynomial time for all pairs $G_i$ and $H_j$, then we can\nsolve \\textsc{ListIso}\\xspace for $G$ and $H$ in polynomial time.\n\\end{lemma}\n\n\\begin{proof}\nLet $G_1,\\dots,G_k$ be the components of $G$ and $H_1,\\dots,H_k$ be the components of $H$. For each\ncomponent $G_i$, we find all components $H_j$ such that there exists a list-compatible isomorphism\nfrom $G_i$ to $H_j$. Notice that a necessary condition is that every vertex in $G_i$ contains one\nvertex of $H_j$ in its list. So we can go through all lists of $G_i$ and find all candidates $H_j$,\nin total time $\\calO(\\ell)$ for all components $G_1,\\dots,G_k$. Let $n' = |V(G_i)|$, $m' = |E(G_i)|$,\nand $\\ell'$ be the total size of lists of $G_i$ restricted to $H_j$. We test existence of a\nlist-compatible isomorphism in time $\\varphi(n',m',\\ell')$. Then we form the bipartite graph $B$\nbetween $G_1,\\dots,G_k$ and $H_1,\\dots,H_k$ such that $G_iH_j \\in E(B)$ if and only if there exists\na list-compatible isomorphism from $G_i$ to $H_j$. There exists a list-compatible isomorphism from\n$G$ to $H$, if and only if there exists a perfect matching in $B$. Using Lemma~\\ref{lem:bipmatch},\nthis can be tested in time $\\calO(\\sqrt k \\ell)$. The total running time depends on the running time of\ntesting \\textsc{ListIso}\\xspace of the components, and we note that the sum of the lengths of lists in these test is at\nmost $\\ell$.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:cycles}\nThe problem \\textsc{ListIso}\\xspace can be solved for cycles in time $\\calO(\\ell)$.\n\\end{lemma}\n\n\\begin{proof}\nWe may assume that $|V(G)| = |V(H)|$. Let $u \\in V(G)$ be a vertex with a smallest list and let $k\n= |\\frakL(u)|$. Since $\\ell = \\calO(kn)$, it suffices to show that we can find a list-compatible\nisomorphism in time $\\calO(kn)$. We test all the $k$ possible mappings $\\pi \\colon G \\to H$ with\n$\\pi(u) \\in \\frakL(u)$. For $u \\in V(G)$ and $v \\in \\frakL(u)$, there are at most two possible isomorphisms\nthat map $u$ to $v$. For each of these isomorphism, we test whether they are list-compatible.\\qed\n\\end{proof}\n\n\\begin{lemma} \\label{lem:max_degree_2}\nThe problem \\textsc{ListIso}\\xspace can be solved for graphs of maximum degree 2 in time $\\calO(\\sqrt n \\ell)$.\n\\end{lemma}\n\n\\begin{proof}\nBoth graphs $G$ and $H$ are disjoint unions of paths and cycles of various lengths. For each two\nconnected components, we can decide in time $\\calO(\\ell')$ whether there exists a list-compatible\nisomorphism between them, where $\\ell'$ is the total size of lists when restricted to these\ncomponents: for paths trivially, and for cycles by Lemma~\\ref{lem:cycles}. The rest follows from\nLemma~\\ref{lem:disconnected}, where the running time is of each test in $\\calO(\\ell')$ where $\\ell'$ is\nthe total length of lists restricted to two components.\\qed\n\\end{proof}\n\n\n\\section{Bounded Genus Graphs} \\label{sec:bounded_genus}\n\nIn this section, we describe an FPT algorithm solving \\textsc{ListIso}\\xspace when parameterized by the Euler genus\n$g$. We modify the recent paper of Kawarabayashi~\\cite{kawarabayashi} solving graph isomorphism in\nlinear time for a fixed genus $g$. The harder part of this paper are structural results, described\nbelow, which transfer to list-compatible isomorphisms without any change. Using these structural\nresults, we can build our algorithm.\n\n\\begin{theorem} \\label{thm:bounded_genus}\nFor every integer $g$, the problem \\textsc{ListIso}\\xspace can be solved on graphs of Euler genus at most $g$ in time\n$\\calO(\\sqrt n \\ell)$.\n\\end{theorem}\n\n\\begin{proof}\nSee~\\cite[p.~14]{kawarabayashi} for overview of the main steps. We show that these steps can be\nmodified to deal with lists. We prove this result by induction on $g$, where the base case for $g=0$\nis Theorem~\\ref{thm:planar}. Next, we assume that both graphs $G$ and $H$ are 3-connected, otherwise\nwe apply Theorem~\\ref{thm:3conn_reduction}. By~\\cite[Theorem 1.2]{kawarabayashi}, if $G$ and $H$\nhave no polyhedral embeddings, then the face-width is at most two.\n\t \n\\emph{Case 1: $G$ and $H$ have polyhedral embeddings.} Following~\\cite[Theorem 1.2]{kawarabayashi},\nwe have at most $f(g)$ possible embeddings of $G$ and $H$. We choose one embedding of $G$ and we\ntest all embeddings of $H$. It is known that the average degree is $\\calO(g)$. Therefore, we can apply\nthe same idea as in the proof of Lemma~\\ref{lem:planar_3conn} and test isomorphism of all these\nembeddings in time $\\calO(\\ell)$. \n\n\\emph{Case 2: $G$ and $H$ have no polyhedral embedding, but have embeddings of face-width exactly two.}\nThen we split $G$ into a pair of graphs $(G',L)$. The graph $L$\nare called \\emph{cylinders} and the graph $G'$ correspond to the remainder of $G$. The following\nproperties hold~\\cite[p.~5]{kawarabayashi}:\n\\begin{packed_itemize}\n\\item We have $G = G' \\cup L$ and for $\\bo L = V(G' \\cap L)$, we have $|\\bo L| = 4$.\n\\item The graph $G'$ can be embedded to a surface of genus at most $g-1$, and $L$ is\nplanar~\\cite[p.~4]{kawarabayashi}.\n\\item This pair $(G',L)$ is canonical, i.e., every isomorphism from $G$ to $H$ maps $(G',L)$ to another\npair $(H',L')$ in $H$.\n\\end{packed_itemize}\nIt is proved~\\cite[Theorem 5.1]{kawarabayashi} that there exists some function $q'(g)$ bounding the\nnumber of these pairs both in $G$ and $H$, and can be found in time $\\calO(n)$. We fix a pair $(G',L)$\nin $G$ and iterate over all pairs $(H'_i,L'_i)$ in $H$. Following~\\cite[p.~36]{kawarabayashi}, we\nget that $G \\cong H$, if and only if there exists a pair $(H'_i,L'_i)$ in $H$ such that $G' \\cong\nH'_i$, $L \\cong L'_i$, and $G' \\cap L$ is mapped to $H'_i \\cap L'_i$. To test this, we run at most\n$2q'(g)$ instances of \\textsc{ListIso}\\xspace on smaller graphs with modified lists.\n\nSuppose that we want to test whether $G' \\cong H'_i$ and $L \\cong L'_i$. First, we modify the lists:\nfor $u \\in V(G')$, put $\\frakL'(u) = \\frakL(u) \\cap H'_i$, and for $v \\in V(L)$, put $\\frakL'(v) = \\frakL(v) \\cap\nL'_i$, and similarly for lists of darts. Further, for all vertices $u \\in \\bo L$ in both $G'$ and\n$L$, we put $\\frakL'(u) = \\frakL(u) \\cap \\bo L$. We test existence of list-compatible isomorphisms from $G'$\nto $H'_i$ and from $L$ to $L'_i$. There exists a list-compatible isomorphism from $G$ to $H$, if and\nonly if these list-compatible isomorphisms exist at least for one pair $(H'_i,L'_i)$.\n\nWe note that when $g=2$, a special case is described in~\\cite[Theorem 5.3]{kawarabayashi}, which is\nslightly easier and can be modified similarly.\n\n\\emph{Case 3: $G$ and $H$ have no polyhedral embedding and have only embeddings of face-width one.}\nLet $V$ be the set of vertices in $G$ such that for each $u \\in V$, there exists a non-contractible\ncurve passing only through $u$. By~\\cite[Lemma 6.3]{kawarabayashi}, $|V| \\le q(g)$ for some function\n$q$. For $u$, the non-contractible curve divides its edges to two sides, so we can cut $G$ at $u$,\nand split the incident edges. We obtain a graph $G'$ which can be embedded to a surface of genus at\nmost $g-1$.\n\nBy~\\cite[Lemma 6.3]{kawarabayashi}, we can find all these vertices $V$ and $V'$ in $G$ and $H$ in\ntime $\\calO(n)$. We choose $u \\in V$ arbitrarily, and we test all possible vertices $v \\in V'$.\nLet $G'$ be constructed from $G$ by splitting $u$ into new vertices $u'$ and $u''$, and similarly\n$H'$ be constructed from $H$ by splitting $v$ into new vertices $v'$ and $v''$. \nIn~\\cite[p.~36]{kawarabayashi}, it is stated that $G \\cong H$, if and only if there exists a choice\nof $v \\in V'$ such that $G' \\cong H'$ and $\\{u',u''\\}$ is mapped to $\\{v',v''\\}$.\nTherefore, we run at most $q(g)$ instances of \\textsc{ListIso}\\xspace on smaller graphs with modified lists.\n\nIf $v \\notin L(u)$, clearly a list-compatible isomorphism is not possible for this choice of $v \\in\nV'$. If $v \\in L(u)$, we put $L'(u') = L'(u'') = \\{v',v''\\}$. Then there exists a list-compatible\nisomorphism from $G$ to $H$, if and only if there exists a list-compatible isomorphism from $G'$ to\n$H'$.\n\nThe correctness of our algorithm follows from~\\cite{kawarabayashi}. It remains to argue the\ncomplexity. Throughout the algorithm, we produce at most $w(g)$ subgraphs of $G$ and $H$, for some\nfunction $w$, for which we test list-compatible isomorphisms. Assuming the induction hypothesis, the\nreduction of graphs to 3-connected graphs can be done in time $\\calO(\\sqrt n \\ell)$. Case 1 can be\nsolved in time $\\calO(\\ell)$. Case 2 can be solved in time $\\calO(\\sqrt n \\ell)$. Case 3 can be solved in\ntime $\\calO(\\sqrt n \\ell)$.\\qed\n\\end{proof}\n\n\\section{Bounded Treewidth Graphs} \\label{sec:bounded_treewidth}\n\nIn this section, we prove that \\textsc{ListIso}\\xspace can be solved in \\hbox{\\rm \\sffamily FPT}\\xspace with respect to the parameter treewidth\n$\\tw(G)$. Unlike in Sections~\\ref{sec:planar_graphs} and~\\ref{sec:intersection}, the difficulty\nof graph isomorphism on bounded treewidth graphs raises from the fact that\ntree decomposition is not uniquely determined. We follow the approach of\nBodlaender~\\cite{iso_xp_treewidth} which describes an \\hbox{\\rm \\sffamily XP}\\xspace algorithm for \\textsc{GraphIso}\\xspace of bounded treewidth\ngraphs, running in time $n^{\\calO(\\tw(G))}$. Then we show that the recent breakthrough by Lokshtanov\net al.~\\cite{iso_fpt_treewidth}, giving an \\hbox{\\rm \\sffamily FPT}\\xspace algorithm for \\textsc{GraphIso}\\xspace, translates as well.\n\n\\begin{definition}\nA \\emph{tree decomposition} of a graph $G$ is a pair ${\\cal T} =\n(\\{B_i\\colon i\\in I\\},T = (I,F)),$ where $T$ is a rooted tree and $\\{B_i\\colon\ni\\in I\\}$ is a family of subsets of $V,$ such that\n\\begin{enumerate}\n\\item for each $v\\in V(G)$ there exists an $i \\in I$ such that $v\\in B_i$,\n\\item for each $e\\in E(G)$ there exists an $i \\in I$ such that $e\\subseteq B_i$,\n\\item for each $v\\in V(G), I_v = \\{i \\in I\\colon v\\in B_i\\}$ induces a subtree of $T.$\n\\end{enumerate} \nWe call the elements $B_i$ the \\emph{nodes}, and the elements of the set $F$ the\ndecomposition edges.\n\\end{definition}\n\nWe define the width of a tree decomposition ${\\cal T} = (\\{B_i\\colon i\\in I\\},\nT)$ as $\\max_{i\\in I}|B_i|-1$ and the \\emph{treewidth} $\\tw(G)$ of a graph $G$\nas the minimum width of a tree decomposition of the graph $G$. \n\n\\heading{Nice Tree Decompositions.}\nIt is common to define a {\\it nice tree decomposition} of the graph~\\cite{kloks}. We naturally\norient the decomposition edges towards the root and for an oriented decomposition edge $(B_j,B_i)$\nfrom $B_j$ to $B_i$ we call $B_i$ the {\\it parent} of $B_j$ and $B_j$ a {\\it child} of $B_i$. If\nthere is an oriented path from $B_j$ to $B_i$ we say that $B_j$ is a {\\it descendant} of $B_i$.\n\nWe also adjust a tree decomposition such that for each decomposition edge $(B_i,B_j)$ it holds that\n$\\big| |B_i|-|B_j| \\big| \\le 1$ (i.e. it joins nodes that differ in at most one vertex). The\nin-degree of each node is at most $2$ and if the in-degree of the node $B_k$ is $2$ then for its\nchildren $B_i,B_j$ holds that $B_i = B_j = B_k$ (i.e. they represent the same vertex set).\n\nWe classify the nodes of a nice decomposition into four classes---namely {\\it introduce nodes}, {\\it\nforget nodes}, {\\it join nodes} and {\\it leaf nodes}. We call the node $B_i$ an introduce node of\nthe vertex $v$, if it has a single child $B_j$ and $B_i\\setminus B_j = \\{v\\}$. We call the node\n$B_i$ a forget node of the vertex $v$, if it has a single child $B_j$ and $B_j\\setminus B_i =\n\\{v\\}$. If the node $B_k$ has two children, we call it a join node (of nodes $B_i$ and $B_j$).\nFinally we call a node $B_i$ a leaf node, if it has no child.\n\n\\heading{Bodlaender's Algorithm.}\nA graph $G$ has treewidth at most $k$ if either $|V(G)| \\le k$, or there exists a cut set $U\n\\subseteq V(G)$ such that $|U| \\le k$ and each component of $G \\setminus U$ together with $U$ has\ntreewidth at most $k$. The set $U$ corresponds to a bag in some tree decomposition of $G$.\nBodlaender's algorithm~\\cite{iso_xp_treewidth} enumerates all possible cut sets $U$ of size at most\n$k$ in $G$ (resp. $H$), we denote these $C_i$ (resp. $D_i$). Furthermore, it enumerates all\nconnected components of $G\\setminus C_i$ as $C_i^j$ (resp.~of $H \\setminus D_i$ as $D_i^j$). We\ndenote by $G[U,W]$ the graph induced by $U \\mathbin{\\dot\\cup} W$. The set $W$ is either a connected component\nor a collection of connected components. We call $U$ the \\emph{border set}.\n\n\\begin{lemma}[\\cite{ACP87:partialktrees,iso_xp_treewidth}]\n\\label{lem:partialKTree}\nA graph $G[U, W]$ with at least $k$ vertices has a treewidth at most $k$ with the border set $U$ if\nand only if there exists a vertex $v\\in W$ such that for each connected component $A$ of $G[W\n\\setminus v]$, there is a $k$-vertex cut $C_s \\subseteq U\\cup\\{v\\}$ such that no vertex in $A$ is\nadjacent to the (unique) vertex in $(U \\cup\\{v\\})\\setminus C_s$, and $G[C_s, A]$ has treewidth at\nmost $k$.\n\\end{lemma}\n\n\\begin{lemma} \\label{lem:xp_treewidth}\nThe problem \\textsc{ListIso}\\xspace can be solved in \\hbox{\\rm \\sffamily XP}\\xspace with respect to the parameter treewidth.\n\\end{lemma}\n\n\\begin{proof}\nWe modify the algorithm of Bodlaender~\\cite{iso_xp_treewidth}. Let $k = \\tw(G) = \\tw(H)$. We\ncompute the sets $C_i, C_i^j$ for $G$ and the sets $D_{i'}, D_{i'}^{j'}$ for $H$; there are\n$n^{\\calO(k)}$ pairs $(C_i,C_i^j)$. The pair $(C_i, C_i^j)$ is \\emph{compatible} if $C_i^j$ is a\nconnected component of $G'\\setminus C_i$ for some $G'\\subseteq G$ that arises during the recursive\ndefinition of treewidth. Let $f \\colon C_i\\to D_{i'}$ be an isomorphism. We say that $(C_i,\nC_i^j)\\equiv_f(D_{i'},D_{i'}^{j'})$ if and only if there exists an isomorphism $\\varphi\\colon\nC_i\\cup C_i^j\\to D_{i'}\\cup D_{i'}^{j'}$ such that $\\varphi|_{C_i} = f$. In other words, $\\varphi$\nis a partial isomorphism from $G$ to $H$. The change for \\textsc{ListIso}\\xspace is that we also require that both $f$\nand $\\varphi$ are list-compatible.\n\nThe algorithm resolves $(C_i, C_i^j)\\equiv_f(D_{i'},D_{i'}^{j'})$ by the dynamic programming,\naccording to the size of $D_{i'}^{j'}$. If $|C_i^j| = |D_{i'}^{j'}| \\le 1$, we can check it\ntrivially in time $k^{\\calO(k)}$. Otherwise, suppose that $|C_i^j| = |D_{i'}^{j'}| > 1$, and let\n$m$ be the number of components of $C_i^j$ (and thus $D_{i'}^{j'}$). We test whether $f : C_i \\to\nD_{i'}$ is a list-compatible isomorphism. Let $v\\in C_i^j$ be a vertex given by\nLemma~\\ref{lem:partialKTree} (with $U = C_i$ and $W = C_i^j$) and let $C_s$ be the corresponding\nextension of $v$ to a cut set. We compute for all $w \\in D_{i'}^{j'} \\cap \\frakL(v)$ all connected\ncomponents $B_q$. From the dynamic programming, we know for all possible extensions $D'$ of $w$ to a\ncut set whether $(C_m, A_p)\\equiv_{f'}(D',B_q)$ with $f'(x) = f(x)$ for $x\\in C_i$ and $f'(v) = w$.\nFinally, we decide whether there exists a perfect matching in the bipartite graph between $(C_m,\nA_p)$'s and $(D',B_q)$'s where the edges are according to the equivalence.\\qed\n\\end{proof}\n\n\\heading{Reducing The Number of Possible Bags.}\nOtachi and Schweitzer~\\cite{GIReductionTechniques} proposed the idea of pruning the family of\npotential bags which finally led to an \\hbox{\\rm \\sffamily FPT}\\xspace algorithm~\\cite{iso_fpt_treewidth}. A family\n$\\mathcal{B}(G)$, whose definition depends on the graph, is called \\emph{isomorphism-invariant} if\nfor an isomorphism $\\phi : G \\to G'$, we get $\\mathcal{B}(G') = \\phi(\\mathcal{B}(G))$, where\n$\\phi(\\mathcal{B}(G))$ denotes the family $\\mathcal{B}(G)$ with all the vertices of $G$ replaced by\ntheir images under $\\phi$. \n\nFor a graph $G$, a pair $(A,B)$ with $A \\cup B = V$ is called a {\\em\nseparation} if there are no edges between $A\\setminus B$ and $B\\setminus A$ in\n$G$. The order of $(A,B)$ is $|A\\cap B|$. For two vertices $u,v \\in V(G)$, by\n$\\mu(u,v)$ we denote the minimum order of separation $(A,B)$ with $u\\in\nA\\setminus B$ and $v\\in B\\setminus A$. We say a graph $G$ is {\\em\n$k$-complemented} if $\\mu_G(u,v) \\ge k) \\implies uv \\in E(G)$ holds for\nevery two vertices $u,v\\in V$. We may canonically modify the input graphs $G$ and $H$ \\textsc{ListIso}\\xspace, by\nadding these additional edges and making them $k$-complemented.\n\n\n\\begin{theorem}[\\cite{iso_fpt_treewidth}, Theorem~5.5] \\label{thm:pruned_bags}\nLet $k$ be a positive integer, and let $G$ be a graph on $n$ vertices that is connected and\n$k$-complemented. There exists an algorithm that computes in time $2^{\\calO(k^5\\log k)}\\cdot n^3$ an\nisomorphism-invariant family of bags $\\mathcal{B}$ with the following properties:\n\\begin{enumerate}\n \\item $|B|\\le \\calO(k^4)$ for each $B\\in\\mathcal{B}$,\n \\item $|\\mathcal{B}|\\le 2^{\\calO(k^5\\log k)}\\cdot n^2$,\n \\item Assuming $\\tw(G) 0 \"damping (D) should be >0\"\n\t@assert H > 0 \"inertia (H) should be >0\" \n\tSOmega_H = SOmega * 2pi \/ H\nend [[Somega, dSomega]] begin\n\tp = real(u * conj(i))\n\tdu = u * im * Somega\n\tdSomega = (P - D*Somega - p)*SOmega_H\nend\n\\end{lstlisting}\nAgain, the left part of the first line provides the name of the new type and the name of the parameters. Lines 2 to 4 are code that should be run only once. In this case these are consistency checks, that the damping and inertia are positive, and the reduction of the rated frequency $\\mathtt{\\Omega}$ and the inertia $\\mathtt{H}$ to a single variable $\\mathtt{\\Omega\\_H}$. In line 5, the variable name of the internal variable $\\mathtt{\\omega}$ and the name for its derivative $\\mathtt{d\\omega}$ are given. Finally, lines 6 to 8 implement \\Cref{eq:swing-u,eq:swing-omega} by simply writing down the mathematical terms.\n\n\\subsection{Instantiating the power grid model}\n\\label{sec:implementation-grid}\n\nIn order to create the grid model, we first have to instantiate the bus models simply by calling them with the corresponding parameter values from \\Cref{tab:ieee14-bus-parameters}, e.g.:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nSwingEq(H=5.148, P=2.32, D=2, SOmega=50) # for bus 1\nPQAlgebraic(S=-0.295-0.166im) # for bus 9\n\\end{lstlisting}\nWithin the actual code we simply loaded the data from a \\texttt{.csv} file into and automatized this instantiation\\footnote{The full source code is not part of this paper as it would be too long, but it will be published along with \\pd{}.}. The instantiated bus models should then be saved in an array called e.g. \\texttt{nodes}. Similarly, the admittance Laplacian should be generated from the line data in \\Cref{tab:ieee14-line-parameters} and saved in a matrix called e.g. \\texttt{LY}. The actual grid model instatiation is then simply one line where the model is saved in the variable \\texttt{g}:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\ng = GridDynamics(node_list, LY)\n\\end{lstlisting}\nNow, \\texttt{g} contains all the information of the power grid and in the following section we will show how to solve it.\n\n\n\\section{Modeling Results}\n\\label{sec:modeling}\n\n\\begin{figure*}[!t]\n\t\\centering\n\t\\subfloat{\\includegraphics[width=\\columnwidth]{ieee14-frequency-perturbation}\n\t\\label{fig:ieee14-frequency-perturbation}}\n\t\\subfloat{\\includegraphics[width=\\columnwidth]{ieee14-line-tripping}\n\t\\label{fig:ieee14-line-tripping}}\n\t\\hfil\n\t\\caption{The two subfigures show the outflowing power (top) at each node and the angular frequency (bottom) for the buses modeled by the swing equation (1, 3, 6, 8) in for the two perturbation scenarios: (left) frequency perturbation at bus 1 and (right) line tripping of line 2 (between bus 1 and 5). The letter in bracets refers to the modeling type, i.e. $G$ for generator \/ synchronous compensator as swing equation, $S$ for the slack bus, and $L$ for a load as algebraic PQ-constraint.}\n\\end{figure*}\n\nWithin this section, we will analyze two simple cases: (a) a frequency perturbation at the largest generator at bus 1 and (b) a line tripping event at of line 2 (between bus 1 and 5) (cmp.\\ \\Cref{fig:ieee14grid}).\n\nBefore we find the normal point of operation for the power grid, i.e.\\ the fixed point of \\Cref{eq:i-definition,eq:u-general,eq:x-general} or synchronous state for the IEEE 14-bus system. For that, we use the grid model \\texttt{g} generated in the previous \\Cref{sec:implementation} and use the function provided by \\pd{}:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nfp = operationpoint(g, ones(SystemSize(g)))\n\\end{lstlisting}\nwhere $\\mathtt{ones(SystemSize(g))}$ is a vector of the correct length for the initial condition of the fixed point search. \\texttt{fp} is now a \\texttt{State} object that we can use as initial condition for the solving the differential equations corresponding to the power grid.\n\n\\subsection{Frequency perturbation}\n\nIn order to model a frequency perturbation, one can simply take a copy of the fixed point as found before and adjust the initial frequency value:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nx0 = copy(fp)\nx0[1, :int, 1] += 0.2 \n\\end{lstlisting}\nThe second line can be read as ``add 0.2 to the 1$^\\text{st}$ internal variable of the 1$^\\text{st}$ node'' which is the frequency $\\omega$ as there is only one internal variable. Note that the first \\texttt{1} refers to the node and then second \\texttt{1} to the internal variable counter. The power grid with the initial condition \\texttt{x0} can then be solved for a time span of 0.5 seconds by calling:\n\\begin{lstlisting}[linewidth=\\columnlistingwidth]\nsol = solve(g, x0, (0.0,.5))\n\\end{lstlisting}\nThe solution is shown in \\Cref{fig:ieee14-frequency-perturbation}. It shows that the system is very stable against frequency perturbations. The actual dynamics is not so exciting as the system is very stable. Please note that this system was taken as an example to present on how easily one can model a power grid using \\pd{}, not to find new exciting dynamics.\n\n\\subsection{Line tripping}\n\nTo show some more dynamic behavior we simulated a line tripping as well. We model this effect by taking the operation point of the full power grid (\\texttt{fp}) as initial condition but defining a new admittance Laplacian where line 2 (between bus 1 and 5, see \\Cref{tab:ieee14-line-parameters,fig:ieee14grid}) has been taken out (i.e. the admittance is set to $0$). Running the model with this new Laplacian yields \\Cref{fig:ieee14-line-tripping}.\n\nIn the frequency plot we identify how the frequency of bus 1, where the line tripping happened, compensates for the momentarily excess power at the bus. The lacking power in the rest of the grid is matched by the synchronous compensators whose frequency decreases in turn. Note that the angular frequency $\\omega$ is shown, so with a division by $2\\pi$ the maximal frequency deviation $f$ is $\\approx 0.05\\,$Hz. After about one second, the system recovers to the normal state of operation.\n\n\n\\section{Conclusion \\& Outlook}\n\\label{sec:conclusion}\n\nWithin this paper, we have seen how one one can use the Open-Source library \\pd{} in order to model the dynamics of a power grid with just a few lines of code. We have seen how the fundamental mathematical equations (given in \\Cref{sec:powerdynamics}) translate to source code that reads exactly the same. We employed \\pd{} for the IEEE 14-bus distribution grid feeder in order to demonstrate how one can easily simulate faults and analyze the transient reaction of the power grid dynamics. As example scenarios we used a frequency perturbation and a line tripping.\n\nFinally, this paper is really just a sneak preview with a simple example the publication of \\pd{} is planned for October 15, 2018. By then, we will have added more inverter control schemes and stochastic descriptions of intermittency due to renewable energy sources.\n\n\n\n\n\n\n\\section*{Acknowledgment}\n\nThis paper was presented at the 19th Wind Integration Workshop and published in the workshop's proceedings.\n\nWe would like to thank the German Academic Exchange Service for the opportunity to participate at the Wind Integration Workshop 2018 in Stockholm via the funding program ``Kongressreisen 2018''.\nThis paper is based on work developed within the Climate-KIC Pathfinder project ``elena -- electricity network analysis'' funded by the European Institute of Innovation \\& Technology.\nThis work was conducted in the framework of the Complex Energy Networks research group at the Potsdam Institute for Climate Impact Research.\n\nWe would like to thank Frank Hellmann and Paul Schultz for the discussions on structuring an Open-Source library for dynamic power grid modeling.\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\\label{intro-sec}\n\\setcounter{equation}{0}\n\nA significant stage in the formation of living systems was the \ntransition from a symmetric chemistry involving mirror-symmetric and \napproximately equal numbers of left- and right-handed chiral species \ninto a system involving just one-handedness of chiral molecules. \n\nIn this paper we focus on mathematical models of one example of a \nphysicochemical system which undergoes such a symmetry-breaking \ntransition, namely the crystal grinding processes investigated by \nViedma \\cite{viedma} and Noorduin {\\em et al.}\\ \\cite{wim}, \nwhich have been recently reviewed by McBride \\& Tully \n\\cite{mcbride-nature}. Our aim is to describe this process by way \nof a detailed microscopic model of the nucleation and growth processes \nand then to simplify the model, retaining only the bare essential \nmechanisms responsible for the symmetry-breaking bifurcation. \n\nWe start by reviewing the processes which are already known to \ncause a symmetry-breaking bifurcation. By this we mean that a system \nwhich starts off in a racemic state (one in which both left-handed and \nright-handed structures occur with approximately equal frequencies) \nand, as the system evolves, the two handednesses grow differently, \nso that at a later time, one handedness is predominant in the system. \n\n\\subsection{Models for homochiralisation}\n\nMany models have been proposed for the emergence of homochirality \nfrom an initially racemic mixture of precursors. Frank \\cite{frank} \nproposed an open system into which $R$ and $S$ particles are \ncontinually introduced, and combine to form one of two possible \nproducts: left- or right-handed species, $X,Y$. Each of these \nproducts acts as a catalyst for its own production (autocatalysis), and \neach combines with the opposing handed product (cross-inhibition) to \nform an inert product ($P$) which is removed from the system at some \nrate. These processes are summarised by the following reaction scheme: \n\\begin{equation} \\begin{array}{rclcrclcl} \n&&&& \\hspace*{-9mm}{\\rm external \\;\\;\\; source} & \\rightarrow &R,S& \n\t\\;\\; & {\\rm input}, k_0, \\\\ \nR+S & \\rightleftharpoons & X && \nR+S & \\rightleftharpoons & Y &\\qquad &\\mbox{slow}, k_1 , \\\\ \nR+S+X & \\rightleftharpoons & 2 X && \nR+S+Y & \\rightleftharpoons & 2 Y &\\quad& \\mbox{fast, autocatalytic}, k_2 \\\\ \n&&&&X + Y & \\rightarrow & P &\\qquad& \\mbox{cross-inhibition}, k_3 , \\\\\n&&&& P &\\rightarrow & & \\qquad & {\\rm removal}, k_4 . \n\\end{array}\\end{equation} \nIgnoring the reversible reactions (for simplicity), \nthis system can be modelled by the differential equations \n\\begin{eqnarray}\n\\frac{{\\rm d} r}{{\\rm d} t} & =& k_0 - 2 k_1 r s - k_2 r s (x\\!+\\!y) \n\t+ k_{-1} (x\\!+\\!y) + k_{-2} (x^2\\!+\\!y^2) , \\\\\n\\frac{{\\rm d} s}{{\\rm d} t} & = & k_0 - 2 k_1 r s - k_2 r s (x\\!+\\!y) \n\t+ k_{-1} (x\\!+\\!y) + k_{-2} (x^2\\!+\\!y^2) , \\\\ \n\\frac{{\\rm d} x}{{\\rm d} t} & = & \n\tk_1 r s + k_2 r s x - k_3 x y - k_{-1} x - k_{-2} x^2 , \\\\ \n\\frac{{\\rm d} y}{{\\rm d} t} & = & \n\tk_1 r s + k_2 r s y - k_3 x y - k_{-1} y - k_{-2} y^2 , \\\\ \n\\frac{{\\rm d} p}{{\\rm d} t} & = & k_{3} x y - k_4 p , \n\\end{eqnarray}\nfrom which we note that at steady-state we have \n\\begin{equation}\nrs=\\frac{k_0+k_{-1}(x+y) + k_{-1}(x^2+y^2)}{2k_1+k_2(x+y)}.\n\\end{equation}\nWe write the absolute enantiomeric excess as $ee=x-y$ and the \ntotal concentration as $\\sigma=x+y$; adding and subtracting \nthe equations for ${\\rm d} x\/{\\rm d} t$ and ${\\rm d} y\/{\\rm d} t$, we find \n\\begin{equation} \n\\sigma^2 = \\frac{2k_0}{k_3} + ee^2 , \n\\end{equation} \n\\begin{equation} \nee \\left[ \n\\frac{k_2(k_{-2}ee^2+k_{-2}\\sigma^2+2k_{-1}\\sigma+2k_0)}\n{2(2k_1+k_2\\sigma)} - k_{-1} - k_{-2} \\sigma \\right] = 0 . \n\\end{equation} \nHence $ee=0$ is always a solution, and there are other solutions with \n$ee\\neq0$ if the rate constants $k_*$ satisfy certain conditions (these \ninclude $k_3>k_{-2}$ and $k_0$ being sufficiently large). \n\nThe important issues to note here are: \n\\begin{description}\n\\item[(i)] this system is {\\em open}, it requires the continual supply of \nfresh $R,S$ to maintain the asymmetric steady-state. Also, the \nremoval of products is required to avoid the input terms \ncausing the total amount of material to increase indefinitely; \n\\item[(ii)] the forcing input term drives the system away from \nan equilibrium solution, into a distinct steady-state solution; \n\\item[(iii)] the system has cross-inhibition which removes equal \nnumbers of $X$ and $Y$, amplifying any differences caused by \nrandom fluctuations in the initial data or in the input rates. \n\\end{description} \n\nSaito \\& Hyuga \\cite{saito} discuss a sequence of toy models \ndescribing homochirality caused by nonlinear autocatalysis and \nrecycling. Their family of models can be summarised by \n\\begin{eqnarray}\n\\frac{{\\rm d} r}{{\\rm d} t} & =& k r^2 (1-r-s) - \\lambda r , \\\\ \n\\frac{{\\rm d} s}{{\\rm d} t} & =& k s^2 (1-r-s) - \\lambda s , \n\\end{eqnarray}\nwhere $r$ and $s$ are the concentrations of the two enantiomers. \nInitially they consider $k_r=k_s=k$ and $\\lambda=0$ and find that \nenantiomeric exess, $r-s$ is constant. Next the case $k_r=kr$, \n$k_s=ks$, $\\lambda=0$ is analysed, wherein the relative enantiomeric \nexcess $\\frac{r-s}{r+s}$ is constant. Then the more complex case of \n$k_r=k r^2$, $k_s=k s^2$, $\\lambda=0$ is analysed, and amplification \nof the enantiomeric excess is obtained. This amplification persists \nwhen the case $\\lambda>0$ is finally analysed. This shows us strong \nautocatalysis may cause homochiralisation, but in any given \nexperiment, it is not clear which form of rate coefficients \n($k_r,k_s,\\lambda$) should be used. \n\nSaito \\& Hyuga (2005) analyse a series of models of crystallisation \nwhich include some of features present in our more general model. \nThey note that a model truncated at tetramers exhibits different \nbehaviour from one truncated at hexamers. In particular, the \nsymmetry-breaking phenomena is {\\em not} present in the tetramer \nmodel, but {\\em is} exhibited by the hexamer model. Hence, later, \nwe will consider models truncated at the tetramer and the hexamer \nlevels and investigate the differences in symmetry-breaking \nbehaviour (Sections \\ref{tetra-sec} and \\ref{hex-sec}). \n\nDenoting monomers by $c$, small and large left-handed clusters by \n$x_1,x_2$ respectively and right-handed by $y_1,y_2$, Uwaha \n\\cite{uwaha} writes down the scheme \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & =& - 2 k_0 z^2 k_1 z (x_1+y_1) + \n\t\\lambda_1(x_2+y_2) + \\lambda_0(x_1+y_1) , \n\\\\ \n\\frac{{\\rm d} x_1}{{\\rm d} t} & = & k_0 z^2 - k_u x_1 x_2 \n\t- k_c x_1^2 + \\lambda_u x_2 + \\lambda_0 x_1 , \n\\\\ \n\\frac{{\\rm d} x_2}{{\\rm d} t} & =& k_1 x_2 c + k_u x_1 x_2 + \n\tk_c x_1^2 - \\lambda_1 x_2 - \\lambda_u x_2 , \n\\\\ \n\\frac{{\\rm d} y_1}{{\\rm d} t} & = & k_0 z^2 - k_u y_1 y_2 \n\t- k_c y_1^2 + \\lambda_u y_2 + \\lambda_0 y_1 , \n\\\\ \n\\frac{{\\rm d} y_2}{{\\rm d} t} & =& k_1 y_2 c + k_u y_1 y_2 + \n\tk_c y_1^2 - \\lambda_1 y_2 - \\lambda_u y_2 , \n\\end{eqnarray}\nwhich models \n\\begin{itemize}\n\\item the formation of small chiral clusters \n\t($x_1,y_1$) from an achiral monomer ($c$) at rate $k_0$, \n\\item small chiral clusters ($x_1,y_1$) of the same handedness \n\tcombining to form larger chiral clusters (rate $k_c$), \n\\item small and larger clusters combining to form larger clusters \n(rate $k_u$), \n\\item large clusters combining with achiral monomers to form more \n\tlarge clusters at the rate $k_1$, \n\\item the break up of larger clusters into smaller clusters (rate $\\lambda_u$), \n\\item the break up of small clusters into achiral monomers (rate $\\lambda_0$), \n\\item the break up of larger clusters into achiral monomers (rate $\\lambda_1$). \n\\end{itemize}\n\nSuch a model can exhibit symmetry-breaking to a solution in which \n$x_1\\neq x_2$ and $x_2\\neq y_2$. Uwaha points out that the \nrecycling part of the model (the $\\lambda_*$ parameters) are crucial \nto the formation of a `completely' homochiral state. One problem with \nsuch a model is that since the variables are all total masses in the \nsystem, the size of clusters is not explicitly included. In asymmetric \ndistributions, the typical size of left- and right- handed clusters may \ndiffer drastically, hence the rates of reactions will proceed differently \nin the cases of a few large crystals or many smaller crystals. \n\nSandars has proposed a model of symmetry-breaking in the \nformation of chiral polymers \\cite{sandars}. His model has an \nachiral substrate ($S$) which splits into chiral monomers $L_1,R_1$ \nboth spontaneously at a slow rate and at a faster rate, when catalysed \nby the presence of long homochiral chains. This catalytic effect has \nboth autocatalytic and crosscatalytic components, that is, for example, \nthe presence of long right-handed chains $R_n$ autocatalyses the \nproduction of right-handed monomers $R_1$ from $S$, (autocatalysis) \nas well as the production of left-handed monomers, $L_1$ (crosscatalysis). \nSandars assumes the growth rates of chains are linear and not \ncatalysed; the other mechanism required to produce a symmetry-breaking \nbifurcation to a chiral state is cross-inhibition, by which chains of \nopposite handednesses interact and prevent either from further \ngrowth. These mechanisms are summarised by \n\\begin{eqnarray}\nS \\rightarrow L_1 , && S \\rightarrow R_1 , \n\t\\qquad \\mbox{slow} , \\nonumber \\\\ \nS \\!+\\! L_n \\rightarrow L_1 \\!+\\! L_n , && \nS \\!+\\! R_n \\rightarrow R_1 \\!+\\! R_n , \\quad \n\\mbox{autocatalytic, rate $\\propto 1\\!+\\!f$} , \\nonumber \n\\\\\nS \\!+\\! R_n \\rightarrow L_1 \\!+\\! R_n , && \nS \\!+\\! L_n \\rightarrow R_1 \\!+\\! L_n , \\quad \n\\mbox{cross-catalytic, rate $\\propto 1\\!-\\!f$} , \\nonumber \n\\\\\nL_n + L_1 \\rightarrow L_{n+1} , && \nR_n + R_1 \\rightarrow R_{n+1} , \\qquad \n\\mbox{chain growth, rate $=a$} , \\nonumber\n\\\\\nL_n + R_1 \\rightarrow Q_{n+1} , && \nR_n + L_1 \\rightarrow P_{n+1} , \\qquad \n\\mbox{cross-inhibition, rate $=a\\chi$} . \\nonumber\n\\end{eqnarray}\nThis model and generalisations of it have been analysed \nby Sandars \\cite{sandars}, Brandenburg {\\em et al.}\\ \\cite{axel,axel3}, \nMultimaki \\& Brandenburg \\cite{axel2}, \nWattis \\& Coveney \\cite{jw-sandars,jw-ch-rna-rev}, \nGleiser \\& Walker \\cite{sara-bup}, Gleiser {\\em et al.}\\ \\cite{sara-punc}. \nTypically a classic pitchfork bifurcation is found when the fidelity \n($f$) of the autocatalysis over the cross-catalysis is increased. \nOne counterintuitive effect is that increasing the cross-inhibition \neffect ($\\chi$) aids the bifurcation, allowing it to occur at lower \nvalues of the fidelity parameter $f$. \n\n\n\\subsection{Experimental results on homochiralisation}\n\nThe Soai reaction was one of the first experiments which \ndemonstrated that a chemical reaction could amplify initial small \nimbalances in chiral balance; that is, a small enantiomeric exess in \ncatalyst at the start of the experiment led to a much larger imbalance \nin the chiralities of the products at the end of the reaction. Soai {\\em et al.}\\ \n\\cite{soai} was able to achieve an enantiomeric exess exceeding \n85\\% in the asymmetric autocatalysis of chiral pyrimidyl alkanol. \n\nThe first work showing that crystallisation experiments could exhibit \nsymmetry breaking was that of Kondepudi \\& Nelson \n\\cite{kon-sci}. Later Kondepudi {\\em et al.}\\ \\cite{kon-jacs} \nshowed that the stirring rate was a good bifurcation parameter to \nanalyse the final distribution of chiralities of crystals emerging from a \nsupersaturated solution of sodium chlorate. With no stirring, there \nwere approximately equal numbers of left- and right-handed crystals. \nAbove a critical (threshold) stirring rate, the imbalance in the numbers \nof each handedness increased, until, at large enough stirring rates, \ntotal chiral purity was achieved. This is due to all crystals in the \nsystem being derived from the same `mother' crystal, which is the first \ncrystal to become established in the system; all other crystals grow \nfrom fragments removed from it (either directly or indirectly). \nBefore this, Kondepudi \\& Nelson \\cite{kon-pla,kon-nat} \nworked on the theory of chiral symmetry-breaking mechanisms \nwith the aim of predicting how parity-violating perturbations could \nbe amplified to give an enantiomeric exess in prebiotic chemistry, \nand the timescales involved. Their results suggest \na timescale of approximately $10^4$ years. More recently, \nKondepudi and Asakura \\cite{kon+a} have summarised \nboth the experimental and theoretical aspects of this work. \n\nViedma \\cite{viedma} was the first to observe that grinding a \nmixture of chiral crystals eventually led to a distribution of crystals \nwhich were all of the same handedness. The crystalline material \nused was sodium chlorate, as used by Kondepudi {\\em et al.}\\ \n\\cite{kon-sci}. \nSamples of L and D crystals are mixed with water in round-bottomed \nflasks and the system is stirred by a magnetic bar (of length 3-20mm) \nat 600rpm. The system is maintained in a supersaturated state; \nsmall glass balls are added to continually crush the crystals. \nThe grinding is thus continuous, and crystals are maintained below \na size of 200 $\\mu$m. The chirality of the resulting crystals was \ndetermined by removing them from the flask, allowing them to grow \nand measuring their optical activity. \nThe results show that, over time, the percentages of left- and \nright-handed crystals steadily change from about 50\/50 to 100\/0 or \n0\/100 -- a state which is described as complete chiral purity. \nWith stirring only and no glass balls, the systems conserve their \ninitial chiral excesses; with glass balls present and stirring, the chiral\nexcess increases, and this occurs more rapidly if more balls are \npresent or the speed of stirring is increased. \n\nMore recently, Noorduin {\\em et al.}\\ \\cite{wim} have observed a \nsimilar effect with amino acids -- a much more relevant molecule \nin the study of origins of life. This work has been reviewed by \nMcBride \\& Tully \\cite{mcbride-nature}, who add to the \nspeculation on the mechanisms responsible for the phenomenon. \nNoorduin {\\em et al.}\\ describe grinding as `dynamic \ndissolution\/crystallization processes that result in the conversion \nof one solid enantiomorph into the other'. They also note that `once \na state of single chirality is achieved, the system is ``locked'' \nbecause primary nucleation to form and sustain new crystals from \nthe opposite enantiomer is kinetically prohibited'. Both these quotes \ninclude the crucial fact that the process evolves {\\em not} towards \nan equilibrium solution (which would be racemic), but towards a \ndifferent, dynamic steady-state solution. As noted by Plasson \n(personal communication, 2008), this nonequilibrium state is \nmaintained due to the constant input of energy into the system \nthrough the grinding process. \n\nMcBride \\& Tully \\cite{mcbride-nature} discuss the growth \nof one enantiomorph, and the dissolution of the other as a type of \nOstwald ripening process; with the large surface area to volume ratio \nof smaller crystals giving a rapid dissolution rate, whilst larger crystals, \nhave a lower surface area to volume ratio meaning that they dissolve \nmore slowly. However appealing such an argument maybe, since \nsurface area arguments can equally well be applied to the growth \nside of the process, it is not clear that this is either necessary or \nsufficient. Infact, the model analysed later in this paper will show \nthat a critical cluster size is not necessary to explain homochiralisation \nthrough grinding. \n\n\\subsection{Our aims}\n\nWe aim to describe the results of the crystal grinding phenomenon \nthrough a model which recycles mass through grinding, which \ncauses crystals to fragment, rather than having explicit mass input \nand removal. Simultaneously we need crystal growth processes \nto maintain a distribution of sizeable crystals. \n\nWe assume that the crystals are solids formed in an aqueous \nenvironment, however, we leave open questions as to whether \nthey are crystals of some mineral of {\\em direct} biological \nrelevance (such as amino acids), or whether they are some other \nmaterial, which after growing, will later provide a chirally \nselective surface for biomolecules to crystallise on, or be a catalyst \nfor chiral polymerisation to occur. Following Darwin's \n\\cite{darwin} ``warm little pond'', an attractive scenario might \nbe a tidal rock pool, where waves agitating pebbles provide the \nenergetic input for grinding. Taking more account of recent work, \na more likely place is a suboceanic hydrothermal vent where \nthe rapid convection of hot water impels growing nucleii into the \nvent's rough walls as well as breaking particles off the walls and \nentraining them into the fluid flow, simultaneously grinding any \ngrowing crystals. \n\nIn Section \\ref{model-sec} we propose a detailed microscopic \nmodel of the nucleation and crystal growth of several species \nsimultaneously. This has the form of a generalised Becker-D\\\"{o}ring \nsystem of equations \\cite{bd}. Due to the complexity of the \nmodel we immediately simplify it, making assumptions on the rate \ncoefficients. Furthermore, to elucidate those processes which \nare responsible for homochiralisation, we remove some processes \ncompletely so as to obtain a simple system of ordinary differential \nequations which can be analysed theoretically. \n\nThe simplest model which might be expected to show \nhomochiralisation is one which has small and large clusters of each \nhandedness. Such a truncated model is considered in Section \n\\ref{tetra-sec} wherein it is shown that such a model might lead \nto amplification of enantiomeric exess in the short time, but that \nin the long-time limit, only the racemic state can be approached. \nThis model has the structure akin to that of Saito \\& Hyuga \n\\cite{saito2} truncated at the tetramer level. \n\nHence, in Section \\ref{hex-sec} we consider a more complex model \nwith a cut-off at larger sizes (one can think of small, medium, and large \nclusters of each handedness). Such a model has a similar structure to \nthe hexamer truncation analysed by Saito \\& Hyuga \\cite{saito2}. \nWe find that such a model does allow a final steady-state in which one \nchirality dominates the system and the other is present only in \nvanishingly small amounts. \n\nHowever, as discussed earlier, there may be subtle effects whereby it \nis not just the {\\em number} of crystals of each type that is important \nto the effect, but a combination of size and number of each handedness \nof crystal that is important to the evolution of the process. Hence, in \nSection \\ref{new-sec} we introduce an alternative reduction of the \nsystem of governing equations. In this, instead of truncating and \nkeeping only clusters of a small size, we postulate a form for the \ndistribution which includes information on both the number and size \nof crystals, and use these two quantities to construct a system of \nfive ordinary differential equations for the system's evolution. \n\nWe discuss the results in Sections \\ref{disc-sec} and \\ref{conc-sec} \nwhich conclude the paper. The Appendix \\ref{app} shows how, \nby removing the symmetry in the growth rates of the two \nhandednesses, the model could be generalised to account for \nthe competitive nucleation of different polymorphs growing from \na common supply of monomer. \n\n\\section{The BD model with dimer interactions and \nan amorphous metastable phase} \n\\label{model-sec}\n\\setcounter{equation}{0}\n\n\\subsection{Preliminaries}\n\nSmoluchowski \\cite{smol} proposed a model in which clusters \nof any sizes could combine pairwise to form larger clusters. Chemically \nthis process is written $C_r + C_s \\rightarrow C_{r+s}$ where $C_r$ \nrepresents a cluster of size $r$. Assuming this process is reversible \nand occurs with a forward rate given by $a_{r,s}$ and a reverse rate \ngiven by $b_{r,s}$, the law of mass action yields the kinetic equations \n\\begin{eqnarray}\n\\frac{{\\rm d} c_r}{{\\rm d} t} &\\!=\\!&\\! \\mbox{$\\frac{1}{2}$} \\sum_{s=1}^{r-1} \n \\left( a_{s,r-s} c_s c_{r-s} - b_{s,r-s} c_r \\right) \n - \\sum_{s=1}^\\infty \\left( a_{r,s} c_r c_s - \n b_{r,s} c_{r+s} \\right) . \\nonumber \\\\ && \\lbl{smol-eq}\n\\end{eqnarray}\nThese are known as the coagulation-fragmentation equations. \nThere are simplifications in which only interactions between clusters \nof particular sizes are permitted to occur, for example when only \ncluster-monomer interactions can occur, the Becker-D\\\"{o}ring \nequations \\cite{bd} are obtained. Da Costa has formulated a \nsystem in which only clusters upto a certain size ($N$) are permitted \nto coalesce with or fragment from other clusters. In the case of \n$N=2$, which is pertinent to the current study, only cluster-monomer \nand cluster-dimer interactions are allowed, for example \n\\begin{equation}\nC_r + C_1 \\rightleftharpoons C_{r+1} , \n\\qquad \nC_r + C_2 \\rightleftharpoons C_{r+2} . \n\\end{equation} \nThis leads to a system of kinetic equations of the form \n\\begin{eqnarray}\n\\frac{{\\rm d} c_r}{{\\rm d} t} & = & J_{r-1} - J_r + K_{r-2} - K_r , \n\t\\qquad (r\\geq3) , \\lbl{gbd-eq1} \\\\ \n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & J_1 - J_2 - K_2 \n\t- \\displaystyle\\sum_{r=1}^\\infty K_r , \\\\ \n\\frac{{\\rm d} c_1}{{\\rm d} t} & = & - J_1 - K_2 \n\t- \\displaystyle\\sum_{r=1}^\\infty J_r , \\\\ \nJ_r &=& a_r c_r c_1 - b_{r+1} c_{r+1} , \\qquad \nK_r = \\alpha_r c_r c_2 - \\beta_{r+2} c_{r+2} . \n\\lbl{gbd-eq4} \\end{eqnarray}\nA simple example of such a system has been analysed \npreviously by Bolton \\& Wattis \\cite{bw-dimers}. \n\nIn the next subsection we generalise the model (\\ref{smol-eq}) to \ninclude a variety of `species' or `morphologies' of cluster, \nrepresenting left-handed, right-handed and achiral clusters. We \nsimplify the model in stages to one in which only monomer and dimer \ninteractions are described, and then one in which only dimer \ninteractions occur. \n\n\\subsection{A full microscopic model of chiral crystallisation}\n\nWe start by outlining all the possible cluster growth, fragmentation \nand transformation processes. We denote the two handed clusters \nby $X_r$, $Y_r$, where the subscript $r$ specifies the size of cluster. \nAchiral clusters are denoted by $C_r$, and we allow clusters to \nchange their morphology spontaneously according to \n\\begin{equation} \\begin{array}{rclclccrclcl}\nC_r & \\rightarrow & X_r & \\quad& {\\rm rate} = \\mu_r , &&\nX_r & \\rightarrow & C_r & \\quad& {\\rm rate} = \\mu_r \\nu_r , \\\\ \nC_r & \\rightarrow & Y_r & \\quad& {\\rm rate} = \\mu_r , && \nY_r & \\rightarrow & C_r & \\quad& {\\rm rate} = \\mu_r \\nu_r . \n\\end{array} \\end{equation}\nWe allow clusters to grow by coalescing with clusters \nof similar handedness or an achiral cluster. In the case of \nthe latter process, we assume that the cluster produced \nis chiral with the same chirality as the parent. Thus \n\\begin{equation} \\begin{array}{rclcl}\nX_r + X_s & \\rightarrow & X_{r+s} , && {\\rm rate} = \\xi_{r,s}, \\\\ \nX_r + C_s & \\rightarrow & X_{r+s} , && {\\rm rate} = \\alpha_{r,s},\\\\ \nC_r + C_s & \\rightarrow & C_{r+s} , && {\\rm rate} = \\delta_{r,s},\\\\ \nY_r + C_s & \\rightarrow & Y_{r+s} , && {\\rm rate} = \\alpha_{r,s},\\\\ \nY_r + Y_s & \\rightarrow & Y_{r+s} , && {\\rm rate} = \\xi_{r,s} . \n\\end{array} \\end{equation}\nWe do not permit clusters of opposite to chirality to merge. \nFinally we describe fragmentation: all clusters may \nfragment, producing two smaller clusters each of \nthe same chirality as the parent cluster \n\\begin{equation} \\begin{array}{rclcl}\nX_{r+s} & \\rightarrow & X_r + X_s && {\\rm rate} = \\beta_{r,s}, \\\\ \nC_{r+s} & \\rightarrow & C_r + C_s && {\\rm rate} = \\epsilon_{r,s}, \\\\ \nY_{r+s} & \\rightarrow & Y_r + Y_s &\\quad& {\\rm rate} = \\beta_{r,s} . \n\\end{array} \\end{equation}\nSetting up concentration variables for each size and each type of \ncluster by defining $c_r(t) = [C_r]$, $x_r(t) = [X_r]$, $y_r(t) = [Y_r]$ \nand applying the law of mass action, we obtain \n\\begin{eqnarray}\n\\frac{{\\rm d} c_r}{{\\rm d} t} &\\!=\\!& -2\\mu_r c_r + \\mu_r\\nu_r(x_r+y_r) \n\t- \\sum_{k=1}^\\infty \\alpha_{k,r} c_r (x_k+y_k) \\\\ && \\nonumber \n\t+ \\mbox{$\\frac{1}{2}$} \\sum_{k=1}^{r-1} \\left( \\delta_{k,r-k} c_k c_{r-k} \n\t- \\epsilon_{k,r-k} c_k c_{r-k} \\right) \n\t- \\sum_{k=1}^\\infty \\left( \\delta_{k,r} c_k c_r \n\t- \\epsilon_{k,r} c_{r+k} \\right) , \n\\lbl{gbd1} \\\\ \n\\frac{{\\rm d} x_r}{{\\rm d} t} &\\!=\\!&\\! \\mu_r c_r \\!-\\! \\mu_r \\nu_r x_r \n\t+ \\sum_{k=1}^{r-1} \\alpha_{k,r-k} c_k x_{r-k} \n\t\\!-\\! \\mbox{$\\frac{1}{2}$} \\sum_{k=1}^{r-1} \\left( \\xi_{k,r-k} x_k x_{r-k} \n\t\\!-\\! \\beta_{k,r\\!-\\!k} x_r \\right) \\nonumber \\\\ && \n\t- \\sum_{k=1}^\\infty \\left( \\xi_{k,r} x_k x_r \n\t- \\beta_{k,r} x_{r+k} \\right) , \n\\\\ \n\\frac{{\\rm d} y_r}{{\\rm d} t} &\\!=\\!&\\! \\mu_r c_r \\!-\\! \\mu_r \\nu_r y_r \n\t+ \\sum_{k=1}^{r-1} \\alpha_{k,r-k} c_k y_{r-k} \n\t\\!-\\! \\mbox{$\\frac{1}{2}$} \\sum_{k=1}^{r-1} \\left( \\xi_{k,r-k} y_k y_{r-k} \n\t\\!-\\! \\beta_{k,r\\!-\\!k} y_r \\right) \\nonumber \\\\ && \n\t- \\sum_{k=1}^\\infty \\left( \\xi_{k,r} y_k y_r \n\t- \\beta_{k,r} y_{r+k} \\right) . \n\\lbl{gbd3} \\end{eqnarray}\nThe main problem with such a model is the vast number \nof parameters that have been introduced ($\\alpha_{r,k}$, \n$\\xi_{r,k}$, $\\beta_{r,k}$, $\\mu_r$, $\\nu_r$, $\\delta_{r,k}$, \n$\\epsilon_{r,k}$, for all $k,r$). \n\nHence we make several simplifications: \n\\begin{description} \n\\item[(i)] \nwe assume that the dominant coagulation and fragmentation \nprocesses are between large and very small clusters (rather \nthan large clusters and other large clusters). Specifically, we \nassume that only coalescences involving $C_1$ and $C_2$ \nneed to be retained in the model, and fragmentation always \nyields either a monomer or a dimer fragment. This assumption \nmeans that the system can be reduced to a generalised \nBecker-D\\\"{o}ring equation closer to the form of \n(\\ref{gbd-eq1})--(\\ref{gbd-eq4}) rather than (\\ref{smol-eq}); \n\n\\item[(ii)] \nwe also assume that the achiral clusters are unstable at larger size, \nso that their presence is only relevant at small sizes. \nTypically at small sizes, clusters are amorphous and do not take \non the properties of the bulk phase, hence at small sizes clusters \ncan be considered achiral. We assume that there is a regime of \ncluster sizes where there is a transition to chiral structures, and \nwhere clusters can take on the bulk structure (which is chiral) as well \nas exist in amorphous form. At even larger sizes, we assume that \nonly the chiral forms exist, and no achiral structure can be adopted; \n\n\\item[(iv)] \nfurthermore, we assume that all rates are independent of cluster size, \nspecifically, \n\\begin{eqnarray}\n\\alpha_{_{k,1}} & = & a , \\qquad \\qquad \n\\alpha_{_{k,2}} = \\alpha , \\qquad \\quad \n\\alpha_{_{k,r}} =0 , \\quad (r\\geq2) \t\\\\ \n\\mu_2 &=& \\mu , \\qquad \\qquad \n\\mu_r=0 , \\quad (r\\geq3) , \t\\\\ \n\\nu_2 &=& \\nu , \\qquad \\qquad \n\\nu_r=0 , \\quad (r\\geq3) , \t\\\\ \n\\delta_{1,1} & = & \\delta , \\qquad \n\\delta_{k,r} = 0 , \\quad ({\\rm otherwise}) \t\\\\ \n\\epsilon_{1,1} & = & \\epsilon , \\qquad \n\\epsilon_{k,r} = 0 , \\quad ({\\rm otherwise})\t\\\\ \n\\xi_{k,2} &=& \\xi_{2,k} = \\xi , \\qquad \n\\xi_{k,r} = 0 , \\quad ({\\rm otherwise})\t\t\\\\\n\\beta_{k,1} & = & \\beta_{1,k} = b , \\qquad \n\\beta_{k,2} = \\beta_{2,k} = \\beta , \\qquad \n\\beta_{k,r} = 0 , \\quad ({\\rm otherwise}), \n\\nonumber \\\\ && \\end{eqnarray}\nUltimately we will set $a=b=0=\\delta=\\epsilon$ so that we have only \nfive parameters to consider ($\\alpha$, $\\xi$, $\\beta$, $\\mu$, $\\nu$). \n\n\\end{description} \n\n\\begin{figure}[!ht]\n\\begin{picture}(500,160)(35,-35)\n\\put(48,50){\\circle{15}}\n\\multiput(80,50)(40,0){6}{\\circle{15}}\n\\multiput(80,10)(40,0){8}{\\circle{15}}\n\\multiput(80,90)(40,0){8}{\\circle{15}}\n\\put(044,48){$c_1$}\n\\put(076,48){$c_2$}\n\\put(116,48){$c_3$}\n\\put(156,48){$c_4$}\n\\put(196,48){$c_5$}\n\\put(236,48){$c_6$}\n\\put(276,48){$c_7$}\n\\put(076,88){$x_2$}\n\\put(116,88){$x_3$}\n\\put(156,88){$x_4$}\n\\put(196,88){$x_5$}\n\\put(236,88){$x_6$}\n\\put(276,88){$x_7$}\n\\put(316,88){$x_8$}\n\\put(356,88){$x_9$}\n\\put(076,08){$y_2$}\n\\put(116,08){$y_3$}\n\\put(156,08){$y_4$}\n\\put(196,08){$y_5$}\n\\put(236,08){$y_6$}\n\\put(276,08){$y_7$}\n\\put(316,08){$y_8$}\n\\put(356,08){$y_9$}\n\\multiput(83,20)(40,0){6}{\\vector(0,1){20}}\n\\multiput(77,40)(40,0){6}{\\vector(0,-1){20}}\n\\multiput(77,60)(40,0){6}{\\vector(0,1){20}}\n\\multiput(83,80)(40,0){6}{\\vector(0,-1){20}}\n\\multiput(69,30)(40,0){6}{$\\mu$}\n\\multiput(85,30)(40,0){6}{$\\nu$}\n\\multiput(69,70)(40,0){6}{$\\mu$}\n\\multiput(85,70)(40,0){6}{$\\nu$}\n\\put(058,53){\\vector(1,0){10}}\n\\put(068,50){\\vector(-1,0){10}}\n\\multiput(095,53)(40,0){5}{\\vector(1,0){10}}\n\\multiput(105,50)(40,0){5}{\\vector(-1,0){10}}\n\\multiput(095,95)(40,0){7}{\\vector(1,0){10}}\n\\multiput(105,90)(40,0){7}{\\vector(-1,0){10}}\n\\multiput(095,15)(40,0){7}{\\vector(1,0){10}}\n\\multiput(105,10)(40,0){7}{\\vector(-1,0){10}}\n\\put(060,59){$\\delta$}\n\\put(060,40){$\\epsilon$}\n\\multiput(097,59)(40,0){5}{$\\delta$}\n\\multiput(097,40)(40,0){5}{$\\epsilon$}\n\\multiput(097,97)(40,0){5}{$a$}\n\\multiput(097,80)(40,0){5}{$b$}\n\\multiput(097,17)(40,0){5}{$a$}\n\\multiput(097,00)(40,0){5}{$b$}\n\\put(118,119){$\\beta$}\n\\put(127,115){\\vector(-1,0){10}}\n\\put(120,100){\\oval(80,30)[t]}\n\\put(120, 0){\\oval(80,30)[b]}\n\\put(127,-15){\\vector(-1,0){10}}\n\\put(118,-27){$\\beta$}\n\\put(158,124){$\\beta$}\n\\put(167,120){\\vector(-1,0){10}}\n\\put(160,100){\\oval(80,40)[t]}\n\\put(160, 0){\\oval(80,40)[b]}\n\\put(167,-20){\\vector(-1,0){10}}\n\\put(158,-32){$\\beta$}\n\\put(198,119){$\\beta$}\n\\put(207,115){\\vector(-1,0){10}}\n\\put(200,100){\\oval(80,30)[t]}\n\\put(200, 0){\\oval(80,30)[b]}\n\\put(207,-15){\\vector(-1,0){10}}\n\\put(198,-27){$\\beta$}\n\\put(238,124){$\\beta$}\n\\put(247,120){\\vector(-1,0){10}}\n\\put(240,100){\\oval(80,40)[t]}\n\\put(240, 0){\\oval(80,40)[b]}\n\\put(247,-20){\\vector(-1,0){10}}\n\\put(238,-32){$\\beta$}\n\\put(278,119){$\\beta$}\n\\put(287,115){\\vector(-1,0){10}}\n\\put(280,100){\\oval(80,30)[t]}\n\\put(280, 0){\\oval(80,30)[b]}\n\\put(287,-15){\\vector(-1,0){10}}\n\\put(278,-27){$\\beta$}\n\\put(318,124){$\\beta$}\n\\put(327,120){\\vector(-1,0){10}}\n\\put(320,100){\\oval(80,40)[t]}\n\\put(320, 0){\\oval(80,40)[b]}\n\\put(327,-20){\\vector(-1,0){10}}\n\\put(318,-32){$\\beta$}\n\\end{picture} \n\\caption{Reaction scheme involving monomer and \ndimer aggregation and fragmentation of achiral clusters \nand those of both handednesses (right and left). \nThe aggregation of achiral and chiral clusters \nis not shown (rates $\\alpha$, $\\xi$). }\n\\label{rec-sch-fig}\n\\end{figure}\n\nThis scheme is illustrated in Figure \\ref{rec-sch-fig}. However, \nbefore writing down a further system of equations, we make one \nfurther simplification. We take the transition region described in (ii), \nabove, to be just the dimers. Thus the only types of achiral cluster \nare the monomer and the dimer ($c_1$, $c_2$); dimers exist in achiral, \nright- and left-handed forms ($c_2$, $x_2$, $y_2$); at larger sizes \nonly left- and right-handed clusters exist ($x_r$, $y_r$, $r\\geq2$). \n\nThe kinetic equations can be reduced to \n\\begin{eqnarray} \n\\frac{{\\rm d} c_1}{{\\rm d} t} & = & 2 \\varepsilon c_2 - 2 \\delta c_1^2 \n\t- \\sum_{r=2}^\\infty ( a c_1 x_r + a c_1 y_r - b x_{r+1} \n\t- b y_{r+1} ) , \\lbl{gbd-c1}\n\t\\\\ \n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & \\delta c_1^2 - \\varepsilon c_2 - 2 \\mu c_2 \n\t+ \\mu\\nu (x_2+y_2) - \\sum_{r=2}^\\infty \\alpha c_2 (x_r+y_r) , \n\t\\\\\n\\frac{{\\rm d} x_r}{{\\rm d} t} & = & a c_1 x_{r-1} - b x_r \n\t- a c_1 x_r + b x_{r+1} + \\alpha c_2 x_{r-2} \n\t- \\alpha c_2 x_r \\nonumber\\\\ && \n\t- \\beta x_r + \\beta x_{r+2} + \\xi x_2 x_{r-2} \n\t- \\xi x_2 x_r , \\qquad \\hfill (r\\geq4) , \n\t\\\\ \n\\frac{{\\rm d} x_3}{{\\rm d} t} & = & a c_1 x_2 - b x_3 - a c_1 x_3 \n\t+ b x_4 - \\alpha c_2 x_3 - \\xi x_2 x_3 + \\beta x_5 , \n\t\\\\ \n\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 + b x_3 \n\t- a c_1 x_2 - \\alpha x_2 c_2 \n\t+ \\beta x_4 \\nonumber \\\\ && + \\sum_{r=2}^\\infty \\beta x_{r+2} \n\t- \\sum_{r=2}^\\infty \\xi x_2 x_r - \\xi x_2^2 , \n\t\\\\ \n\\frac{{\\rm d} y_r}{{\\rm d} t} & = & a c_1 y_{r-1} - b y_r \n\t- a c_1 y_r + b y_{r+1} + \\alpha c_2 y_{r-2} \n\t- \\alpha c_2 y_r \\nonumber\\\\ && \n\t- \\beta y_r + \\beta y_{r+2} + \\xi y_2 y_{r-2} \n\t- \\xi y_2 y_r , \\qquad \\hfill (r\\geq4), \n\t\\\\ \n\\frac{{\\rm d} y_3}{{\\rm d} t} & = & a c_1 y_2 - b y_3 - a c_1 y_3 \n\t+ b y_4 - \\alpha c_2 y_3 - \\xi y_2 y_3 + \\beta y_5 , \n\t\\\\ \n\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 + b y_3 \n\t- a c_1 y_2 - \\alpha y_2 c_2 \n\t+ \\beta y_4 \\nonumber \\\\ && + \\sum_{r=2}^\\infty \\beta y_{r+2} \n\t- \\sum_{r=2}^\\infty \\xi y_2 y_r - \\xi y_2^2 . \\lbl{gbd-y2}\n\\end{eqnarray} \n\n\\subsection{Summary and simulations of the macroscopic model}\n\\label{macro-sec} \n\nThe advantage of the above simplifications is that certain sums \nappear repeatedly; by defining new quantities as these sums, \nthe system can be written in a simpler fashion. We define $N_x = \n\\sum_{r=2}^\\infty x_r$, $N_y = \\sum_{r=2}^\\infty y_r$, then \n\\begin{eqnarray}\n\\frac{{\\rm d} c_1}{{\\rm d} t} & = & 2 \\varepsilon c_2 - 2 \\delta c_1^2 \n\t- a c_1 (N_x+N_y) + b (N_x-x_2+N_y-y_2) , \\lbl{macro-c1}\\\\ \n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & \\delta c_1^2 - \\varepsilon c_2 - 2 \\mu c_2 \n\t+ \\mu\\nu (x_2+y_2) - \\alpha c_2(N_x+N_y) ,\\\\\n\\frac{{\\rm d} N_x}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 \n\t+ \\beta (N_x-x_3-x_2) - \\xi x_2 N_x , \\\\ \n\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 + b x_3 \n\t- a c_1 x_2 - \\alpha x_2 c_2 + \\beta (x_4+N_x-x_2-x_3) \n\t\\nonumber \\\\ && -\\xi x_2^2 - \\xi x_2 N_x , \\\\ \n\\frac{{\\rm d} N_y}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 \n\t+ \\beta (N_y-y_3-y_2) - \\xi y_2 N_y , \\\\ \n\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 + b y_3 \n\t- a c_1 y_2 - \\alpha y_2 c_2 + \\beta (y_4+N_y-y_2-y_3) \n\t\\nonumber\\\\ && - \\xi y_2^2 - \\xi y_2 N_y . \\lbl{macro-y2}\n\\end{eqnarray} \nHowever, such a system of equations is not `closed'. The equations \ncontain $x_3,y_3,x_4,y_4$, and yet we have no expressions for \nthese; reintroducing equations for $x_3,y_3$ would introduce \n$x_5,y_5$ and so an infinite regression would be entered into. \n\n\\begin{figure}[!ht]\n\\vspace*{65mm}\n\\special{psfile=fig-fullt.eps \n\thscale=85 vscale=60 hoffset=-80 voffset=-160}\n\\caption{ Plot of the concentrations $c_1$, $c_2$, \n$N_x$, $N_y$, $N=N_x+N_y$, $\\varrho_x$, $\\varrho_y$, \n$\\varrho_x+\\varrho_y$ and $\\varrho_x+\\varrho_y+2c_2+c1$ \nagainst time, $t$ on a logarithmic timescale. Since model equations \nare in nondimensional form, the time units are arbitrary. \nParameter values $\\mu=1.0$, $\\nu=0.5$, $\\delta=1$, $\\varepsilon=5$, \n$a=4$, $b=0.02$, $\\alpha=10$, $\\xi=10$, $\\beta=0.03$, with \ninitial conditions $c_2=0.49$, $x_4(0)=0.004$, $y_4(0)=0.006$, \nand all other concentrations zero. }\n\\label{fig-fullt}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{73mm}\n\\special{psfile=fig-fullxy.eps \n\thscale=80 vscale=60 hoffset=-80 voffset=-145}\n\\caption{Plot of the cluster size distribution at \n$t=0$ (dashed line), $t=112$ (dotted line) and \n$t=9.4\\times10^5$. Parameters and initial conditions \nas in Figure \\protect\\ref{fig-fullt}. }\n\\label{fig-fullxy}\n\\end{figure}\n\nHence we need to find some suitable alternative expressions for \n$x_3,y_3,x_4,y_4$; or an alternative way of reducing the system to \njust a few ordinary differential equations that can easily be analysed. \nSuch systems are considered in Sections \\ref{tetra-sec}, \n\\ref{hex-sec} and \\ref{new-sec}. Before that, however, we illustrate \nthe behaviour of the system by briefly presenting the results of some \nnumerical simulations. In Figures \\ref{fig-fullt} and \\ref{fig-fullxy} \nwe show the results of a simulation of (\\ref{macro-c1})--(\\ref{macro-y2}). \nThe former shows the evolution of the concentrations $c_1$ which \nrises then decays, $c_2$ which decays since the parameters have \nbeen chosen to reflect a cluster-dominated system. Also plotted are \nthe numbers of clusters $N_x,N_y$ and the mass of material in \nclusters $\\varrho_x$, $\\varrho_y$ defined by \n\\begin{equation} \n\\varrho_x = \\sum_{j=2}^K j x_j , \\qquad \\varrho_y = \\sum_{j=2}^K j y_j . \n\\end{equation} \nNote that under this definition $\\varrho_x+\\varrho_y+c_1+2c_2$ \nis conserved, and this is plotted as {\\sl rho}. Both the total number \nof clusters, $N_x+N_y$, and total mass of material in handed \nclusters $\\varrho_x+\\varrho_y$ appear to equilibrate by $t=10^2$, \nhowever, at a much later time ($t\\sim 10^4 - 10^5$) a \nsymmetry-breaking bifurcation occurs, and the system changes from \nalmost racemic (that is, symmetric) to asymmetric. This is more \nclearly seen in Figure \\ref{fig-fullxy}, where we plot the cluster size \ndistribution at three time points. At $t=0$ there are only dimers \npresent (dashed line), and we impose a small difference in the \nconcentrations of $x_2$ and $y_2$. At a later time, $t=112$ \n(dotted line), there is almost no difference between the $X$- and \n$Y$-distributions, however by the end of the simulation ($t\\sim10^6$, \nsolid line) one distribution clearly completely dominates the other. \n\n\\subsection{Simplified macroscopic model}\n\nTo obtain the simplest model which involves three polymorphs \ncorresponding to right-handed and left-handed chiral clusters and \nachiral clusters, we now aim to simplify the processes of cluster \naggregation and fragmentation in (\\ref{macro-c1})--(\\ref{macro-y2}). \nOur aim is to retain the symmetry-breaking phenomenon but \neliminate physical processes which are not necessary for it to occur. \n\nOur first simplification is to remove all clusters of odd size from the \nmodel, and just consider dimers, tetramers, hexamers, {\\em etc}. \nThis corresponds to putting $a=0$, $b=0$ which removes $x_3$ \nand $y_3$ from the system. \nFurthermore, we put $\\varepsilon=0$ and make $\\delta$ large, so that the \nachiral monomer is rapidly and irreversibly converted to achiral dimer. \nSince the monomers do not then influence the evolution of any of \nthe other variables, we further simplify the system by ignoring $c_1$ \n(or, more simply, just impose initial data in which $c_1(0)=0$). \nThus we are left with \n\\begin{eqnarray}\n\\!\\!\\!\\frac{{\\rm d} c_2}{{\\rm d} t} & \\!=\\! & - 2 \\mu c_2 \n\t+ \\mu\\nu (x_2+y_2) - \\alpha c_2(N_x+N_y) , \\lbl{smm2} \\\\\n\\!\\!\\!\\frac{{\\rm d} N_x}{{\\rm d} t} & \\!=\\! & \\mu c_2 - \\mu\\nu x_2 \n\t+ \\beta (N_x-x_2) - \\xi x_2 N_x , \\\\ \n\\!\\!\\!\\frac{{\\rm d} x_2}{{\\rm d} t} & \\!=\\! & \\!\\mu c_2 - \\mu\\nu x_2 - \\alpha x_2 c_2 \n\t+ \\beta (N_x\\!-\\!x_2 \\!+\\! x_4 ) - \\xi x_2^2 - \\xi x_2 N_x , \\\\ \n\\!\\!\\!\\frac{{\\rm d} N_y}{{\\rm d} t} & \\!=\\! & \\mu c_2 - \\mu\\nu y_2 \n\t+ \\beta (N_y-y_2) - \\xi y_2 N_y , \\\\ \n\\!\\!\\!\\frac{{\\rm d} y_2}{{\\rm d} t} & \\!=\\! & \\!\\mu c_2 - \\mu\\nu y_2 - \\alpha y_2 c_2 \n\t+ \\beta (N_y\\!-\\!y_2 \\!+\\! y_4) - \\xi y_2^2 - \\xi y_2 N_y . \\lbl{smmy2}\n\\end{eqnarray} \nSince we have removed four parameters from the model, \nand halved the number of dependent variables, we show a couple \nof numerical simulations just to show that the system \nabove does still exhibit symmetry-breaking behaviour. \n\n\\begin{figure}[!ht]\n\\vspace*{69mm}\n\\special{psfile=fig-dimt.eps \n\thscale=85 vscale=58 hoffset=-80 voffset=-150}\n\\caption{ Plot of the concentrations $c_1$, $c_2$, \n$N_x$, $N_y$, $N=N_x+N_y$, $\\varrho_x$, $\\varrho_y$, \n$\\varrho_x+\\varrho_y$ and $\\varrho_x+\\varrho_y+2c_2+c_1$ \nagainst time, $t$ on a logarithmic timescale. Since model \nequations are in nondimensional form, the time units are \narbitrary. Parameter values $\\mu=1$, $\\nu=0.5$, \n$\\alpha=10$, $\\xi=10$, $\\beta=0.03$, with initial \nconditions $c_2=0.49$, $x_4(0)=0.004$, $y_4(0)=0.006$, \nall other concentrations zero. }\n\\label{fig-dimt}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{71mm}\n\\special{psfile=fig-dimxy.eps \n\thscale=80 vscale=58 hoffset=-80 voffset=-140}\n\\caption{Plot of the cluster size distribution at \n$t=0$ (dashed line), $t=250$ (dotted line) and \n$t=6\\times10^5$. Parameters and initial conditions \nas in Figure \\protect\\ref{fig-dimt}. }\n\\label{fig-dimxy}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{70mm}\n\\special{psfile=fig-dimtt.eps \n\thscale=80 vscale=60 hoffset=-80 voffset=-150}\n\\caption{ Plot of the concentrations $c_1$, $c_2$, $N_x$, $N_y$, \n$N=N_x+N_y$, $\\varrho_x$, $\\varrho_y$, $\\varrho_x+\\varrho_y$ and \n$\\varrho_x+\\varrho_y+2c_2+c_1$ against time, $t$ on a logarithmic timescale. \nParameters and initial conditions as in Figure \\protect\\ref{fig-dimt}. }\n\\label{fig-dimtt}\n\\end{figure}\n\nFigure \\ref{fig-dimt} appears similar to Figure \\ref{fig-fullt}, \nsuggesting that removing the monomer interactions has changed the \nunderlying dynamics little. We still observe the characteristic \nequilibration of cluster numbers and cluster masses as $c_2$ decays, \nand then a period of quiesence ($t\\sim10$ to $10^4$) before a later \nsymmetry-breaking event, around $t\\sim10^5$. \nAt first sight, the distribution of $X$- and $Y$-clusters displayed in \nFigure \\ref{fig-dimxy} is quite different to Figure \\ref{fig-fullxy}; this \nis due to the absence of monomers from the system, meaning that \nonly even-sized clusters can now be formed. If one only looks at the \neven-sized clusters in Figure \\ref{fig-dimxy}, we once again see only \na slight difference at $t=0$ (dashed line), almost no difference at \n$t\\approx250$ (dotted line) but a significant difference at \n$t=6\\times10^5$ (solid line). \nWe include one further graph here, Figure \\ref{fig-dimtt} similar to \nFigure \\ref{fig-dimt} but on a linear rather than a logarithmic timescale. \nThis should be compared with Figures such as Figures 3 and 4 of \nViedma \\cite{viedma} and Figure 1 of Noorduin {\\em et al.}\\ \\cite{wim}. \n\n\\section{The truncation at tetramers} \n\\label{tetra-sec} \n\\setcounter{equation}{0}\n\n\\begin{figure}[!ht]\n\\begin{picture}(500,120)(0,-10)\n\\put(80,50){\\circle{17}}\n\\put(077,48){$c_2$}\n\\put(080,10){\\circle{17}}\n\\put(077,08){$y_2$}\n\\put(080,90){\\circle{17}}\n\\put(077,88){$x_2$}\n\\put(140,10){\\circle{20}}\n\\put(137,08){$y_4$}\n\\put(140,90){\\circle{20}}\n\\put(137,88){$x_4$}\n\\put(77,20){\\vector(0,1){20}}\n\\put(83,40){\\vector(0,-1){20}}\n\\put(83,60){\\vector(0,1){20}}\n\\put(77,80){\\vector(0,-1){20}}\n\\put(63,30){$\\nu\\mu$}\n\\put(85,30){$\\mu$}\n\\put(63,70){$\\nu\\mu$}\n\\put(85,70){$\\mu$}\n\\put(107,97){$\\beta$}\n\\put(125,95){\\vector(-1,0){30}}\n\\put(095,90){\\vector(1,0){30}}\n\\put(107,81){$\\xi$}\n\\put(110,85){\\oval(30,20)[b]}\n\\put(125,85){\\vector(0,1){5}}\n\\put(113,58){$\\alpha$}\n\\put(090,52){\\vector(3,+2){40}}\n\\put(090,48){\\vector(3,-2){40}}\n\\put(113,38){$\\alpha$}\n\\put(125,17){\\vector(0,-1){5}}\n\\put(110,17){\\oval(30,20)[t]}\n\\put(107,12){$\\xi$}\n\\put(095,10){\\vector(1,0){30}}\n\\put(125,05){\\vector(-1,0){30}}\n\\put(107,-5){$\\beta$}\n\\end{picture} \n\\caption{Simplest possible reaction scheme which might \nexhibit chiral symmetry-breaking. }\n\\label{simp-rec-sch-fig}\n\\end{figure}\n\nThe simplest possible reaction scheme of the form \n(\\ref{gbd-c1})--(\\ref{gbd-y2}) which we might expect to exhibit \nsymmetry-breaking to homochirality is the system truncated at \ntetramers, namely \n\\begin{eqnarray} \n\\displaystyle\\frac{{\\rm d} c_2}{{\\rm d} t} & = & - 2\\mu c_2 + \\mu\\nu (x_2+y_2) \n\t-\\alpha c_2(x_2+y_2) , \\lbl{dim-c2dot} \\\\ \n\\displaystyle\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu x_2 \n\t- \\alpha c_2 x_2 - 2 \\xi x_2^2 + 2 \\beta x_4 , \\\\ \n\\displaystyle\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu\\nu y_2 \n\t- \\alpha c_2 y_2 - 2 \\xi y_2^2 + 2 \\beta y_4 , \\\\ \n\\displaystyle\\frac{{\\rm d} x_4}{{\\rm d} t} & = & \\alpha x_2 c_2 + \\xi x_2^2 - \\beta x_4 , \\\\ \n\\displaystyle\\frac{{\\rm d} y_4}{{\\rm d} t} & = & \\alpha y_2 c_2 + \\xi y_2^2 - \\beta y_4 . \n\\lbl{dim-y4dot} \\end{eqnarray} \n\nWe investigate the symmetry-breaking by transforming \nthe variables $x_2$, $x_4$, $y_2$, $y_4$ according to \n\\begin{eqnarray}\nx_2 = \\mbox{$\\frac{1}{2}$} z (1+\\theta) , &\\quad& y_2 = \\mbox{$\\frac{1}{2}$} z (1-\\theta) , \n\\lbl{tet-ztheta} \\\\ \nx_4 = \\mbox{$\\frac{1}{2}$} w (1+\\phi) , && y_4 = \\mbox{$\\frac{1}{2}$} w (1-\\phi) , \n\\lbl{tet-wphi} \n\\end{eqnarray}\nwhere $z=x_2+y_2$ is the total concentration of chiral dimers, \n$w=x_4+y_4$ is the total tetramer concentration, $\\theta=(x_2-y_2)\/z$ \nis the relative chirality of the dimers, $\\phi=(x_4-y_4)\/w$ is the \nrelative chirality of tetramers. Hence \n\\begin{eqnarray}\n\\frac{{\\rm d} c_2}{{\\rm d} t} & = & - 2\\mu c_2 + \\mu\\nu z - \\alpha c_2 z , \n\t\\lbl{c23} \\\\ \n\\frac{{\\rm d} z}{{\\rm d} t} & = & 2 \\mu c_2 - \\mu\\nu z - \\alpha c_2 z \n\t- \\xi z^2 (1+\\theta^2) + 2 \\beta w , \\\\ \n\\frac{{\\rm d} w}{{\\rm d} t} & = & \\alpha z c_2 + \\mbox{$\\frac{1}{2}$} \\xi z^2 (1+\\theta^2) \n\t- \\beta w , \\lbl{w3}\\\\ \n\\frac{{\\rm d} \\theta}{{\\rm d} t} & = & - \\theta \\left( \\frac{2\\mu c}{z} + \n\t\\frac{2\\beta w}{z}+ \\xi z (1-\\theta^2) \\right) + \n\t\\frac{2\\beta w\\phi}{z} , \\\\\n\\frac{{\\rm d} \\phi}{{\\rm d} t} & = & \\theta \\frac{z}{w} ( \\alpha c + \\xi z ) \n\t- \\left( \\alpha c + \\mbox{$\\frac{1}{2}$} \\xi z (1+\\theta^2) \\right) \\frac{z}{w} \\phi . \n\\end{eqnarray}\nThe stability of the evolving symmetric-state ($\\theta=\\phi=0$) \nis given by the eigenvalues ($q$) of the matrix \n\\begin{equation}\n\\left( \\begin{array}{cc} \n- \\left( \\frac{2\\mu c}{z} + \\frac{2\\beta w}{z} + \\xi z \\right) & \n\\frac{2\\beta w}{z} \\\\ \n(\\alpha c + \\xi z) \\frac{z}{w} & \n- (\\alpha c + \\mbox{$\\frac{1}{2}$} \\xi z) \\frac{z}{w} \n\\end{array} \\right) , \n\\end{equation}\nwhich are given by\n\\begin{eqnarray}\nq^2 + q \\left( \\frac{\\alpha c z}{w} + \\frac{\\xi z^2}{w} \n+ \\frac{2\\mu c}{z} + \\xi z + \\frac{2\\beta w}{z} \\right) + && \\nonumber \\\\ \n\\frac{1}{w} \\left( 2\\mu c \\alpha c + \\mu c \\xi z + \n\\alpha c \\xi z^2 + \\mbox{$\\frac{1}{2}$} \\xi^2 z^3 - \\beta \\xi z w \\right) &=&0 . \n\\end{eqnarray}\nHence there is an instability if \n\\begin{equation}\n\\beta \\xi z w > 2\\mu c \\alpha c + \\mu c \\xi z + \n\\alpha c \\xi z^2 + \\mbox{$\\frac{1}{2}$} \\xi^2 z^3 , \n\\lbl{crude-instab}\n\\end{equation}\nusing the steady-state result that $2\\beta w = z(2\\alpha c + \\xi z)$ \nand factorising ($2\\alpha c + \\xi z$) out of the result, reduces the \ninstability (\\ref{crude-instab}) to the contradictory $\\xi z^2 > \n\\xi z^2 + 2\\mu c$. Hence the racemic steady-state of the system \nis stable for all choices of parameter values and is approached \nfrom all initial conditions. However, initial perturbations, \nmay be amplified due to the presence of nonlinear terms. \n\n\\begin{figure}[!ht]\n\\vspace*{75mm}\n\\special{psfile=fig-tet3.eps \nhscale=80 vscale=67 hoffset=-60 voffset=-175}\n\\caption{The concentrations $c_2$, $z$ and $w$ \n(\\protect\\ref{tet-ztheta})--(\\protect\\ref{tet-wphi}) plotted against \ntime, for the tetramer-truncated system with the two sets of initial \ndata (\\protect\\ref{tet-ics}). Since model equations \nare in nondimensional form, the time units are arbitrary. The \nparameter values are $\\mu=1$, $\\nu=0.5$, $\\alpha=\\xi=10$, \n$\\beta=0.1$. }\n\\label{fig-tet3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{75mm}\n\\special{psfile=fig-tet4.eps \nhscale=80 vscale=67 hoffset=-60 voffset=-175}\n\\caption{The chiralities $\\theta$, $\\phi$ \n(\\protect\\ref{tet-ztheta})--(\\protect\\ref{tet-wphi}) plotted against \ntime, for the tetramer-truncated system with \nthe two sets of initial data (\\protect\\ref{tet-ics}). Since model equations \nare in nondimensional form, the time units are arbitrary. The \nparameter values are the same as in Figure \\ref{fig-tet3}.}\n\\label{fig-tet4}\n\\end{figure}\n\nEvolution from two sets of initial conditions of the system \n(\\ref{dim-c2dot})--(\\ref{dim-y4dot}) are shown in each of Figures \n\\ref{fig-tet3}, \\ref{fig-tet4}. The continuous and dotted lines \ncorrespond to the initial data \n\\begin{equation} \\begin{array}{c} \nc_2(0) = 0.29 , \\quad x_2(0) = 0.0051, \\quad y_2(0) = 0.0049, \\\\ \nx_4(0) = 0.051 , \\quad y_4(0) = 0.049 ; \\quad {\\rm and} \\\\ \nc_2(0) = 0 , \\quad x_2(0) = 0.051 \\quad y_2(0) = 0.049, \\\\ \nx_4(0) = 0.1 , \\quad y_4(0) = 0.1 ; \n\\end{array} \\lbl{tet-ics} \\end{equation}\nrespectively. \nIn the former case, the system starts with considerable amount \nof amorphous dimer, which is converted into clusters, and initially \nthere is a slight chiral imbalance in favour of $x_2$ and $x_4$ over \n$y_2$ and $y_4$. Over time this imbalance reduces (see figure \n\\ref{fig-tet4}); although there is a region around $t=1$ where \n$\\theta$ increases, both $\\theta$ and $\\phi$ eventually \napproach the zero steady-state. \n\nFor both sets of initial conditions we note that the chiralities evolve \nover a significantly longer timescale than the concentrations, the \nlatter having reached steady-state before $t=10$ and the former \nstill evolving when $t={\\cal O}(10^2)$. In the second set of initial \ndata, there is no $c_2$ present initially and there are exactly equal \nnumbers of the two chiral forms of the larger cluster, but a slight \nexess of $x_2$ over $y_2$. In time an imbalance in larger clusters \nis produced, but over larger timescales, both $\\theta$ and $\\phi$ \nagain approach the zero steady-state. \n\nHence, we observe that the truncated system \n(\\ref{dim-c2dot})--(\\ref{dim-y4dot}) does {\\em not} yield a chirally \nasymmetric steady-state. Even though in the early stages of the \nreaction chiral perturbations may be amplified, at the end of the \nreaction there is a slower timescale over which the system returns \nto a racemic state. In the next section we consider a system \ntruncated at hexamers to investigate whether that system allows \nsymmetry-breaking of the steady-state. \n\n\\section{The truncation at hexamers}\n\\label{hex-sec} \\setcounter{equation}{0}\n\nThe above analysis has shown that the truncation of the model \n(\\ref{gbd-c1})--(\\ref{gbd-y2}) to (\\ref{dim-c2dot})--(\\ref{dim-y4dot}) \nresults in a model which always ultimately approaches the \nsymmetric (racemic) steady-state. In this section, we show that a \nmore complex model, the truncation at hexamers retains enough \ncomplexity to demonstrate the symmetry-breaking bifurcation which \noccurs in the full system. In this case the governing equations are \n\\begin{eqnarray} \n\\displaystyle\\frac{{\\rm d} c_2}{{\\rm d} t} &=& -2 \\mu c_2 + \\mu \\nu (x_2+y_2) \n- \\alpha c_2 (x_2+y_2) - \\alpha c_2 (x_4+y_4) , \n\\lbl{hex-c2} \\\\ \n\\displaystyle\\frac{{\\rm d} x_2}{{\\rm d} t} & = & \\mu c_2 - \\mu \\nu x_2 \n- \\alpha c_2 x_2 - 2 \\xi x_2^2 - \\xi x_2 x_4 + 2\\beta x_4 \n+ \\beta x_6 , \n\\\\ \n\\displaystyle\\frac{{\\rm d} x_4}{{\\rm d} t} & = & \\alpha x_2 c_2 + \\xi x_2^2 \n- \\beta x_4 - \\alpha c_2 x_4 - \\xi x_2 x_4 + \\beta x_6 , \n\\\\\n\\displaystyle\\frac{{\\rm d} x_6}{{\\rm d} t} & = & \\alpha x_4 c_2 + \\xi x_2 x_4 \n- \\beta x_6 , \n\\\\\n\\displaystyle\\frac{{\\rm d} y_2}{{\\rm d} t} & = & \\mu c_2 - \\mu \\nu y_2 \n- \\alpha c_2 y_2 - 2 \\xi y_2^2 - \\xi y_2 y_4 + 2\\beta y_4 \n+ \\beta y_6 , \n\\\\ \n\\displaystyle\\frac{{\\rm d} y_4}{{\\rm d} t} & = & \\alpha y_2 c_2 + \\xi y_2^2 \n- \\beta y_4 - \\alpha c_2 y_4 - \\xi y_2 y_4 + \\beta y_6 , \n\\\\\n\\displaystyle\\frac{{\\rm d} y_6}{{\\rm d} t} & = & \\alpha y_4 c_2 + \\xi y_2 y_4 \n- \\beta y_6 . \n\\lbl{hex-y6} \\end{eqnarray}\n\nTo analyse the symmetry-breaking in the system we transform the \ndependent coordinates from $x_2,x_4,x_6,y_2,y_4,y_6$ to total \nconcentrations $z,w,u$ and relative chiralities $\\theta,\\phi,\\psi$ \naccording to \n\\begin{equation} \\begin{array}{rclcrclcrcl}\nx_2 &=& \\mbox{$\\frac{1}{2}$} z (1 + \\theta) , & \\quad & \nx_4 &=& \\mbox{$\\frac{1}{2}$} w (1 + \\phi) , & \\quad & \nx_6 &=& \\mbox{$\\frac{1}{2}$} u (1 + \\psi) , \\\\[2ex] \ny_2 &=& \\mbox{$\\frac{1}{2}$} z (1 - \\theta) , & \\quad & \ny_4 &=& \\mbox{$\\frac{1}{2}$} w (1 - \\phi) , & \\quad & \ny_6 &=& \\mbox{$\\frac{1}{2}$} u (1 - \\psi) . \n\\end{array} \\end{equation}\n\nWe now separate the governing equations for the total concentrations \nof dimers ($c,z$), tetramers ($w$) and hexamers ($u$)\n\\begin{eqnarray}\n\\displaystyle\\frac{{\\rm d} c}{{\\rm d} t} & = & - 2 \\mu c \n+ \\mu \\nu z - \\alpha c z - \\alpha c w , \n\\lbl{hex-cdot}\\\\ \n\\displaystyle\\frac{{\\rm d} z}{{\\rm d} t} & =& 2\\mu c - \\mu \\nu z \n- \\alpha c z - \\xi z^2 (1+\\theta^2) - \\mbox{$\\frac{1}{2}$} z w (1+\\theta\\phi) \n+ \\beta u + 2 \\beta w , \\nonumber \\\\ && \n\\\\ \n\\displaystyle\\frac{{\\rm d} w}{{\\rm d} t} & = & \\alpha c z \n+ \\mbox{$\\frac{1}{2}$} \\xi z^2 (1+\\theta^2) - \\beta w + \\beta u \n- \\alpha c w - \\mbox{$\\frac{1}{2}$} \\xi z w (1+\\theta\\phi) , \n\\\\ \n\\displaystyle\\frac{{\\rm d} u}{{\\rm d} t} & = & \\alpha c w \n+ \\mbox{$\\frac{1}{2}$} \\xi z w (1+\\theta\\phi) - \\beta u , \n\\lbl{hex-udot}\\end{eqnarray}\nfrom those for the chiralities \n\\begin{eqnarray}\n\\displaystyle \\frac{{\\rm d} \\psi}{{\\rm d} t} & =& \n\\frac{\\alpha c w}{u} (\\phi-\\psi) + \n\\frac{\\xi z w}{2u} ( \\theta+\\phi-\\psi-\\psi\\phi\\theta )\n\\lbl{hex-psi-dot} \\\\ \n\\displaystyle \\frac{{\\rm d} \\phi}{{\\rm d} t} & = & \n\\frac{\\alpha c z }{w} (\\theta-\\phi) + \n\\frac{\\xi z^2}{2w} ( 2\\theta -\\phi-\\phi\\theta^2) + \n\\frac{\\beta u}{w} (\\psi-\\phi) - \n\\mbox{$\\frac{1}{2}$} \\xi z \\theta (1-\\phi^2) , \\nonumber \\\\ && \n\\\\ \n\\displaystyle \\frac{{\\rm d} \\theta}{{\\rm d} t} & = & \n-\\frac{2\\mu c \\theta}{z} - \\xi z \\theta(1\\!-\\!\\theta^2) - \n\\mbox{$\\frac{1}{2}$} \\xi w \\phi (1\\!-\\!\\theta^2) + \\frac{\\beta u\\psi}{z} - \n\\frac{\\beta u \\theta}{z} \\nonumber \\\\ && + \\frac{2\\beta w\\phi}{z} \n- \\frac{2\\beta w \\theta}{z} . \\lbl{hex-th-dot}\n\\end{eqnarray}\n\nIn applications, we expect $\\nu<1$, so that the small amorphous \nclusters (dimers) prefer to adopt one of their chiral states rather \nthan the achiral structure. In addition, we note that the grinding \nprocess observed in experiments is much longer than the \ncrystallisation process, and that there are many larger, macroscopic \ncrystals hence we consider two limits in which $\\beta \\ll \\alpha \\xi$. \nWe will consider the case of small $\\beta$ with all other parameters \nbeing ${\\cal O}(1)$ and then the case where $\\alpha\\sim\\xi\\gg1$ and \nall other parameters are ${\\cal O}(1)$. \n\n\\subsection{Symmetric steady-state for the concentrations}\n\nFirstly, let us solve for the symmetric steady-state. \nIn this case we assume $\\theta=0=\\phi=\\psi$, simplifying \nequations (\\ref{hex-cdot})--(\\ref{hex-udot}). One of \nthese is a redundant equation, hence we have the solution \n\\begin{equation}\nw = \\frac{z}{\\beta}(\\alpha c + \\mbox{$\\frac{1}{2}$} \\xi z) , \\qquad \nu = \\frac{z}{\\beta^2}(\\alpha c+\\mbox{$\\frac{1}{2}$}\\xi z)^2 , \n\\lbl{hex-wu-sol} \\end{equation}\n\\begin{equation}\nc = \\frac{1}{\\alpha} \\left(\\sqrt{ \\left( \\frac{\\beta}{2} + \n\\frac{\\beta\\mu}{\\alpha z} + \\frac{\\xi z}{4} \\right)^2 \n+ \\beta\\mu\\nu} - \\frac{\\beta}{2} - \\frac{\\beta\\mu}\n{\\alpha z} - \\frac{\\xi z}{4} \\right) , \n\\lbl{hex-c-sol} \\end{equation}\nwith $z$ being determined by conservation of total mass in the system \n\\begin{equation} \n2c + 2 z + 4 w + 6 u = \\varrho . \\lbl{hex-roo} \n\\end{equation} \n\nIn the case of small grinding, ($\\beta\\ll1$), with $\\varrho$ \nand all other parameters being ${\\cal O}(1)$, we find \n\\begin{equation} \\begin{array}{rclcrcl}\nz & = & \\left( \\displaystyle\\frac{2\\varrho \\beta^2}{3 (\\alpha\\nu+\\xi)^2} \n\t\\right)^{1\/3} , &\\qquad& \nc & = & \\nu \\left( \\displaystyle\\frac{\\varrho \\beta^2}{12 (\\alpha\\nu+\\xi)^2} \n\t\\right)^{1\/3} , \\\\ \nw & = & \\left( \\displaystyle\\frac{\\varrho^2 \\beta}{18 (\\alpha\\nu+\\xi)} \n\t\\right)^{1\/3} , &\\qquad& \nu & = & \\displaystyle\\frac{\\varrho}{6} . \n\\end{array} \\lbl{hex-ssss-asymp} \\end{equation} \nIn this case most of the mass is in hexamers with a little in \ntetramers and very little in dimers. \n\nIn the asymptotic limit of $\\alpha \\sim \\xi \\gg 1$ \nand all other parameters ${\\cal O}(1)$, we find \n\\begin{eqnarray} & \nc = \\displaystyle\\frac{\\mu\\nu}{\\alpha} \\left( \n\\displaystyle\\frac{12\\beta}{\\varrho\\xi} \\right)^{1\/3} , \\quad \nz = \\left( \\displaystyle\\frac{2\\beta^2\\varrho}{3\\xi^2} \\right)^{1\/3} , \\quad \nw = \\left( \\displaystyle\\frac{\\beta\\varrho^2}{18\\xi} \\right)^{1\/3} , \\quad\nu = \\displaystyle\\frac{\\varrho}{6} . \n& \\nonumber \\\\ && \\lbl{hex-sss2-asymp} \n\\end{eqnarray} \nThis differs significantly from the other asymptotic scaling as, not only \nare $c$ and $z$ both small, they are now different orders of magnitude, \nwith $c\\ll z$. We next analyse the stability of these symmetric states. \n\n\\subsection{Stability of symmetric state}\n\nIn deriving the above solutions (\\ref{hex-wu-sol})--(\\ref{hex-c-sol}), \nwe have assumed chiral symmetry, that is, $\\theta=0=\\psi=\\phi$. \nWe now turn to analyse the validity of this assumption. Linearising \nthe system of equations (\\ref{hex-psi-dot})--(\\ref{hex-th-dot}) \nwhich govern the chiralities, we determine whether the symmetric \nsolution is stable from \n\\begin{equation} \\!\\! \n\\frac{{\\rm d} }{{\\rm d} t} \\!\\! \\left( \\!\\! \\begin{array}{c} \n\\psi \\\\ \\phi \\\\ \\theta \\end{array} \\!\\! \\right) \\!=\\! \n\\left( \\begin{array}{ccc} \n\\!\\!\\!- \\displaystyle\\frac{\\alpha c w}{u} \\!-\\! \\displaystyle\\frac{\\xi z w}{2u} \\!\\!& \n\\displaystyle\\frac{\\alpha c w}{u} \\!+\\! \\displaystyle\\frac{\\xi z w}{2u} & \n\\displaystyle\\frac{\\xi z w}{2u} \\\\[2ex] \n\\displaystyle\\frac{\\beta u}{w} & -\\displaystyle\\frac{\\alpha c z}{w} \n\\!-\\! \\displaystyle\\frac{\\xi z^2}{2w} \\!-\\! \\displaystyle\\frac{\\beta u}{w} & \n\\displaystyle\\frac{\\alpha c z}{w} \\!+\\! \\displaystyle\\frac{\\xi z^2}{w} \n\\!-\\! \\mbox{$\\frac{1}{2}$} \\xi z \\\\[2ex] \n\\displaystyle\\frac{\\beta u}{z} & \n\\displaystyle\\frac{2\\beta w}{z} \\!-\\! \\displaystyle\\frac{\\xi w}{2} & \\!\\!\n- \\displaystyle\\frac{2\\mu c}{z} \\!-\\! \\xi z \\!-\\! \\frac{\\beta u}{z} \n\\!-\\! \\displaystyle\\frac{2\\beta w}{z} \\!\\!\n\\end{array} \\! \\right) \\!\\!\n\\left( \\!\\begin{array}{c} \\psi \\\\ \\phi \\\\ \\theta \n\\end{array} \\!\\right) \\!.\\!\\!\n\\lbl{hex-stab-mat} \\end{equation}\nFor later calculations it is useful to know the determinant of this matrix. \nUsing the steady-state solutions (\\ref{hex-wu-sol}), the determinant \nsimplifies to \n\\begin{equation}\nD = \\frac{3 c}{4 \\beta \\rho} ( 2 \\alpha c + \\xi z )^2 \n( \\alpha \\xi z^2 - 4 \\beta \\mu ) . \n\\end{equation}\n\nFor general parameter values, the signs of the real parts of the \neigenvalues of the matrix in (\\ref{hex-stab-mat}) are not clear. \nHowever, using the asymptotic result (\\ref{hex-ssss-asymp}), \nfor $\\beta\\ll1$, we obtain the simpler matrix \n\\begin{equation}\n\\left( \\!\\!\\begin{array}{ccc} \n-\\beta & \\beta & \\displaystyle \\frac{\\beta\\xi}{\\xi\\!+\\!\\alpha\\nu} \n\\\\[2ex] \n\\left( \\displaystyle\\frac{\\beta^2 \\varrho (\\xi\\!+\\!\\alpha\\nu) }{12} \\right)^{1\/3} & \n- \\left( \\displaystyle\\frac{\\beta^2 \\varrho (\\xi\\!+\\!\\alpha\\nu) }{12} \\right)^{1\/3} & \n-\\frac{\\xi}{2} \\left( \\displaystyle\\frac{2\\beta^2\\varrho}{3(\\xi\\!+\\!\\alpha\\nu)^2} \\right) ^{1\/3} \n\\\\[2ex] \n\\beta^{1\/3} \\left( \\displaystyle\\frac{\\xi\\!+\\!\\alpha\\nu}{12\\varrho} \\right)^{2\/3} & \n- \\frac{\\xi}{2} \\left( \\displaystyle\\frac{\\beta\\varrho^2}{18(\\xi\\!+\\!\\alpha\\nu)} \\right)^{1\/3} & \n- \\mu \\nu - \\beta^{1\/3} \\left( \\displaystyle\\frac{\\xi\\!+\\!\\alpha\\nu}{12\\varrho} \\right)^{2\/3} \n\\end{array} \\!\\!\\right) \\! , \\lbl{hex-asy1-mat}\n\\end{equation}\nwhose characteristic polynomial is \n\\begin{equation} \n0 = q^3 + \\mu\\nu q^2 + \\mu\\nu \\left( \\rec{12} \\beta^2 \\varrho \n(\\xi\\!+\\!\\alpha\\nu) \\right)^{1\/3} q - D , \\lbl{hex-asy1-cp} \n\\end{equation} \nFormally $D$ is the determinant of the matrix in (\\ref{hex-asy1-mat}), \nwhich is zero, giving a zero eigenvalue, which indicates marginal \nstability. Hence, we return to the more accurate matrix in \n(\\ref{hex-stab-mat}), which gives $D \\sim -\\beta^2\\mu\\nu$. \nThe polynomial (\\ref{hex-asy1-cp}) thus has roots \n\\begin{equation} \nq_1 \\sim -\\mu\\nu, \\quad \nq_2 \\sim - \\left( \\frac{ \\beta^2 \\varrho (\\xi\\!+\\!\\alpha\\nu)}{12} \\right)^{1\/3} , \n\\quad \nq_3 \\sim - \\left( \\frac{12 \\beta^4}{\\varrho(\\alpha\\nu\\!+\\!\\xi)} \\right)^{1\/3} . \n\\lbl{hex-ev1} \\end{equation} \nThis means that the symmetric state is always linearly stable \nfor this asymptotic scaling. We expect to observe evolution on \nthree distinct timescales, one of ${\\cal O}(1)$, one of \n${\\cal O}(\\beta^{-2\/3})$ and one of ${\\cal O}(\\beta^{-4\/3})$. \n\nWe now consider the other asymptotic limit, namely, \n$\\alpha\\sim\\xi\\gg1$ and all other parameters are ${\\cal O}(1)$. \nIn this case, taking the leading order terms in each row, the stability \nmatrix in (\\ref{hex-stab-mat}) reduces to \n\\begin{equation}\n\\left( \\begin{array}{ccc} \n-6 \\mu \\nu \\left( \\frac{12\\beta}{\\varrho\\xi} \\right)^{2\/3} & \n6 \\mu \\nu \\left( \\frac{12\\beta}{\\varrho\\xi} \\right)^{2\/3} & 0\n\\\\\n\\left( \\frac{\\beta^2\\varrho\\xi}{12}\\right)^{1\/3} & \n-\\left( \\frac{\\beta^2\\varrho\\xi}{12}\\right)^{1\/3} & \n-\\left( \\frac{\\beta^2\\varrho\\xi}{12}\\right)^{1\/3} \n\\\\ \n\\left( \\frac{\\beta\\varrho^2 \\xi^2}{144} \\right)^{1\/3} & \n- \\left( \\frac{\\beta\\varrho^2 \\xi^2}{144} \\right)^{1\/3} & \n- \\left( \\frac{\\beta\\varrho^2 \\xi^2}{144} \\right)^{1\/3} \n\\end{array} \\right) , \n\\end{equation}\nwhich again formally has a zero determinant. \nThe characteristic polynomial is \n\\begin{equation}\n0 = q^3 + q^2 + 6 \\beta \\mu\\nu q - D , \n\\end{equation}\nwherein we again take the more accurate determinant \nobtained from a higher-order expansion of \n(\\ref{hex-stab-mat}), namely $D=\\beta^2\\mu\\nu$. \nThe eigenvalues are then given by \n\\begin{equation}\nq_1 \\sim - \\left( \\frac{\\beta\\varrho^2\\xi^2}{144} \\right)^{1\/3} , \n\\qquad \nq_{2,3} \\sim \\pm \\sqrt{\\beta\\mu\\nu} \n\\left( \\frac{12\\beta}{\\varrho\\xi} \\right)^{1\/3} . \n\\lbl{hex-ev2} \\end{equation}\nWe now observe that there is always one stable and \ntwo unstable eigenvalues, so we deduce that the system \nbreaks symmetry in the case $\\alpha \\sim \\xi \\gg 1$. \nThe first eigenvalue corresponds to a faster timescale \nwhere $t\\sim {\\cal O}(\\xi^{-2\/3})$ whilst the latter two \ncorrespond to the slow timescale where $t={\\cal O}(\\xi^{1\/3})$. \n\n\\subsection{Simulation results} \n\n\\begin{figure}[!ht] \n\\vspace*{64mm} \n\\special{psfile=fig-hex1.eps \n\thscale=60 vscale=45 hoffset=00 voffset=-96} \n\\caption{Illustration of the evolution of the total concentrations \n$c_2,z,w,u$ for a numerical solution of the system truncated \nat hexamers (\\ref{hex-c2})--(\\ref{hex-y6}) in the limit \n$\\alpha\\sim\\xi \\gg1$. Since model equations \nare in nondimensional form, the time units are arbitrary. \nThe parameters are $\\alpha=\\xi=30$, $\\nu=0.5$, $\\beta=\\mu=1$, \nand the initial data is $x_6(0)=y_6(0)=0.06$, \n$x_4(0)=y_4(0)=0.01$, $x_2(0) = 0.051$, $y_2(0) = 0.049$, \n$c_2(0) = 0$. Note the time axis has a logarithmic scale. } \n\\label{fig-hex-1} \n\\end{figure}\n\n\\begin{figure}[!ht]\n\\vspace*{64mm}\n\\special{psfile=fig-hex2.eps \n\thscale=60 vscale=45 hoffset=00 voffset=-96}\n\\caption{Graph of the evolution of the chiralities against time \non a log-log scale; results of numerical simulation of the same \nhexamer-truncated system, with identical initial data and \nparameters as in Figure \\protect\\ref{fig-hex-1}. }\n\\label{fig-hex-2}\n\\end{figure}\n\nWe briefly review the results of a numerical simulation of \n(\\ref{hex-c2})--(\\ref{hex-y6}) in the case $\\alpha\\sim\\xi\\gg1$ \nto illustrate the symmetry-breaking observed therein. \nAlthough the numerical simulation used the variables $x_k$ and \n$y_k$ ($k=2,4,6$) and $c_2$, we plot the total concentrations \n$z,w,u$ in Figure \\ref{fig-hex-1}. The initial conditions have a \nslight imbalance in the handedness of small crystals ($x_2,y_2$). \nThe chiralities of small ($x_2,y_2,z$), medium ($x_4,y_4,w$), and \nlarger ($x_6,y_6,u$) are plotted in Figure \\ref{fig-hex-2} on a \nlog-log scale. Whilst Figure \\ref{fig-hex-1} shows the \nconcentrations in the system has equilibrated by $t=10$, at this \nstage the chiralities are in a metastable state, that is, a long plateau \nin the chiralities between $t=10$ and $t=10^3$ where little appears \nto change. There then follows a period of equilibration of chirality \non the longer timescale when $t\\sim 10^4$. We have observed \nthis significant delay between the equilibration of concentrations \nand that of chiralities in a large number of simulations. The reason \nfor this difference in timescales is due to the differences in the sizes \nof the eigenvalues in (\\ref{hex-ev1}). \n\nWe have also investigated the case $\\beta \\ll1$ with all other \nparameters ${\\cal O}(1)$ to verify that this case does indeed approach \nthe racemic state at large times (that is, $\\theta,\\phi,\\zeta \\rightarrow0$ \nas $t\\rightarrow\\infty$). However, once again the difference in \ntimescales can be observed, with the concentrations reaching \nequilibration on a faster timescale than the chiralities, due to the \ndifferent magnitudes of eigenvalues (\\ref{hex-ev2}).\n\n\\section{New simplifications of the system}\n\\label{new-sec}\n\\setcounter{equation}{0}\n\nWe return to the equations (\\ref{smm2})--(\\ref{smmy2}) \nin the case $\\delta=0$, now writing $x_2=x$ and $y=y_2$ \nto obtain \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & = & - 2 \\mu c \n\t+ \\mu\\nu (x+y) - \\alpha c(N_x+N_y) , \\lbl{newcdot} \\\\\n\\frac{{\\rm d} x}{{\\rm d} t} & = & \\mu c - \\mu\\nu x - \\alpha x c \n\t+ \\beta (N_x-x + x_4) - \\xi x^2 - \\xi x N_x , \\lbl{newxdot} \\\\ \n\\frac{{\\rm d} y}{{\\rm d} t} & = & \\mu c - \\mu\\nu y - \\alpha y c \n\t+ \\beta (N_y-y + y_4) - \\xi y^2 - \\xi y N_y , \\lbl{newydot}\\\\ \n\\frac{{\\rm d} N_x}{{\\rm d} t} & = & \\mu c - \\mu\\nu x \n\t+ \\beta (N_x-x) - \\xi x N_x , \\lbl{newnxdot} \\\\ \n\\frac{{\\rm d} N_y}{{\\rm d} t} & = & \\mu c - \\mu\\nu y \n\t+ \\beta (N_y-y) - \\xi y N_y , \\lbl{newnydot} \n\\end{eqnarray} \nwhich are not closed, since $x_4,y_4$ appear \non the {\\sc rhs}'s of (\\ref{newxdot}) and (\\ref{newydot}), \nhence we need to find formulae to determine $x_4$ and $y_4$ \nin terms of $x,y,N_x,N_y$.\n\nOne way of achieving this is to expand the system to include other \nproperties of the distribution of cluster sizes. For example, equations \ngoverning the mass of crystals in each chirality can be derived as \n\\begin{equation}\n\\frac{{\\rm d} \\varrho_x}{{\\rm d} t}=2\\mu c-2\\mu\\nu x+2\\alpha c N_x ,\n\\quad\n\\frac{{\\rm d} \\varrho_y}{{\\rm d} t}=2\\mu c-2\\mu\\nu y+2\\alpha c N_y . \n\\lbl{new-roxy-dot} \\end{equation}\nThese introduce no more new new quantities into the \nmacroscopic system of equations, and do not rely on knowing \n$x_4$ or $y_4$, (although they do require knowledge of $x$ and $y$). \n\nIn the remainder of this section we consider various potential formulae \nfor $x_4$, $y_4$ in terms of macroscopic quantities so that a \nmacroscopic system can be constructed. We then analyse such \nmacroscopic systems in two specific limits to show that predictions \nrelating to symmetry-breaking can be made. \n\n\\subsection{Reductions}\n\nThe equations governing the larger cluster sizes $x_k$, $y_k$, are \n\\begin{equation} \n\\frac{{\\rm d} x_{2k}}{{\\rm d} t} = \\beta( x_{2k+2} - x_{2k} ) \n- (x_{2k}-x_{2k-2})(\\alpha c + \\xi x) ; \n\\lbl{5xkdot} \\end{equation} \nin general this has solutions of the form $x_{2k} = \\sum_j A_j(t) \n\\Lambda_j^{k-1}$, \nwhere $\\Lambda_j$ are parameters (typically taking values between \nunity (corresponding to a steady-state in which mass is being \nadded to the distribution) and \n$\\mfrac{\\alpha c+\\xi x}{\\beta}$ (the equilibrium value); and $A_j(t)$ \nare time-dependent; for some $\\Lambda_j$, $A_j$ will be constant. \n\nWe assume that the distribution of each chirality of cluster is given by \n\\begin{equation} \nx_{2k} = x \\left( 1 - \\frac{1}{\\lambda_x} \\right)^{k-1} ,\\qquad\\qquad\ny_{2k} = y \\left( 1 - \\frac{1}{\\lambda_y} \\right)^{k-1} , \n\\end{equation}\nsince solutions of this form may be steady-states of the governing \nequations (\\ref{5xkdot}). However, in our approximations for \n$x_4$ and $y_4$ the parameters $\\lambda_x$, $\\lambda_y$ are \npermitted to vary with time in some way that depends on other \nquantities in the model equations. The resulting expressions for \nthe macroscopic number and mass quantities are \n\\begin{eqnarray}\nN_x = \\sum_{k=1}^\\infty x_{2k} = x \\lambda_x , &\\qquad& \nN_y = \\sum_{k=1}^\\infty y_{2k} = y \\lambda_y , \\\\ \n\\varrho_x = \\sum_{k=1}^\\infty 2 k x_{2k} = 2 x \\lambda_x^2 , &\\qquad& \n\\varrho_y = \\sum_{k=1}^\\infty 2 k y_{2k} = 2 y \\lambda_y^2 . \n\\end{eqnarray}\nOur aim is to find a simpler expression for the terms $x_4$ \nand $y_4$ which occur in (\\ref{newxdot})--(\\ref{newydot}), \nthese are given by $x_4=x(1-1\/\\lambda_x)$ where \n\\begin{equation}\n\\lambda_x = \\frac{N_x}{x} = \\frac{\\varrho_x}{2N_x} \n= \\sqrt{\\frac{\\varrho_x}{2x}} , \\lbl{lambda-eqs}\n\\end{equation}\nhence\n\\begin{equation}\nx_4 = x - \\frac{x^2}{N_x} , \\quad \nx_4 = x - \\frac{2 x N_x}{\\varrho_x} ,\\quad \n{\\rm or} \\;\\;\\; \nx_4 = x - x\\sqrt{\\frac{2x}{\\varrho_x}} . \n\\end{equation}\n\nThere are thus three possible reductions of the equations \n(\\ref{newcdot})--(\\ref{newnydot}), each eliminating one of \n$x,N_x,\\varrho_x$ (and the corresponding $y,N_y,\\varrho_y$). We consider \neach reduction in turn in the following subsections. Since some \nof these reductions involve $\\varrho_x, \\varrho_y$, we also use the \nevolution equations (\\ref{new-roxy-dot}) for these quantities. \n\n\\subsection{Reduction 1: to $x,y,N_x,N_y$}\n\nHere we assume $\\lambda_x = N_x\/x$, $\\lambda_y = N_y\/y$, \nso, in addition to (\\ref{newcdot}), (\\ref{newnxdot})--(\\ref{newnydot}) \nthe equations of motion are \n\\begin{eqnarray}\n\\frac{{\\rm d} x}{{\\rm d} t} & = & \\mu c - \\mu \\nu x + \\beta N_x \n\t- \\frac{\\beta x^2}{N_x} - \\xi x^2 - \\xi x N_x , \\\\\n\\frac{{\\rm d} y}{ {\\rm d} t} & = & \\mu c - \\mu \\nu y + \\beta N_y \n\t- \\frac{\\beta y^2}{N_y} - \\xi y^2 - \\xi y N_y ; \n\\end{eqnarray}\nwe have no need of the densities $\\varrho_x,\\varrho_y$ in this formulation. \n\nThe disadvantage of this reduction is that, due to \n(\\ref{lambda-eqs}), the total mass is given by \n\\begin{equation}\n\\varrho = 2c + \\varrho_x+\\varrho_y = 2 c + \\frac{2 N_x^2}{x} \n+ \\frac{2 N_y^2}{y} , \n\\end{equation}\nand there is no guarantee that this will be conserved. \n\nWe once again consider the system in terms of total concentrations \nand relative chiralities by applying the transformation \n\\begin{eqnarray}&\nx = \\mbox{$\\frac{1}{2}$} z (1\\!+\\!\\theta) , \\quad \ny = \\mbox{$\\frac{1}{2}$} z (1\\!-\\!\\theta) , \\quad \nN_x = \\mbox{$\\frac{1}{2}$} N (1\\!+\\!\\phi) , \\quad \nN_y = \\mbox{$\\frac{1}{2}$} N (1\\!-\\!\\phi) , & \\nonumber \\\\ && \n\\end{eqnarray}\nto obtain the equations\n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & =& - 2 \\mu c + \\mu \\nu z - \\alpha c N , \n\\lbl{R1cdot} \\\\ \n\\frac{{\\rm d} z}{{\\rm d} t} & =& 2\\mu c - \\mu \\nu z - \\alpha c z + \\beta N \n\t-\\frac{\\beta z^2(1+\\theta^2-2\\theta\\phi)}{N(1-\\phi^2)} \\nonumber \\\\ && \n\t- \\mbox{$\\frac{1}{2}$} \\xi z^2(1+\\theta^2) - \\mbox{$\\frac{1}{2}$} \\xi z N (1+\\theta\\phi) , \n\t\\lbl{R1zdot} \\\\\n\\frac{{\\rm d} N}{{\\rm d} t} & = & 2\\mu c - \\mu \\nu z + \\beta N - \\beta z \n\t- \\mbox{$\\frac{1}{2}$} \\xi z N (1+\\theta\\phi) . \\lbl{R1Ndot} \\\\ \n\\frac{{\\rm d} \\theta}{{\\rm d} t} & = & \n\t- \\left( \\mu \\nu + \\alpha c + \\xi z + \\mbox{$\\frac{1}{2}$} \\xi N \n\t\t+ \\frac{2\\beta z}{N(1\\!-\\!\\phi^2)} \n\t\t+ \\frac{1}{z} \\frac{{\\rm d} z}{{\\rm d} t} \\right) \\theta \\nonumber \\\\ && \n\t+ \\left( \\frac{\\beta N}{z} - \\mbox{$\\frac{1}{2}$} \\xi N \n\t+ \\frac{\\beta z (1\\!+\\!\\theta^2)}{N(1\\!-\\!\\phi^2)} \n\t\\right) \\phi , \\\\ \n\\frac{{\\rm d} \\phi}{{\\rm d} t} & =& \n\t- \\left( \\mu\\nu + \\beta + \\mbox{$\\frac{1}{2}$} \\xi N \\right) \\frac{z}{N} \\theta \n\t+ \\left( \\beta - \\mbox{$\\frac{1}{2}$} \\xi z - \\frac{1}{N}\\frac{{\\rm d} N}{{\\rm d} t} \\right) \\phi . \n\t\\nonumber \\\\ && \n\\end{eqnarray}\nThese equations have the symmetric steady-state given by \n$\\theta=0=\\phi$ and $c,z,N$ satisfying \n\\begin{equation}\nc = \\frac{\\mu\\nu z}{2\\mu+\\alpha N} , \\qquad\nz = \\frac{2\\beta N (2\\mu+\\alpha N) }\n{(2\\beta + \\xi N)(2\\mu+\\alpha N)+2\\alpha\\mu\\nu N} , \n\\lbl{R1sss} \\end{equation}\nfrom (\\ref{R1cdot}) and (\\ref{R1Ndot}). Note that the steady \nstate value of $N$ will depend upon the initial conditions, it is \nnot determined by (\\ref{R1zdot}). This is because the \nsteady-state equations obtained by setting the time derivatives \nin (\\ref{R1cdot})--(\\ref{R1Ndot}) are not independent. \nThe difference (\\ref{R1zdot})--(\\ref{R1Ndot}) is equal \nto $z\/N$ times the sum (\\ref{R1cdot})$+$(\\ref{R1Ndot}). \n\nIn subsections \\ref{5R1A1-sec} and \\ref{5R1A2-sec} below, \nso as to discuss the stability of a solution in the two asymptotic \nregimes $\\beta\\ll1$ and $\\alpha\\sim\\xi\\gg1$, we augment the \nsteady-state equations (\\ref{R1cdot})--(\\ref{R1Ndot}) with the \ncondition $\\varrho=2N^2\/z$, with $\\varrho$ assumed to be ${\\cal O}(1)$.\n\nThe linear stability of $\\theta=0=\\phi$ is given by assuming \n$\\theta$ and $\\phi$ are small, yielding the system \n\\begin{equation} \n\\frac{{\\rm d}}{{\\rm d} t} \\!\\!\\left(\\!\\! \\begin{array}{c} \n\\theta \\\\[2ex] \\phi \\end{array}\\!\\! \\right) \\!\n= \\! \\left( \\begin{array}{cc} \n- \\left( \\displaystyle\\frac{2\\mu c}{z} + \\displaystyle\\frac{\\xi z}{2} \n\t+ \\displaystyle\\frac{\\beta z}{N} \n\t+ \\displaystyle\\frac{\\beta N}{z} \\right) &\n\\left(\\displaystyle\\frac{\\beta N}{z} + \\displaystyle\\frac{\\beta z}{N} \n\t- \\displaystyle\\frac{\\xi N}{2} \\right) \\\\ \n- ( \\mu \\nu + \\beta + \\mbox{$\\frac{1}{2}$} \\xi N ) \\displaystyle\\frac{z}{N} & \n\\left( \\beta + \\mu\\nu - \\displaystyle\\frac{2\\mu c}{z} \\right) \n\\displaystyle\\frac{z}{N} \\end{array} \\right) \\!\\! \\left( \\!\\!\n\\begin{array}{c} \\theta \\\\[2ex] \\phi \\end{array} \\!\\!\\right)\\! . \n\\lbl{5stabmat} \\end{equation}\nAn instability of the symmetric solution is indicated by the \ndeterminant of this matrix being negative. Substituting (\\ref{R1sss}) \ninto the determinant, yields \n\\begin{equation}\n\\mbox{det} = \\frac{ \\beta \\mu \\nu ( 4 \\beta \\mu - \\alpha \\xi N^2 )}\n{4\\beta\\mu + 2 \\alpha \\beta N + 2 \\mu \\xi N + 2 \\alpha \\mu \\nu N \n+ \\alpha \\xi N^2} . \n\\lbl{5detsimp} \\end{equation}\nHence we find that the symmetric (racemic) state is unstable if \n$N > 2 \\sqrt{ \\mu\\beta \/ \\alpha \\xi }$, that is, large aggregation rates \n($\\alpha,\\xi$) and slow grinding ($\\beta$) are preferable for \nsymmetry-breaking. \n\nWe consider two specific asymptotic limits of parameter values \nso as to derive specific results for steady-states and conditions on \nstability. In both limits, we have that the aggregation rates dominate \nfragmentation ($\\alpha \\sim \\xi \\gg \\beta$), so that the system is \nstrongly biased towards the formation of crystals and the dimer \nconcentrations are small. In the first case we assume that the \nfragmentation is small and the aggregation rates are of a similar \nscale to the interconversion of dimers ($\\beta \\ll \\mu \\sim \\alpha \\sim \n\\xi = {\\cal O}(1)$); whilst the second has a fragmentation rate of \nsimilar size to the dimer conversion rates and larger aggregation rates \n($\\alpha \\sim \\xi \\gg \\mu \\sim \\beta = {\\cal O}(1)$). \n\n\\subsubsection{Asymptotic limit 1: $\\beta\\ll1$} \n\\label{5R1A1-sec} \n\nIn the case of asymptotic limit 1, $\\beta\\ll1$, we find the \nsteady-state solution \n\\begin{equation}\nN \\sim \\sqrt{\\frac{\\beta\\varrho}{\\xi+\\alpha\\nu}} , \\quad \nz \\sim \\frac{2\\beta}{\\xi+\\alpha\\nu} , \\quad \nc \\sim \\frac{\\beta\\nu}{\\xi+\\alpha\\nu} . \n\\end{equation}\nFrom (\\ref{5detsimp}), we find an instability if $\\varrho > \\varrho_c := \n4 \\mu (\\xi+\\alpha\\nu) \/ \\alpha\\xi$. That is, larger masses ($\\varrho$) \nfavour symmetry-breaking, as do larger aggregation rates \n($\\alpha,\\xi$). The eigenvalues of (\\ref{5stabmat}) in this limit are \n$q_1 = -\\mu\\nu$ -- a fast stable mode of the dynamics and \n\\begin{equation}\nq_2 = \\frac{\\alpha \\xi \\beta^{3\/2}}{2\\mu \\sqrt{\\varrho} (\\xi+\\alpha\\nu)^{3\/2}} \n\\left( \\varrho - \\frac{4\\mu(\\xi+\\alpha\\nu)}{\\alpha\\xi} \\right) , \n\\end{equation}\nwhich indicates a slowly growing instability when $\\varrho>\\varrho_c$. Hence \nthe balace of achiral to chiral morphologies of smaller clusters ($\\nu$) \nalso influences the propensity for non-racemic solution. However, \nsince the dynamics described by this model does not conserve total \nmass, the results from this should be treated with some caution, \nand we now analyse models which do conserve total mass. \n\n\\subsubsection{Asymptotic limit 2: $\\alpha\\sim\\xi\\gg1$}\n\\label{5R1A2-sec}\n\nIn this case we find the steady-state solution is given by \n\\begin{equation}\nN \\sim \\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \\quad \nz \\sim \\frac{2\\beta}{\\xi} , \\quad \nc \\sim \\frac{4\\mu\\nu}{\\alpha} \\sqrt{\\frac{\\beta}{\\xi\\varrho}} . \n\\end{equation}\nThe condition following from (\\ref{5detsimp}) then implies that we \nhave an instability if $\\varrho>\\varrho_c := 4\\mu\/\\alpha \\ll 1$. The eigenvalues \nof the stability matrix are $q_1 = - \\mbox{$\\frac{1}{2}$} \\sqrt{\\beta\\varrho\\xi}$, which is \nlarge and negative, indicating attraction to some lower dimensional \nsolution over a relatively fast timescale; the eigenvector being \n$(1,0)^T$ showing that $\\theta\\rightarrow0$. The other eigenvalue \nis $q_2 = 2\\mu\\nu \\sqrt{\\beta\/\\varrho\\xi} \\ll 1$, and corresponds to a slow \ngrowth of the chirality of the solution, since it relates to the \neigenvector $(0,1)^T$. Assuming the system is initiated near its \nsymmetric solution ($\\theta=\\phi=0$), this shows that the distribution \nof clusters changes its chirality first, whilst the dimer concentrations \nremain, at least to leading order, racemic. We expect that at a later \nstage the chirality of the dimers too will become nonzero. \n\n\\subsection{Reduction 2: to $x,y,\\varrho_x,\\varrho_y$}\n\nHere we eliminate $x_4=x(1-1\/\\lambda_x)$, \n$y_4=y(1-1\/\\lambda_y)$ together with $N_x$ and $N_y$ using \n\\begin{equation}\n\\lambda_x=\\sqrt{\\frac{\\varrho_x}{2x}}, \\quad \n\\lambda_y=\\sqrt{\\frac{\\varrho_y}{2y}}, \\quad \nN_x = \\sqrt{\\frac{x\\varrho_x}{2}}, \\quad \nN_y = \\sqrt{\\frac{y\\varrho_y}{2}}, \n\\end{equation}\nleaving a system of equations for $(c,x,y,\\varrho_x,\\varrho_y)$ \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & = & \\mu\\nu(x+y) - 2\\mu c \n- \\sqrt{2} \\alpha c \\left( \\sqrt{x\\varrho_x} + \\sqrt{y \\varrho_y} \\right) , \\\\ \n\\frac{{\\rm d} x}{{\\rm d} t} & =& \\mu c - \\mu \\nu x - \\alpha c x - \n\\xi x^2 - \\xi x \\sqrt{\\frac{x\\varrho_x}{2}} + \\beta \\sqrt{\\frac{x\\varrho_x}{2}} \n- \\beta x \\sqrt{\\frac{2x}{\\varrho_x}} , \\nonumber \\\\ && \\\\ \n\\frac{{\\rm d} \\varrho_x}{{\\rm d} t} & = & - 2 \\mu \\nu x + 2 \\mu c \n+ 2 \\alpha c \\sqrt{\\frac{x\\varrho_x}{2}} , \n\\end{eqnarray} \nwith similar equations for $y,\\varrho_y$. Transforming to total \nconcentrations and relative chiralities by way of \n\\begin{eqnarray}&\nx = \\mbox{$\\frac{1}{2}$} z (1+\\theta) , \\quad \ny = \\mbox{$\\frac{1}{2}$} z (1-\\theta) , \\quad \n\\varrho_x = \\mbox{$\\frac{1}{2}$} R (1+\\zeta) , \\quad \n\\varrho_y = \\mbox{$\\frac{1}{2}$} R (1-\\zeta) , \n&\\nonumber\\\\&&\n\\end{eqnarray}\nwe find \n\\begin{eqnarray}\n\\frac{{\\rm d} c}{{\\rm d} t} & =& \\mu \\nu z - 2 \\mu c \n\t- \\frac{\\alpha c \\sqrt{z R}}{2\\sqrt{2}} \\left[ \n\t\\sqrt{(1\\!+\\!\\theta)(1\\!+\\!\\zeta)} + \n\t\\sqrt{(1\\!-\\!\\theta)(1\\!-\\!\\zeta)} \\right] , \\lbl{new-r2-cdot} \n\\\\ \n\\frac{{\\rm d} z}{{\\rm d} t} & = & 2\\mu c - \\mu \\nu z - \\alpha c z \n\t- \\mbox{$\\frac{1}{2}$} \\xi z^2 (1\\!+\\!\\theta^2) \\nonumber \\\\ && \n\t+ \\frac{\\beta \\sqrt{zR}}{2\\sqrt{2}} \n\t\\left[ \\sqrt{(1\\!+\\!\\theta)(1\\!+\\!\\zeta)} + \\sqrt{(1\\!-\\!\\theta)(1\\!-\\!\\zeta)} \n\t\\right] \\nonumber \\\\ && \n\t- \\frac{\\xi z^{3\/2} R^{1\/2}}{4\\sqrt{2}} \\left[ \n\t(1\\!+\\!\\theta)^{3\/2} (1\\!+\\!\\zeta)^{1\/2} + (1\\!-\\!\\theta)^{3\/2} \n\t(1\\!-\\!\\zeta)^{1\/2} \\right] \\nonumber \\\\ && \n\t- \\frac{\\beta z^{3\/2} }{\\sqrt{2R}} \n\t\\left[ \\frac{(1\\!+\\!\\theta)^{3\/2}}{(1\\!+\\!\\zeta)^{1\/2}} + \n\t\\frac{(1\\!-\\!\\theta)^{3\/2}}{(1\\!-\\!\\zeta)^{1\/2}} \\right] , \n\\lbl{new-r2-zdot} \n\\\\ \n\\frac{{\\rm d} R}{{\\rm d} t} & = & - 2\\mu\\nu z + 4 \\mu c \n\t+ \\mbox{$\\frac{1}{2}$} \\alpha c \\sqrt{2zR} \\left[ \n\t\\sqrt{(1\\!+\\!\\theta)(1\\!+\\!\\zeta)} + \n\t\\sqrt{(1\\!-\\!\\theta)(1\\!-\\!\\zeta)} \\right] , \n\\nonumber \\\\ && \\lbl{new-r2-Rdot} \n\\end{eqnarray}\ntogether with the equations \n(\\ref{new-r2-thetadot})--(\\ref{new-r2-zetadot}) for the relative \nchiralities $\\theta$ and $\\zeta$, which will be analysed later. \n\nSince the equations for ${\\rm d} R\/dd t$ and ${\\rm d} c\/{\\rm d} t$ are \nessentially the same, we obtain a third piece of information from \nthe requirement that the total mass in the system is unchanged \nfrom the initial data, hence the new middle equation above. \nSolving these we find $c=\\mbox{$\\frac{1}{2}$} (\\varrho-R)$ and use this in place \nof the equation for $c$. \n\nIn the symmetric case ($\\theta=\\zeta=0$) we obtain the \nsteady-state conditions \n\\begin{eqnarray}\n0 & = & 2\\mu\\nu z - 4\\mu c - \\alpha c \\sqrt{2zR} , \\qquad\\qquad \n\\varrho \\; = \\; R + 2 c , \\lbl{R2-ssss1} \\\\ \n0 & = & 2\\mu c - \\mu \\nu z - \\alpha c z - \\mbox{$\\frac{1}{2}$} \\xi z^2 \n\t+ \\mbox{$\\frac{1}{2}$} \\beta \\sqrt{2zR} - \\beta z \\sqrt{\\frac{2z}{R}} \n\t- \\frac{\\xi z}{2} \\sqrt{\\frac{zR}{2}} . \\nonumber \\\\ && \\lbl{R2-ssss2} \n\\end{eqnarray}\nFor small $\\theta,\\zeta$, the equations for the chiralities \ncan be approximated by \n\\begin{eqnarray}\n\\frac{{\\rm d} \\theta}{{\\rm d} t} & = & - \\left( \\frac{2\\mu c}{z} \n\t+ \\mbox{$\\frac{1}{2}$} \\xi z + \\mbox{$\\frac{1}{2}$} \\beta \\sqrt{\\frac{R}{2z}} \n\t+ \\mbox{$\\frac{1}{2}$} \\beta \\sqrt{\\frac{2z}{R}} \n\t+ \\rec{4} \\xi \\sqrt{\\frac{zR}{2}} \\right) \\theta \\nonumber \\\\ && \n\t+ \\left( \\frac{\\beta(R+2z)}{2\\sqrt{2zR}} \n\t- \\frac{\\xi}{4} \\sqrt{\\frac{Rz}{2}} \\right) \\zeta , \n\\lbl{new-r2-thetadot} \\\\ \n\\frac{{\\rm d} \\zeta}{{\\rm d} t} & = & \\left( \\frac{2\\mu\\nu z}{R} \n- \\alpha c \\sqrt{\\frac{zR}{2}} \\right) \\theta \n- \\left( \\frac{2\\mu\\nu z}{R} - \\frac{4\\mu c}{R} \\right) \\zeta , \n\\lbl{new-r2-zetadot} \n\\end{eqnarray}\nWe analyse the stability of the symmetric (racemic) state in the two \nlimits $\\beta\\ll1$ and $\\alpha\\sim\\xi\\gg1$ in the next subsections. \n\n\\subsubsection{Asymptotic limit 1: $\\beta\\ll1$}\n\nIn this case, solving the conditions \n(\\ref{R2-ssss1})--(\\ref{R2-ssss2}) asymptotically, \nwe find \n\\begin{equation}\nz \\sim \\frac{2\\beta}{\\xi+\\alpha\\nu} , \\qquad \nc \\sim \\frac{\\beta\\nu}{\\xi+\\alpha\\nu} , \\qquad \nR \\sim \\varrho - 2c . \n\\end{equation}\nSubstituting these values into the differential equations \nwhich determine the stability of the racemic state leads to \n\\begin{equation}\n\\frac{{\\rm d} }{{\\rm d} t} \n\\left( \\begin{array}{c} \\theta \\\\[3ex] \\zeta \\end{array} \\right) \n\\left( \\begin{array}{cc} \n-\\mu\\nu & \n\\displaystyle\\frac{\\alpha\\nu}{4} \\sqrt{\\displaystyle\\frac{\\beta\\varrho}{\\xi+\\alpha\\nu}}\\\\ \n-\\displaystyle\\frac{4\\beta\\mu\\nu}{\\varrho(\\xi+\\alpha\\nu)} & \n\\displaystyle\\frac{\\alpha\\nu\\beta^{3\/2}}{(\\xi+\\alpha\\nu)^{3\/2} \\sqrt{\\varrho}} \n\\end{array} \\right) \n\\left( \\begin{array}{c} \\theta \\\\[3ex] \\zeta \\end{array} \\right) . \n\\end{equation}\nFormally this matrix has eigenvalues of zero and $-\\mu\\nu$. \nSince the zero eigenvalue indicates marginal stability of the \nracemic solution, we need to consider higher-order terms to \nobtain a more definite result. \n\nGoing to higher order, gives the determinant of the resulting matrix \nas $-\\alpha \\xi \\nu \/ (\\alpha\\nu+\\xi)^2$ hence the eigenvalues are \n\\begin{equation}\nq_1 = -\\mu\\nu , \\qquad {\\rm and} \\quad \nq_2 = \\frac{ \\alpha \\xi }{\\mu (\\alpha\\nu+\\xi)^2 } , \n\\end{equation}\nthe former indicating a rapid decay of $\\theta$ (corresponding to the \neigenvector $(1,0)^T$), and the latter showing a slow divergence from \nthe racemic state in the $\\zeta$-direction, at leading order, according to \n\\begin{equation}\n\\left( \\begin{array}{c} \\theta \\\\ \\zeta \\end{array} \\right) \n\\sim C_1 \\left( \\begin{array}{c} 0 \\\\ 1 \\end{array} \\right) \n\\exp \\left( \\frac{ \\alpha \\xi t }{\\mu (\\alpha\\nu+\\xi)^2 } \\right) . \n\\end{equation}\nHence in the case $\\beta\\ll1$, we find an instability of the \nsymmetric solution for all other parameter values. \n\n\\subsubsection{Asymptotic limit 2: $\\alpha\\sim\\xi\\gg1$}\n\nIn this case, solving the conditions \n(\\ref{R2-ssss1})--(\\ref{R2-ssss2}) asymptotically, we find \n\\begin{equation} \nz \\sim \\frac{2\\beta}{\\xi} , \\qquad \nc \\sim \\frac{2\\mu\\nu}{\\alpha} \\sqrt{\\frac{\\beta}{\\varrho\\xi}} , \\qquad \nR \\sim \\varrho - 2c . \n\\end{equation} \nSubstituting these values into the differential equations \n(\\ref{new-r2-thetadot})--(\\ref{new-r2-zetadot}) \nwhich determine the stability of the racemic state leads to \n\\begin{equation} \n\\frac{{\\rm d} }{{\\rm d} t} \n\\left( \\begin{array}{c} \\theta \\\\[1ex] \\zeta \\end{array} \\right) \n\\left( \\begin{array}{ccc} \n- \\mbox{$\\frac{1}{2}$} \\sqrt{\\beta\\xi\\varrho} && o(\\sqrt{\\xi}) \\\\[1ex] \n- \\displaystyle\\frac{4\\beta\\mu\\nu}{\\varrho\\xi} && \\displaystyle\\frac{4\\beta\\mu\\nu}{\\varrho\\xi} \n\\end{array} \\right) \n\\left( \\begin{array}{c} \\theta \\\\[1ex] \\zeta \\end{array} \\right) , \n\\end{equation} \nhence the eigenvalues are $q_1=-\\mbox{$\\frac{1}{2}$}\\sqrt{\\beta\\varrho\\xi}$ and \n$q_2 = 4\\mu\\nu\\beta\/\\varrho\\xi$, (in the above $o(\\sqrt{\\xi})$ means \na quantity $q$ satisfying $q\\ll\\sqrt{\\xi}$ as $\\xi\\rightarrow\\infty$). \nWhilst the former indicates the existence of a stable manifold (with \na fast rate of attraction), the latter shows that there is also an unstable \nmanifold. Although the timescale associated with this is much slower, \nit shows that the symmetric (racemic) state is unstable. \n\n\\subsection{Reduction 3: to $N_x,N_y,\\varrho_x,\\varrho_y$}\n\nIn this case our aim is to retain only information on the number \nand typical size of crystal distribution, so we eliminate the dimer \nconcentrations $x,y$, using \n\\begin{equation} \n\\lambda_x = \\frac{\\varrho_x}{2 N_x} , \\quad \n\\lambda_y = \\frac{\\varrho_y}{2 N_y} , \\quad \nx = \\frac{2 N_x^2}{\\varrho_x} , \\quad \ny = \\frac{2 N_y^2}{\\varrho_y} . \n\\end{equation} \nThese transformations reformulate the governing equations \n(\\ref{newcdot})--(\\ref{new-roxy-dot}) to \n\\begin{eqnarray}\n\\frac{{\\rm d} N_x}{{\\rm d} t} & = & \\mbox{$\\frac{1}{2}$} \\mu (\\varrho -R) + \\beta N_x \n\t- 2 (\\mu\\nu+\\beta) \\frac{N_x^2}{\\varrho_x} \n\t- \\frac{2\\xi N_x^3}{\\varrho_x} , \\lbl{r3-nxdot} \\\\ \n\\frac{{\\rm d} N_y}{{\\rm d} t} & = & \\mbox{$\\frac{1}{2}$} \\mu (\\varrho - R) + \\beta N_y\n\t- 2 (\\mu\\nu+\\beta) \\frac{N_y^2}{\\varrho_y} \n\t- \\frac{2\\xi N_y^3}{\\varrho_y} , \\\\ \n\\frac{{\\rm d} \\varrho_x}{{\\rm d} t} & = & (\\varrho-R)(\\mu+\\alpha N_x) \n\t- \\frac{4\\mu\\nu N_x^2}{\\varrho_x} , \\lbl{r3-roxdot} \\\\ \n\\frac{{\\rm d} \\varrho_y}{{\\rm d} t} & = & (\\varrho-R)(\\mu+\\alpha N_y) \n\t- \\frac{4\\mu\\nu N_y^2}{\\varrho_y} , \\lbl{r3-roydot}\n\\end{eqnarray}\nwhere $R := \\varrho_x + \\varrho_y$. \nWe now transform to total concentrations ($N$, $R$) \nand relative chiralities ($\\phi$ and $\\zeta$) {\\em via} \n\\begin{equation} \nN_x = \\mbox{$\\frac{1}{2}$} N (1+\\phi) , \\quad \nN_y = \\mbox{$\\frac{1}{2}$} N (1-\\phi) , \\quad \n\\varrho_x = \\mbox{$\\frac{1}{2}$} R (1+\\zeta) , \\quad \n\\varrho_y = \\mbox{$\\frac{1}{2}$} R (1-\\zeta) , \n\\end{equation}\ntogether with $c = \\mbox{$\\frac{1}{2}$} (\\varrho - R)$, to obtain \n\\begin{eqnarray}\n\\frac{{\\rm d} R}{{\\rm d} t} & = & (\\varrho-R)(2\\mu+ \\alpha N) \n- \\frac{4\\mu\\nu N^2(1+\\phi^2-2\\phi\\zeta)}{R (1-\\zeta^2)} , \n\\lbl{r3Rd} \\\\ \\lbl{r3Nd} \n\\frac{{\\rm d} N}{{\\rm d} t} & = & \\!\\!\\mu (\\varrho \\! - \\! R) + \\beta N \n\t\\\\ && \\! - \\frac{N^2}{R(1\\!-\\!\\zeta^2)} \\left[ \n\t2(\\mu\\nu\\!+\\!\\beta) (1\\!+\\!\\phi^2\\!-\\!2\\phi\\zeta) + \n\t\\xi N (1\\!+\\!3\\phi^2\\!-\\!3\\phi\\zeta\\!-\\!\\phi^3\\zeta) \\right] ,\n\\nonumber \\\\ \n\\frac{{\\rm d}\\phi}{{\\rm d} t} &=& \\beta\\phi - \\frac{1}{N}\\frac{{\\rm d} N}{{\\rm d} t}\\phi\n\t\\\\&& \\!\\!- \\frac{N}{R(1\\!-\\!\\zeta^2)} \\left[ \n\t2(\\beta\\!+\\!\\mu\\nu)(2\\phi\\!-\\!\\zeta\\!-\\!\\phi^2\\zeta)\n\t+ \\xi N (3\\phi\\!-\\!\\zeta\\!+\\!\\phi^3\\!-\\!3\\phi^2\\zeta) \\right] , \\nonumber \n\\\\ \n\\frac{{\\rm d} \\zeta}{{\\rm d} t} & =& \\frac{\\alpha (\\varrho-R) N \\phi}{R} \n\t- \\frac{1}{R}\\frac{{\\rm d} R}{{\\rm d} t} \\zeta - \\frac{4\\mu\\nu N^2 \n\t(2\\phi-\\zeta-\\phi^2\\zeta)}{R^2 (1-\\zeta^2)} . \n\\end{eqnarray}\nWe now analyse this system in more detail, since this set of \nequations conserves mass, and is easier to analyse than \n(\\ref{new-r2-cdot})--(\\ref{new-r2-Rdot}) due to the absence of \nsquare roots. We consider the two asymptotic limits ($\\beta\\ll1$ \nand $\\alpha\\sim\\xi\\gg1$) in which, at steady-state, the majority of \nmass is in the form of clusters. \n\n\\subsubsection{The symmetric steady-state}\n\nPutting $\\zeta=0=\\phi$, we find the symmetric steady-state is given by \n\\begin{eqnarray} \n0 &=& (\\varrho-R)(2\\mu+\\alpha N) - \\frac{4\\mu\\nu N^2}{R} , \n\\lbl{r3ssss1} \\\\ \n0 &=& \\mu (\\varrho-R) + \\beta N \n- 2(\\mu\\nu+\\beta)\\frac{N^2}{R} - \\frac{\\xi N^3}{R} . \n\\lbl{r3ssss2} \\end{eqnarray} \nthe former is solved by one of \n\\begin{equation}\nR = \\mbox{$\\frac{1}{2}$} \\varrho \\left( 1 \\pm \\sqrt{ 1 - \\frac{16\\mu\\nu N^2} \n{ (2\\mu+\\alpha N) \\varrho^2 } } \\right) , \\qquad \n\\end{equation}\n\\begin{equation}\nN = \\frac{\\alpha R(\\varrho-R)}{8\\mu\\nu} \\left( 1 + \n\\sqrt{1 + \\frac{32\\mu^2\\nu}{\\alpha^2 R(\\varrho-R)}} \\right) . \n\\end{equation}\nMore complete asymptotic solutions will be derived in Sections \n\\ref{r3-a1-sec} and \\ref{r3-a2-sec}. \n\n\\subsubsection{Stability of the symmetric state} \n\nWe now consider the stability of the \nsymmetric steady-state. For small $\\phi,\\zeta$ we have \n\\begin{eqnarray} \\!\\!\\!\\!\\!& \n\\displaystyle\\frac{R}{N} \\displaystyle\\frac{{\\rm d}}{{\\rm d} t} \\!\\!\n\\left( \\!\\!\\begin{array}{c} \\phi \\\\ \\\\ \\zeta \\end{array} \\!\\!\\right) \n\\!=\\!\\! \\left( \\!\\!\\begin{array}{cc} \n\\!\\! - \\! 2\\beta \\!-\\! 2\\mu\\nu \\!-\\! 2 \\xi N \n\t\\!-\\! \\displaystyle\\frac{\\mu (\\varrho\\!-\\!R) R}{N^2} \\!\\!&\\! \n2\\beta \\!+\\! 2\\mu\\nu \\!+\\! \\xi N \n\\\\ \n\\left( \\alpha (\\varrho\\!-\\!R) \n\t\\!-\\! \\displaystyle\\frac{8\\mu\\nu N}{R} \\right) \\! \\!&\\! \n\\!8\\mu\\nu \\!-\\! \\displaystyle\\frac{(\\varrho\\!\\!-\\!\\!R)(2\\mu\\!\\!+\\!\\!\\alpha N)R}{N^2} \\! \n\\end{array} \\!\\!\\right) \\!\\!\\!\n\\left(\\!\\! \\begin{array}{c} \\phi \\\\ \\\\ \\zeta \\end{array} \\!\\!\\right) \\!\\!,\\!\\! \n& \\nonumber \\\\ \\!\\!\\!\\!\\!\\! && \\!\\!\\!\\!\\lbl{r3-stab}\n\\end{eqnarray}\nand this is unstable if the determinant of this matrix is negative. \nNow we consider the two asymptotic limits in more detail. \n\n\\subsubsection{Asymptotic limit 1: $\\beta \\ll1$}\n\\label{r3-a1-sec}\n\nWhen fragmentation is slow, that is, $\\beta\\ll1$, at steady-state we \nhave $N={\\cal O}(\\sqrt{\\beta})$ and $R = \\varrho - {\\cal O}(\\beta)$. \nBalancing terms in (\\ref{r3ssss1})--(\\ref{r3ssss2}) we find the same \nleading order equation twice, namely $2\\nu N^2=\\beta\\varrho(\\varrho-R) $. \nTaking the difference of the two yields an independent equation \nfrom higher order terms, hence we obtain \n\\begin{equation} \nN \\sim \\sqrt{\\frac{\\beta \\varrho}{\\xi+\\alpha\\nu}} , \\qquad \nR \\sim \\varrho - \\frac{2\\nu\\beta}{\\xi+\\alpha\\nu} . \n\\end{equation} \nNote that this result implies that the dimer concentrations \nare small, with $c\\sim z$ and $c \\sim \\beta\\nu \/ (\\xi+\\alpha\\nu)$, \n$z\\sim 2\\beta\/(\\xi+\\alpha\\nu)$. \n\nSubstituting these expressions into those for the stability of the \nsymmetric steady-state (\\ref{r3-stab}), we find \n\\begin{equation}\n\\frac{R}{4\\mu\\nu N} \\frac{{\\rm d}}{{\\rm d} t} \n\\left( \\begin{array}{c} \\phi \\\\[1ex] \\zeta \\end{array} \\right) = \n\\left( \\begin{array}{cc} -1 & \\quad \\frac{1}{2} \\\\ \n-2\\sqrt{\\displaystyle\\frac{\\beta}{\\varrho(\\xi\\!+\\!\\alpha\\nu)}} & \\quad 1 \n\\end{array} \\right) \n\\left( \\begin{array}{c} \\phi \\\\[1ex] \\zeta \\end{array} \\right) . \n\\end{equation}\nThis matrix has one stable eigenvalue (corresponding to \n$(1,0)^T$ and hence the decay of $\\phi$ whilst $\\zeta$ remains \ninvariant), the unstable eigenvector is $(1,4)^T$, hence we find \n\\begin{equation}\n\\left( \\begin{array}{c} \\phi(t) \\\\ \\zeta(t) \\end{array} \\right) \\sim C \n\\left( \\begin{array}{c} 1 \\\\ 4 \\end{array} \\right) \\exp \\left( \n\\frac{4\\mu\\nu t \\sqrt{\\beta}}{\\sqrt{\\varrho(\\xi+\\alpha\\nu)}} \\right) . \n\\lbl{r3-chir-rate} \\end{equation} \nIf we compare the timescale of this solution to that over which the \nconcentrations $N,R$ vary, we find that symmetry-breaking \noccurs on a slower timescale than the evolution of cluster masses \nand numbers. This is illustrated in the numerical simulation of \nequations (\\ref{r3-nxdot})--(\\ref{r3-roydot}) shown in Figure \n\\ref{fig-r3alpha}. More specifically, the time-scale increases with \nthe mass in the system, and with the ratio of aggregation to \nfragmentation rates, $(\\alpha\\nu+\\xi)\/\\beta$, and is inversely related \nto the chiral switching rate of small clusters ($\\mu\\nu$). \n\n\\begin{figure}[!ht]\n\\vspace*{68mm}\n\\special{psfile=fig-r3beta.eps \n\thscale=80 vscale=60 hoffset=-70 voffset=-150}\n\\caption{Graph of concentrations $N_x,N_y,\\varrho_x,\\varrho_y,c$ \nagainst time on a logarithmic time for the asymptotic limit 1, \nwith initial conditions $N_x=0.2=N_y$, $\\varrho_x=0.45$, $\\varrho_y=0.44$, \nother parameters given by $\\alpha=1=\\xi=\\mu$, $\\beta=0.01$ , \n$\\varrho=8$. Since model equations are in nondimensional form, \nthe time units are arbitrary. }\n\\label{fig-r3alpha}\n\\end{figure}\n\n\\subsubsection{Asymptotic limit 2: $\\alpha \\sim \\xi \\gg 1$}\n\\label{r3-a2-sec}\n\nIn this case we retain the assumptions that $\\mu,\\nu={\\cal O}(1)$, \nhowever, we now impose $\\beta={\\cal O}(1)$ and \n$\\alpha \\sim \\xi \\gg1$. For a steady-state, we require the scalings \n$N ={\\cal O}(1\/\\sqrt{\\xi})$ and $\\varrho-R={\\cal O}(1\/\\xi^{3\/2})$. \nSpecifically, solving (\\ref{r3ssss1})--(\\ref{r3ssss2}) we find \n\\begin{equation}\nN \\sim \\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \\qquad \nR \\sim \\varrho - \\frac{4\\mu\\nu}{\\alpha\\varrho} \\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \n\\lbl{r3a2-sss} \\end{equation}\nhence the dimer concentrations $c = \\mbox{$\\frac{1}{2}$} (\\varrho-R) \\sim N^3 = \n{\\cal O}(1\/\\xi^{3\/2})$ and $z = 2 N^2\/\\varrho \\sim N^2 = {\\cal O}(1\/\\xi)$. \nMore precisely, $c\\sim (2\\mu\\nu\/\\alpha)\\sqrt{\\beta\/\\varrho\\xi}$ and \n$z\\sim 2\\beta\/\\xi$, in contrast with the previous asymptotic scaling \nwhich gave $z\\sim N^2$). \n\nTo determine the timescales for crystal growth and dissolution, \nwe use (\\ref{r3a2-sss}) to define\n\\begin{equation} \nN \\sim n(t) \\sqrt{\\beta \\varrho\/\\xi} , \\quad \nR \\sim \\varrho - \\frac{4\\mu\\nu r(t)}{\\alpha \\varrho} \n\\sqrt{\\frac{\\beta\\varrho}{\\xi}} , \n\\end{equation} \nand so rewrite the governing equations (\\ref{r3Rd})--(\\ref{r3Nd}) as \n\\begin{eqnarray}\n\\frac{{\\rm d} n}{{\\rm d} t} & = & \\beta n \\left( 1 - n^2 - \n\\frac{2 n (\\beta+\\mu\\nu)}{\\sqrt{\\varrho\\xi\\beta}} \\right) , \\\\ \n\\frac{{\\rm d} r}{{\\rm d} t} & = & \\alpha \\sqrt{\\frac{\\beta\\varrho}{\\xi}} \n\\left( n^2 -r - \\frac{2\\mu r}{\\alpha} \n\\sqrt{\\frac{\\xi}{\\beta\\varrho}} \\right) . \n\\end{eqnarray}\nHere, the former equation for $n(t)$ corresponds to the \nslower timescale, with a rate $\\beta$, the rate of \nequilibration of $r(t)$ being $\\alpha \\sqrt{\\beta\\varrho\/\\xi}$. \n\nThe stability of the symmetric state is determined by \n\\begin{equation}\n\\frac{R}{N} \\frac{{\\rm d} }{{\\rm d} t} \n\\left( \\begin{array}{c} \\phi(t) \\\\ \\zeta(t) \\end{array} \\right) = \n\\left( \\begin{array}{cc} -2 \\sqrt{\\beta\\varrho\\xi} & \\sqrt{\\beta\\varrho\\xi} \\\\ \n-4\\mu\\nu \\sqrt{\\beta \/ \\xi \\varrho} & 4\\mu\\nu \\end{array} \\right) \n\\left( \\begin{array}{c} \\phi \\\\ \\zeta \\end{array} \\right) . \n\\lbl{r3a2-phi-zeta-sys} \\end{equation} \nThis matrix has one large negative eigenvalue ($\\sim -2\\sqrt{\\beta\\varrho\\xi}$) \nand one (smaller) positive eigenvalue ($\\sim 4\\mu\\nu$); the former \ncorresponds to $(1,0)^T$ hence the decay of $\\phi$, whilst the latter \ncorresponds to the eigenvector $(1,2)^T$. Hence the system \n(\\ref{r3a2-phi-zeta-sys}) has the solution \n\\begin{equation} \n\\left( \\begin{array}{c} \\phi \\\\ \\zeta \\end{array} \\right) \\sim \nC \\left( \\begin{array}{c} 1 \\\\ 2 \\end{array} \\right) \n\\exp \\left( 4 \\mu \\nu t \\sqrt{ \\frac{\\beta}{\\varrho\\xi}} \\right) . \n\\lbl{r3a2urate} \\end{equation} \nThe chiralities evolve on two timescales, the faster being \n$2\\beta$ corresponding to the stable eigenvalue of \n(\\ref{r3a2-phi-zeta-sys}) and the slower unstable rate \nbeing $4\\mu\\nu\\sqrt{\\beta\/\\xi\\varrho}$. This timescale is \nsimilar to (\\ref{r3-chir-rate}), being dependent on mass and \nthe ratio of aggregation to fragmentation, and inversely \nproportional to the chiral switching rate of dimers ($\\mu\\nu$). \n\n\\begin{figure}[!ht]\n\\vspace*{68mm}\n\\special{psfile=fig-r3alpha.eps \n\thscale=80 vscale=60 hoffset=-70 voffset=-150}\n\\caption{Graph of the concentrations $N_x,N_y,\\varrho_x,\\varrho_y,c$ \nagainst time on a logarithmic time for the asymptotic limit 2, \nwith initial conditions $N_x=0.2=N_y$, $\\varrho_x=0.45$, $\\varrho_y=0.44$, \nother parameters given by $\\alpha=10=\\xi$, $\\beta=1=\\mu$, \n$\\nu=0.5$, $\\varrho=2$. Since model equations \nare in nondimensional form, the time units are arbitrary. }\n\\label{fig-r3beta}\n\\end{figure}\n\n\\subsection{The asymmetric steady-state}\n\nSince the symmetric state can be unstable, there must be some \nother large-time asymmetric attractor(s) for the system, \nwhich we now aim to find. From (\\ref{r3-nxdot}) and \n(\\ref{r3-roxdot}), at steady-state, we have \n\\begin{equation}\n2c_2 (2\\mu+\\alpha N_x) = \\frac{4\\mu\\nu N_x^2}{\\varrho_x} , \n\\qquad \\mu c_2 + \\beta N_x = \n2 (\\mu\\nu+\\beta+\\xi N_x) \\frac{N_x^2}{\\varrho_x} . \n\\lbl{r3a-eqs} \\end{equation} \nTaking the ratio of these we find a single quadratic \nequation for $N_x$\n\\begin{equation}\n0 = \\alpha \\xi N_x^2 - \\left( \\frac{\\beta\\mu\\nu}{c_2} \n- \\alpha\\beta - \\alpha\\mu\\nu - \\xi\\mu \\right) N_x + \\beta\\mu , \n\\lbl{r3a-Neq}\n\\end{equation}\nwith an identical one for $N_y$. Hence there is the possibility \nof distinct solutions for $N_x$ and $N_y$ if both roots of \n(\\ref{r3a-Neq}) are positive; this occurs if \n\\begin{equation}\nc_2 < \\frac{\\beta\\mu\\nu}{\\alpha\\beta + \\xi\\mu \n+ \\alpha\\mu\\nu + 2\\sqrt{\\alpha\\beta\\xi\\mu} } . \n\\lbl{r3a-ineq} \\end{equation}\nGiven $N_x$ ($N_y$), we then have to solve one of \n(\\ref{r3a-eqs}) to find $\\varrho_x$ ($\\varrho_y$), {\\em via} \n\\begin{equation}\n\\varrho_x = \\frac{2 \\mu \\nu N_x^2}{c_2 (\\mu+\\alpha N_x)} , \n\\lbl{r3a-a1-rox} \\end{equation} \nand then satisfy the consistency condition that \n$\\varrho_x + \\varrho_y + 2 c_2 = \\varrho$. After some algebra, \nthis condition reduces to \n\\begin{eqnarray}\n\\mbox{$\\frac{1}{2}$} \\alpha^2 \\xi c_2^2 (\\beta \\!-\\! \\alpha c_2 ) (\\varrho\\!-\\!2c_2) \n&\\!=\\!& \\beta^2\\mu^2\\nu^2 - \\beta\\mu\\nu c_2 [ \\alpha\\beta \n+ 2\\alpha\\mu\\nu + 2\\xi\\mu ] \\nonumber \\\\ && \n+ \\mu c_2^2 [ \\mu (\\alpha\\nu\\!+\\!\\xi)^2 + \\alpha\\beta (\\alpha\\nu\\!-\\!\\xi) ] . \n\\lbl{r3a-consistency} \\end{eqnarray}\nBeing a cubic, it is not straightforward to write down explicit \nsolutions of this equation, hence we once again consider the two \nasymptotic limits ($\\beta\\ll1$ and $\\alpha\\sim\\xi\\gg1$). \n\n\\subsubsection{Asymptotic limit 1: $\\beta \\ll 1$}\n\nIn this case, $c_2 = {\\cal O}(\\beta)$ hence we put $c_2=\\beta C$ \nand the consistency condition (\\ref{r3a-consistency}) yields \n\\begin{equation} \n{\\cal O}(\\beta^3) = \\beta^2 \\left[ \\nu - (\\alpha\\nu+\\xi) C \\right]^2 , \n\\lbl{r3a-a1-C} \\end{equation} \nhence, to leading order, $C=\\nu\/(\\alpha\\nu+\\xi)$ . Unfortunately, the \nresulting value for $c_2$ leads to all the leading order terms in the \nlinear equation (\\ref{r3a-Neq}) for $N_x$ to cancel. We thus have to \nfind higher order terms in the expansion for $c_2$; due to the form of \n(\\ref{r3a-a1-C}), the next correction term is ${\\cal O}(\\beta^{3\/2})$. \nPutting $c_2=\\beta C(1+\\tilde C \\sqrt{\\beta})$, we find \n\\begin{equation} \n\\tilde C^2 = \\frac{\\alpha\\xi \\,\\left[ \\, \\alpha\\xi\\varrho + 4 \\mu (\\alpha\\nu+\\xi) \n\\, \\right] }{2\\mu^2 (\\alpha\\nu+\\xi)^3} . \n\\end{equation} \nIn order to satisfy the inequality (\\ref{r3a-ineq}), we require the \nnegative root, that is, $\\tilde C<0$. \n\nAlthough the formulae for $N_x,N_y$ are lengthy, \ntheir sum and products simplify to \n\\begin{equation}\n\\Sigma = N_x + N_y = \n\\frac{\\mu \\tilde C \\sqrt{\\beta} (\\alpha\\nu+\\xi)}{\\alpha\\xi} , \\qquad \n\\Pi = N_x N_y = \\frac{\\beta\\mu}{\\alpha\\xi} . \n\\end{equation}\nThe chirality $\\phi$ can be simplified using $\\phi^2=1-4\\Pi\/\\Sigma^2$ \nwhich implies \n\\begin{equation}\n\\phi^2 = \\frac{\\alpha\\varrho \\xi - 4\\mu(\\alpha\\nu+\\xi)}\n{\\alpha\\varrho\\xi+4\\mu (\\alpha\\nu+\\xi)} . \n\\end{equation}\nHence we require $\\varrho > \\varrho_c := 4\\mu(\\alpha\\nu+\\xi)\/\\alpha\\xi$ in \norder for the system to have nonsymmetric steady-states, that is, the \nsystem undergoes a symmetry-breaking bifurcation as $\\varrho$ increases \nthrough $\\varrho=\\varrho_c$. As the mass in the system increases further, \nthe chirality $\\phi$ approaches ($\\pm$) unity, indicating a state in \nwhich one handedness of crystal completely dominates the other. \n\n\\subsubsection{Asymptotic limit 2: $\\alpha \\sim \\xi \\gg 1$}\n\nIn this case, the left-hand side of the consistency condition \n(\\ref{r3a-consistency}) is ${\\cal O}(\\alpha^2\\xi c_2^2)$ whilst the \nright-hand side is ${\\cal O}(1)+{\\cal O}(\\alpha c_2^2)$, which implies \nthe balance $c_2={\\cal O}(\\xi^{-3\/2})$. Solving for $c_2$ leads to \n\\begin{equation} \nc_2 \\sim \\frac{\\mu\\nu}{\\alpha} \\sqrt{ \\frac{2\\beta}{\\varrho\\xi} } . \n\\end{equation} \nThe leading order equation for $N_x,N_y$ is then \n\\begin{equation} \n0 = \\alpha\\xi N^2 - \\alpha N \\sqrt{\\mbox{$\\frac{1}{2}$}\\beta\\varrho\\xi} + \\beta\\mu , \n\\end{equation} \nhence we find the roots \n\\begin{equation} \nN_x,N_y \\sim \\sqrt{\\frac{\\beta\\varrho}{2\\xi}} , \n\\frac{2\\mu}{\\alpha} \\sqrt{\\frac{\\beta}{2\\xi\\varrho}} , \\qquad \n\\varrho_x , \\varrho_y \\sim \\varrho , \\frac{2\\mu}{\\alpha} . \n\\end{equation} \nSince we have either $\\varrho_x \\gg N_x \\gg \\varrho_y \\gg N_y$ \nor $\\varrho_y \\gg N_y \\gg \\varrho_x \\gg N_x$, in this asymptotic limit, \nthe system is completely dominated by one species or the other. \nPutting $\\Sigma=N_x+N_y$ and $\\Pi=N_xN_y$ we have \n$\\phi^2=1-4\\Pi\/\\Sigma^2 \\sim 1 - 8 \\mu\/\\alpha\\varrho$. \n\n\\section{Discussion}\n\\label{disc-sec}\n\\setcounter{equation}{0}\n\nWe now try to use the above theory and experimental \nresults of Viedma \\cite{viedma} to estimate the relevant \ntimescales for symmetry-breaking in a prebiotic world. \nExtrapolating the data of time against grinding rate in rpm \nfrom Figure 2 of Viedma \\cite{viedma} \nsuggests times of $2\\times10^5$ hours using a straight line \nfit to log(time) against log(rpm) or 1000--3000 hours if \nlog(time) against rpm or time against log(rpm) is fitted. \nA reduction in the speed of grinding in prebiotic circumstances \nis expected since natural processes such as water waves \nare much more likely to operate at the order of a few seconds$^{-1}$ \nor minutes$^{-1}$ rather than 600 rpm. \n\nSimilar extrapolations on the number and mass of balls \nused to much lower amounts gives a further reduction \nof about 3, using a linear fit to log(time) against mass of balls \nfrom Figure 1 of Viedma \\cite{viedma}. There is an \nequally good straight line fit to time against log(ball-mass) \nbut it is then difficult to know how small a mass of balls \nwould be appropriate in the prebiotic scenario. \nThere is an additional factor due to the experiments \nof Viedma being on a small volume of 10 ml, whereas a \nsensible volume for prebiotic chemistry is 1000 l, \ngiving an additional factor of $10^5$. \nCombining these three factors ($10^3$, 3, and $10^5$) with \nthe 10 days of the original experiment, we estimate that the \ntimescale for prebiotic symmetry breaking is ${\\cal O}(3\\times10^9)$ \ndays, which is equivalent to the order of about ten million years. \n\nThis extrapolation ignores the time required to arrive \nat the initial enantiomeric excesses of 5\\% used by Viedma \n\\cite{viedma} from a small asymmetry caused by \neither a random fluctuation or by the parity-violation. \nAlthough the observed chiral structures are the minimum energy \nconfigurations as predicted by parity violation, there is an evens \nprobability that the observed handedness could simply be the result \nof a random fluctuation which was amplified by the same mechanisms. \nIn order to perform an example calculation, we take a random \nfluctuation of the size predicted by parity violation, which is \nof the order of $10^{-17}$, as suggested by Kondepudi \\& Nelson \n\\cite{kon-pla}. Our goal is now to find the time taken to \namplify this to an ${\\cal O}(1)$ (5\\%) enantiomeric excess. \n\nThe models derived in this paper, for example in Section \n\\ref{r3-a2-sec}, predict that the chiral excess grows \nexponentially in time. Assuming, from (\\ref{r3a2urate}), \nthat $\\phi(t_0)=10^{-17}$ and $\\phi(t_1)= 0.1$, then \nthe timescale for the growth of this small perturbation is \n\\[ t_1 - t_0 = \\frac{1}{4\\mu\\nu} \\sqrt{\\frac{\\xi\\varrho}{\\beta}} \n\\log \\frac{10^{-1}}{10^{-17}} . \\] \nSince the growth of enantiomeric excess is exponential, \nit only takes 16 times as long for the perturbation to grow \nfrom $10^{-17}$ to $10^{-1}$ as from $10^{-1}$ to 1. \nHence we only need to increase our estimate of the timescale \nby one power of ten, to 100 million years. \n\nThis estimate should be taken as a very rough estimate, since it \nrelies on extrapolating results by many orders of magnitude. \nAlso, given the vast differences in temperature from the \nputative subzero prebiotic world to a tentative hot hydrothermal \nvent, there could easily be changes in timescale by a factor of \nseveral orders of magnitude. \n\n\\section{Conclusions}\n\\label{conc-sec}\n\\setcounter{equation}{0}\n\nAfter summarising the existing models of chiral symmetry-breaking \nprocesses we have systematically derived a model in which through \naggregation and fragmentation chiral clusters compete for achiral \nmaterial. The model is closed, in that there is no input of mass \ninto the system, although the form of the aggregation and \nfragmentation rate coefficients mean that there is an input of energy, \nkeeping the system away from equilibrium. Furthermore, there is no \ndirect interaction of clusters of opposite handedness; rather \njust through a simple competition for achiral substrate, the system \ncan spontaneously undergo chiral symmetry-breaking. This model \nhelps explain the experimental results of Viedma \\cite{viedma} \nand Noorduin {\\em et al.}\\ \\cite{wim}. \n\nThe microscopic model originally derived has been simplified \nsuccessively to a minimalistic model, which, numerical results show, \nexhibits symmetry-breaking. Even after this reduction, the model is \nextremely complex to analyse due to the large number of cluster \nsizes retained in the model. Hence we construct two truncated \nmodels, one truncated at tetramers, which shows no \nsymmetry-breaking and one at hexamers which shows \nsymmetry-breaking under certain conditions on the parameter values. \nAlternative reductions are proposed: instead of retaining the \nconcentrations of just a few cluster sizes, we retain information \nabout the shape of the distribution, such as the number of clusters \nand the total mass of material in clusters of each handedness. \nThese reduced models are as simple to analyse as truncated models \nyet, since they more accurately account for the shape of the \nsize-distribution than a truncated model, are expected to give models \nwhich more easily fit to experimental data. Of course, other \nansatzes for the shape of the size distributions could be made, \nand will lead to modified conditions for symmetry-breaking; \nhowever, we believe that the qualitative results outlined here will \nnot be contradicted by analyses of other macroscopic reductions. \n\nOne noteworthy feature of the results shown herein is that the \nsymmetry-breaking is inherently a product of the two handednesses \ncompeting for achiral material. The symmetry-breaking does not rely \non critical cluster sizes, which are a common feature of theories of \ncrystallisation, or on complicated arguments about surface area to \nvolume ratios to make the symmetric state unstable. We do not \ndeny that these aspects of crystallisation are genuine, these features \nare present in the phenomena of crystal growth, but they are not \nthe fundamental cause of chiral symmetry-breaking. \n\nMore accurate fitting of the models to experimental data could be \nacheived if one were to fit the generalised Becker-D\\\"{o}ring model \n(\\ref{gbd1})--(\\ref{gbd3}) with realistic rate coefficients. Questions \nto address include elucidating how the number and size distribution \nat the start of the grinding influences the end state. For example, if \none were to start with a few large right-handed crystals and many \nsmall left-handed crystals, would the system convert to entirely \nleft- or entirely right-handed crystals ? Answers to these more \ncomplex questions may rely on higher moments of the size distributions, \nsurface area to volume ratios and critical cluster nuclei sizes. \n\n\\subsection*{Acknowledgments}\n\nI would particularly like to thank Professors Axel Brandenburg and \nRaphael Plasson for inviting me to an extended programme of \nstudy on homochirality at Nordita (Stockholm, Sweden) in February 2008. \nThere I met and benefited greatly from discussions with Professors \nMeir Lahav, Mike McBride, Wim Noorduin, as well as many others. \nThe models described here are a product of the stimulating \ndiscussions held there. I am also grateful for funding under \nEPSRC springboard fellowship EP\/E032362\/1. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\\label{sec:intro}\n\n\\emph{Seriation} is an important ordering problem\nwhose aim is to find the best enumeration order of a set of units, according\nto a given correlation function. The desired order can be characteristic of the\ndata, a chronological order, a gradient or any sequential structure of the\ndata. \n\nThe concept of seriation has been formulated in many different ways and appears\nin various fields, such as archaeology, anthropology, psychology, and \nbiology~\\cite{brusco06,genome,matharcheo,mirkin1984}.\nIn this paper we use the archaeological setting as a metaphor for the seriation\nproblem.\nAn important aim of archaeological investigation is to date excavation\nsites on the basis of found objects and determine their relative chronology,\ni.e., a dating which indicates if a given site is chronologically preceding or\nsubsequent to another.\nIn general, relative chronologies are devoid of a direction, in the sense that\nthe units are placed in a sequence which can be read in both directions.\nRelative dating methods can be used where absolute dating methods, such as\ncarbon dating, cannot be applied.\n\nThe available data are usually represented by a \\emph{data matrix}, in which\nthe rows are the archaeological units (e.g., the sites) and the columns\nrepresent the types (the archaeological finds).\nEach unit is characterized by the presence of certain artefacts, which are in\nturn classified in types. In~\\cite{ps05}, the authors refer to the data matrix as either \\emph{incidence matrix} or \\emph{abundance\nmatrix}, depending on the archaeological data representation. In the first case, the data are reported by using a binary\nrepresentation, i.e., an element in the position $(i,j)$ is equal to $1$ if\ntype $j$ is present in the unit $i$, and $0$ otherwise.\nIn the second second case, the data matrix reports the number of objects\nbelonging to a certain type in a given unit, or its percentage. In this paper,\nwe will follow the usual terminology used in \\emph{complex networks theory} and\nwe will refer to a binary representation as an \\emph{adjacency matrix}, an\nexample of which is given in Table~\\ref{tab:adjacency}. More details can be\nfound in Section~\\ref{sec:mathback}. If\nthe data matrix represents types of found objects as columns and the locations\n(graves, pits, etc.) in which they are found as rows, we can find a\nchronological order for the locations by assuming that the types were produced,\nor were ``fashionable'', only for a limited period of time. In the light of\nthis assumption, the purpose of determining a relative chronology results in\nobtaining an ordering of the rows and columns of the data matrix that places\nthe nonzero entries close to the diagonal of the data matrix.\n\n\\begin{table}\n\\caption{Adjacency matrix for archaeological data originated from female\nburials at the Bornholm site, Germany; see~\\cite{ps05} and the references\ntherein. The rows report the names of the tombs, the columns the identification\ncodes of the found \\emph{fibulae}.}\n\\label{tab:adjacency}\n\\scriptsize\n\\begin{center}\n\\begin{tabular}{rcccccccccccc}\n\\hline\\noalign{\\smallskip}\n&G3&F27&S1&F26&N2&F24&P6&F25&P5&P4&N1&F23\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\textsf{Mollebakken 2} & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\\textsf{Kobbea 11}&0&1&1&0&1&1&0&0&0&0&0&0\\\\ \n\\textsf{Mollebakken 1}&1&1&0&1&1&0&1&1&0&0&0&0\\\\ \n\\textsf{Levka 2}&0&1&1&0&1&0&0&1&1&0&0&0\\\\ \n\\textsf{Grodbygard 324}&0&0&0&0&1&1&0&0&0&1&0&0\\\\ \n\\textsf{Melsted 8}&0&0&1&1&0&0&1&1&0&1&0&0\\\\ \n\\textsf{Bokul 7}&0&0&0&0&0&0&1&1&0&0&1&0\\\\ \n\\textsf{Heslergaard 11}&0&0&0&0&0&0&0&1&0&1&0&0\\\\ \n\\textsf{Bokul 12}&0&0&0&0&0&0&0&1&1&0&0&1\\\\ \n\\textsf{Slamrebjerg 142}&0&0&0&0&0&0&0&0&0&1&0&1\\\\ \n\\textsf{Nexo 6} &0&0&0&0&0&0&0&0&0&1&1&1\\\\ \n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nA closely related problem is the \\emph{consecutive ones problem}\n(C1P)~\\cite{fulkerson65,or09}, whose aim is to find all the permutations of the\nrows of a binary matrix that place the $1$'s consecutively in each column. If\nsuch permutations exist, then the matrix is said to have the \\emph{consecutive\nones property} for columns. The equivalent property for rows can be similarly\ndefined. The problem of verifying if a matrix possesses this property has\napplications in different fields, such as computational biology and recognition\nof interval graphs~\\cite{booth1976testing,cor98}.\nThe connection between C1P and seriation has been investigated by Kendall \nin~\\cite{kendall69}.\n\nThe first systematic formalization of the seriation problem was made by Petrie\nin 1899~\\cite{petrie}, even if the term seriation was used before in\narchaeology. The subject was later resumed by Breinerd and\nRobinson~\\cite{brainerd1951place,robinson1951method}, who also proposed a\npractical method for its solution, and by\nKendall~\\cite{kendall63,kendall69,kendall70}.\nNice reviews on seriation are given in~\\cite{liiv2010},~\\cite{ps05}, \nand~\\cite{bl02}, where its application in stratigraphy is discussed, \nwhile~\\cite{bb15,matharcheo} describe other applications of mathematics in\narchaeology.\n\nGiven the variety of applications, some software packages have been developed in\nthe past to manipulate seriation data. Some of these packages have not undergo a\nregular maintenance, and does not seem to be easily usable on modern computers,\nlike the Bonn Archaeological Software Package (BASP)\n(\\url{http:\/\/www.uni-koeln.de\/~al001\/}).\nA software specifically designed for the seriation problem in bioinformatics\nhas been developed by Caraux and Pinloche~\\cite{permutsoft}, and is available\nfor free download.\n\nOther implementations of the spectral algorithm from~\\cite{atkins1998spectral}\nhave been discussed in~\\cite{fogel2014}, \\cite{hahsler2008}, which describes an\nR package available at \\url{http:\/\/cran.r-project.org\/web\/packages\/seriation\/},\nand~\\cite{seminaroti2016}.\nThe paper~\\cite{fogel2015} proposes an interesting method, based on quadratic\nprogramming, aimed at treating ``noisy cases''.\n\nIn this paper we present a Matlab implementation of a spectral method for the\nsolution of the seriation problem which appeared in~\\cite{atkins1998spectral},\nbased on the use of the Fiedler vector of the Laplacian associated to the\nproblem, and which describes the results in terms of a particular data\nstructure called a PQ-tree.\nWe further develop some numerical aspects of the algorithm, concerning the\ndetection of equal components in the Fiedler vector and the computation of the\neigensystem of the Laplacian associated to a large scale problem.\nWe also provide a parallel version of the method.\nThe package, named the \\texttt{PQser} toolbox, also defines a data structure to store\na PQ-tree and provides the Matlab functions to manipulate and visualize it.\nFinally, we discuss the implications of the presence of a multiple Fiedler\nvalue, an issue which has been disregarded up to now, and we illustrate some\nnumerical experiments.\n\nThe plan of the paper is the following. Section \\ref{sec:mathback} reviews the\nnecessary mathematical background and sets up the terminology to be used in the\nrest of the paper. Section~\\ref{sec:pqtrees} describes the data structures\nused to store the solutions of the seriation problem.\nThe spectral algorithm is discussed in Section~\\ref{sec:seriation} and the\nspecial case of a multiple Fiedler values is analyzed in\nSection~\\ref{sec:mult}.\nSection~\\ref{sec:numexp} reports some numerical results and\nSection~\\ref{sec:last} contains concluding remarks. \n\n\n\\section{Mathematical background}\\label{sec:mathback}\n\nWith the aim of making this paper self-contained, we review some mathematical\nconcepts that will be used in the following.\nWe denote matrices by upper case roman letters and their elements by\nlower case double indexed letters.\n\nLet $G$ be a simple graph formed by $n$ nodes.\nEach entry $f_{ij}$ of the adjacency matrix $F\\in{\\mathbb{R}}^{n\\times n}$ associated to\n$G$ is taken to be the weight of the edge connecting node $i$ to node $j$. \nIf the two nodes are not connected, then $f_{ij}=0$.\nA graph is unweighted if the weights are either 0 or 1.\nThe adjacency matrix is symmetric if and only if the graph is undirected.\n\nThe (unnormalized) \\emph{graph Laplacian} of a symmetric, matrix \n$F\\in{\\mathbb{R}}^{n\\times n}$ is the symmetric, positive semidefinite matrix \n$$\nL=D-F,\n$$\nwhere $D=\\diag(d_1,\\ldots,d_n)$ is the \\emph{degree matrix}, whose $i$th\ndiagonal element equals the sum of the weights of the edges starting from node\n$i$ in the undirected network defined by $F$, that is, \n$d_i=\\sum_{j=1}^n f_{ij}$.\nIn the case of an unweighted graph, $d_i$ is the number of nodes connected to\nnode $i$.\n\nSetting $\\bm{e}=[1,\\dots,1]^T\\in{\\mathbb{R}}^n$, it is immediate to observe that\n$$\nL\\bm{e} = (D-F)\\bm{e}= \\bm{0},\n$$\nwhere $\\bm{0}\\in{\\mathbb{R}}^n$ is the zero vector.\nHence, $0$ is an eigenvalue of the graph Laplacian with eigenvector $\\bm{e}$.\n\nThe Gershgorin circle theorem implies all the eigenvalues are non-negative, so\nwe order them as $\\lambda_1=0\\leq\\lambda_2\\leq\\dots\\leq\\lambda_n$, with\ncorresponding eigenvectors $\\bm{v}_1=\\bm{e},\\bm{v}_2,\\dots, \\bm{v}_n$.\nThe smallest eigenvalue of $L$ with associated eigenvector orthogonal to\n$\\bm{e}$ is called the \\emph{Fiedler value}, or the \\emph{algebraic\nconnectivity}, of the graph described by $F$.\nThe corresponding eigenvector is the \\emph{Fiedler\nvector}~\\cite{fiedler1973algebraic,fiedler1975property,fiedler1989laplacian}.\n\nAlternatively, the Fiedler value may be defined by\n$$\n\\min_{\\bm{x}^{T}\\bm{e}=0,\\ \\bm{x}^{T}\\bm{x}=1}\\bm{x}^{T}L\\bm{x}.\n$$\nThen, a Fiedler vector is any vector $\\bm{x}$ that achieves the minimum.\n\nFrom the Kirchhoff matrix-tree theorem it follows that the Fiedler value is\nzero if and only if the graph is not connected~\\cite{de2007old}; in particular,\nthe number of times 0 appears as an eigenvalue of the Laplacian is the number\nof connected components of the graph.\nSo, if the considered adjacency matrix is irreducible, that is, if the\ngraph is connected, the Fiedler vector corresponds to the first non-zero\neigenvalue of the Laplacian matrix.\n\nA \\emph{bipartite graph} $G$ is a graph whose vertices can be divided into two\ndisjoint sets $U$ and $V$ containing $n$ and $m$ nodes, respectively, such\nthat every edge connects a node in $U$ to one in $V$.\n\nIn our archaeological metaphor, the nodes sets $U$ (units) and $V$\n(types) represent the excavation sites and the found artifacts, respectively.\nThen, as already outlined in the Introduction, \nthe associated adjacency matrix $A$, of size $n\\times m$, is obtained by\nsetting $a_{ij}=1$ if the unit $i$ contains objects of type $j$, and 0\notherwise. \nIf the element $a_{ij}$ takes value different from 1, we consider it as a\nweight indicating the number of objects of type $j$ contained in unit $i$, or\ntheir percentage. In this case, we denote $A$ as the \\emph{abundance matrix}.\n\nThe first mathematical definition of seriation was based on the construction of\na symmetric matrix $S$ known as \\emph{similarity\nmatrix}~\\cite{brainerd1951place,robinson1951method}, where $s_{ij}$ describes,\nin some way, the likeness of the nodes $i,j\\in U$. One possible definition is\nthrough the product $S=AA^T$, being $A$ the adjacency matrix of the problem.\nIn this case, $s_{ij}$ equals the number of types shared between unit $i$ and\nunit $j$. \nThe largest value on each row is the diagonal element, which reports the\nnumber of types associated to each unit. By permuting the rows and columns of\n$S$ in order to cluster the largest values close to the main diagonal, one \nobtains a permutation of the corresponding rows of $A$ that places closer the\nunits similar in types.\nIt is worth noting that this operation of permuting rows and columns of $S$ is\nnot uniquely defined.\n\nThe \\emph{Robinson method}~\\cite{robinson1951method} is a statistical technique\nbased on a different similarity matrix. It is based on the concept that each\ntype of artifact used in a certain period eventually decreases in popularity\nuntil it becomes forgotten. This method is probably the first documented\nexample of a practical procedure based on the use of the similarity matrix, so\nits description is interesting in a historical perspective.\n\nThe method, starting from an abundance matrix $A \\in {\\mathbb{R}}^{n \\times m}$ whose\nentries are in percentage form (the sum of each row is 100), computes the\nsimilarity matrix $S$ by a particular rule, leading to a symmetric matrix of\norder $n$ with entries between $0$ (rows with no types in common) and $200$,\nwhich corresponds to units containing exactly the same types.\nThen, the method searches for a permutation matrix $P$ such that $PSP^T$ has\nits largest entries as close as possible to the main diagonal. The same\npermutation is applied to the rows of the data matrix $A$ to obtain a\nchronological order for the archaeological units. Since, as already remarked,\nthe sequence can be read in both directions, external information must be used\nto choose an orientation.\n\nThe procedure of finding a permutation matrix $P$ is not uniquely specified.\nOne way to deal with it is given by the so called \\emph{Robinson's form}, which\nplaces larger values close to the main diagonal, and lets off-diagonal entries\nbe nonincreasing moving away from the main diagonal.\nMore in detail, a symmetric matrix $S$ is in Robinson's form, or is an\nR-matrix, if and only if\n\\begin{eqnarray}\ns_{ij}\\leqslant s_{ik}, \\quad \\text{if } j\\leqslant k\\leqslant i, \n\\label{rmatrix1} \\\\\ns_{ij}\\geqslant s_{ik}, \\quad \\text{if } i\\leqslant j\\leqslant k.\n\\label{rmatrix2} \n\\end{eqnarray}\nA symmetric matrix is pre-$R$ if and only if there exists a simultaneous\npermutation of its rows and columns which transforms it in Robinson's form, so\nit corresponds to a well-posed ordering problem.\nFor other interesting references on R-matrices and their detection, \nsee~\\cite{chepoi1997,laurent2017lex,laurent2017,prea2014,seston2008}.\n\n\n\n\\section{PQ-trees}\\label{sec:pqtrees}\n\nA \\emph{PQ-tree} is a data structure introduced by Booth and\nLueker~\\cite{booth1976testing} to encode a family of permutations of a\nset of elements, and solve problems connected to finding admissible permutations\naccording to specific rules.\n\nA PQ-tree $T$ over a set $U = \\{u_1,u_2,\\dots,u_n\\}$ is a rooted tree whose\nleaves are elements of $U$ and whose internal (non-leaf) nodes are\ndistinguished as either P-nodes or Q-nodes. \nThe only difference between them is the way in which their children are\ntreated: for Q-nodes only one order and its reverse are allowed, whereas in the\ncase of P-node all possible permutations of the children leaves are permitted.\nThe root of the tree can be either a P or a Q-node.\n\nWe will represent graphically a P-node by a circle, and a Q-node by a\nrectangle. \nThe leaves of $T$ will be displayed as triangles, and labeled by the elements\nof $U$. The frontier of $T$ is one possible permutation of the elements of $U$,\nobtained by reading the labels of the leaves from left to right.\n\nWe recall two definitions from~\\cite{booth1976testing}.\n\n\\begin{definition}\\label{def:proper}\nA PQ-tree is \\textit{proper} when the following conditions hold:\n\\begin{itemize}\n\\item[i)] every $u_{i}\\in U$ appears precisely once as a leaf;\n\\item[ii)] every P-node has at least two children;\n\\item[iii)] every Q-node has at least three children. \n\\end{itemize}\n\\end{definition}\n\nAs we observed above, the only difference between a P-node and a Q-node is the\ntreatment of their children, and in the case of exactly two children there is\nno real distinction between a P-node and a Q-node.\nThis justifies the second and third conditions of Definition~\\ref{def:proper}.\n\n\\begin{definition}\\label{def:equiv}\nTwo PQ-trees are said to be \\textit{equivalent} if one can be transformed into\nthe other by applying a sequence of the following two transformations:\n\\begin{itemize}\n\\item[i)] arbitrarily permute the children of a P-node;\n\\item[ii)] reverse the children of a Q-node.\n\\end{itemize} \n\\end{definition}\n\nA PQ-tree represents permutations of the elements of a set through\nadmissible reorderings of its leaves.\nEach transformation in Definition~\\ref{def:equiv} specifies an admissible\nreordering of the nodes within a PQ-tree.\nFor example, a tree with a single P-node represents the equivalence\nclass of all permutations of the elements of $U$, while a tree with a single\nQ-node represents both the left-to-right and right-to-left orderings of the\nleaves.\nA tree with a mixed P-node and Q-node structure represents the equivalence\nclass of a constrained permutation, where the exact structure of the tree\ndetermines the constraints.\nFigure~\\ref{fig:pqtree} displays a PQ-tree and the admissible permutations\nit represents.\n\n\\begin{figure}\n\\begin{minipage}{.52\\textwidth}\n\\includegraphics[width=\\textwidth]{pqtree}\n\\end{minipage}\n\\begin{minipage}{.45\\textwidth}\n\\tiny\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\hline\\noalign{\\smallskip}\n1 & 2 & 3 & 4 & 5 & 6 \\\\\n1 & 2 & 3 & 6 & 5 & 4 \\\\\n1 & 3 & 2 & 4 & 5 & 6 \\\\\n1 & 3 & 2 & 6 & 5 & 4 \\\\\n2 & 1 & 3 & 4 & 5 & 6 \\\\\n2 & 1 & 3 & 6 & 5 & 4 \\\\\n2 & 3 & 1 & 4 & 5 & 6 \\\\\n2 & 3 & 1 & 6 & 5 & 4 \\\\\n3 & 1 & 2 & 4 & 5 & 6 \\\\\n3 & 1 & 2 & 6 & 5 & 4 \\\\\n3 & 2 & 1 & 4 & 5 & 6 \\\\\n3 & 2 & 1 & 6 & 5 & 4 \\\\\n4 & 5 & 6 & 1 & 2 & 3 \\\\\n4 & 5 & 6 & 1 & 3 & 2 \\\\\n4 & 5 & 6 & 2 & 1 & 3 \\\\\n4 & 5 & 6 & 2 & 3 & 1 \\\\\n4 & 5 & 6 & 3 & 1 & 2 \\\\\n4 & 5 & 6 & 3 & 2 & 1 \\\\\n6 & 5 & 4 & 1 & 2 & 3 \\\\\n6 & 5 & 4 & 1 & 3 & 2 \\\\\n6 & 5 & 4 & 2 & 1 & 3 \\\\\n6 & 5 & 4 & 2 & 3 & 1 \\\\\n6 & 5 & 4 & 3 & 1 & 2 \\\\\n6 & 5 & 4 & 3 & 2 & 1 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\caption{On the left, a PQ-tree over the set $U=\\{1,\\ldots,6\\}$; on the right,\nthe 24 admissible permutations encoded in the tree.}\n\\label{fig:pqtree}\n\\end{figure}\n\nThe PQ-tree data structure has been exploited in a variety of applications,\nfrom archaeology and chronology reconstruction~\\cite{atkins1998spectral} to\nmolecular biology with DNA mapping and sequence\nassembly~\\cite{greenberg1995physical}.\nThe first problem to which it was applied is the consecutive ones property\n(C1P) for matrices~\\cite{booth1976testing}, mentioned in\nSection~\\ref{sec:intro}.\n\nGiven a pre-R matrix, the spectral algorithm from~\\cite{atkins1998spectral},\nthat will be discussed in Section~\\ref{sec:seriation}, constructs a PQ-tree\ndescribing the set of all the permutations of rows and columns that lead to an\nR-matrix.\n\n\n\\subsection{Implementation of PQ-trees}\n\nThe \\texttt{PQser} toolbox for Matlab is available as a compressed archive that can be downloaded from the authors webpages (see, e.g., \n\\url{http:\/\/bugs.unica.it\/~gppe\/soft\/}).\nBy uncompressing it, the directory \\texttt{PQser} will be created. \nIt must be added to Matlab search path, either by the command \\ttt{addpath} or\nusing the graphical interface menus.\nThe sub-directory \\texttt{demo} contains a tutorial for the toolbox and the\nscripts which were used to construct the examples reported in the paper.\nThe installation procedure and the toolbox content are described in detail in\nthe README.txt file, which can be found in the main directory.\n\nIn the \\texttt{PQser} toolbox, a PQ-tree $T$ is a \\ttt{struct} variable (i.e., a\nrecord) composed by two fields.\nThe first field, \\ttt{T.type}, specifies the type of the node, i.e., P, Q,\nor a leaf, in the case of a trivial tree.\nThe second field, \\ttt{T.value}, is a vector which provides a list of PQ-trees,\nrecursively defined.\nIn the case of a leaf, this field contains the index of the unit it represents.\n\nFor example, the graph in Figure~\\ref{fig:pqtree} was obtained by the following\npiece of code\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nv(1) = pnode([1 2 3]);\nv(2) = qnode([4 5 6]);\nT = pnode(v);\t\npqtreeplot(T)\t\n\\end{verbatim}\n\\end{quote}\nthe resulting data structure for the PQ-tree is\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nT = \n struct with fields:\n type: 'P'\n value: [1x2 struct]\n\\end{verbatim}\n\\end{quote}\nand the permutations encoded in $T$ are computed by\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nperms_matrix = pqtreeperms(T)\n\\end{verbatim}\n\\end{quote}\nThese instructions are contained in the script \\texttt{graf1.m}, in the\n\\texttt{demo} sub-directory.\n\n\\begin{table}[htb]\n\\footnotesize\n\\centering\n\\begin{tabular}{ll}\n\\hline\n\\ttt{pnode} & create a P-node \\\\\n\\ttt{qnode} & create a Q-node \\\\\n\\ttt{lnode} & create a leaf \\\\\n\\ttt{mnode} & create an M-node \\\\\n\\ttt{pqtreeplot} & plot a PQ-tree \\\\\n\\ttt{pqtreeNperm} & number of admissible permutations in a PQ-tree \\\\\n\\ttt{pqtreeperms} & extract all admissible permutations from a PQ-tree \\\\\n\\ttt{pqtree1perm} & extract one admissible permutation from a PQ-tree \\\\\n\\ttt{pqtreegetnode} & extract a subtree from a PQ-tree \\\\\n\\ttt{pqtreenodes} & converts a PQ-tree to Matlab \\ttt{treeplot} format \\\\\n\\hline\n\\end{tabular}\n\\caption{Functions in the \\texttt{PQser} toolbox devoted to the manipulation of\nPQ-trees.}\n\\label{tab:pqfuncs}\n\\end{table}\n\nThe functions intended for creating and manipulating a PQ-tree are listed in\nTable~\\ref{tab:pqfuncs}.\nThe function \\ttt{mnode} creates an additional type of node, an M-node, which\nis intended to deal with multiple Fiedler values; we will comment on it in\nSection~\\ref{sec:mult};\n\\ttt{pqtreegetnode} and \\ttt{pqtreenodes} are utility functions for\n\\ttt{pqtreeplot}, they are not intended to be called directly by the user.\nAll the functions are documented via the usual Matlab \\ttt{help} command, e.g.,\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nhelp pnode\nhelp pqtreeplot\n\\end{verbatim}\n\\end{quote}\n\nAs an example, we report in Algorithm~\\ref{alg:pqtreeNperm} the structure of\n\\ttt{pqtreeNperm}, a function which returns the number $N$ of all the\npermutations contained in the tree whose root $T$ is given in input.\nIn the particular case of a leaf, only one permutation is possible \n(line~\\ref{l1}--\\ref{l2}).\nOtherwise, we consider the vector $\\bm{c}$ of size $k$, containing the children\nnodes of the root of T (line~\\ref{l5}).\nThe algorithm calls itself recursively on each component of $\\bm{c}$\n(line~\\ref{l8}).\nIn the case of a Q-node the number of permutations is doubled, because only one\nordering and its reverse are admissible, whereas for a P-node the number is\nmultiplied by the factorial of $k$, since in this case all the possible\npermutations of the children are allowed.\nThe same procedure is applied to an M-node; see Section~\\ref{sec:mult} for\ndetails.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $N = \\pqtreeNperm(T)$\n\\IF $T$ is a leaf\n\\label{l1}\n\\STATE $N =1$\n\\label{l2}\n\\ELSE \n\t\\STATE $\\bm{c}=T\\ttt{.value}$, $k=\\length(\\bm{c})$\n\\label{l5}\n\t\\STATE $p=1$\n\t\\FOR $i = 1,\\dots,k$\n\t\t\\STATE $p = p*\\pqtreeNperm(c_i)$\n\\label{l8}\n\t\\ENDFOR\n\t\\IF $T$ is a Q-node\n\t\t\\STATE $N = 2*p$\n\t\\ELSE \n\t\t\\STATE $N = \\factorial(k)*p$\n\t\\ENDIF\n\\ENDIF\t \n\\end{algo}\n\\caption{Compute the number of admissible permutations in a PQ-tree.}\n\\label{alg:pqtreeNperm}\n\\end{algorithm}\n\nThe toolbox includes an interactive graphical tool for exploring a PQ-tree $T$.\nAfter displaying $T$ by \\ttt{pqtreeplot}, it is possible to extract a subtree\nby clicking on one node with the left mouse button. In this case, the\ncorresponding subtree is extracted, it is plotted in a new figure, and it is\nsaved to the variable \\ttt{PQsubtree} in the workspace.\nThis feature is particularly useful when analyzing a large PQ-tree.\nThe function \\ttt{pqtreeplot} allows to set some attributes of the plot; see\nthe help page.\n\n\n\n\\section{A spectral algorithm for the seriation problem}\\label{sec:seriation}\n\nIn this section we briefly review the spectral algorithm for the seriation\nproblem introduced in~\\cite{atkins1998spectral}, and describe our\nimplementation.\n\nGiven the set of units $U=\\{u_1,u_2,\\dots,u_n\\}$, we will write \n$i\\preccurlyeq j$ if $u_i$ precedes $u_j$ in a chosen ordering.\nIn~\\cite{atkins1998spectral}, the authors consider a symmetric bivariate\n\\emph{correlation function} $f$ reflecting the desire for units $i$ and $j$ to\nbe close to each other in the sought sequence.\nThe point is to find all index permutation vectors\n${\\boldsymbol{\\pi}}=(\\pi_1,\\ldots,\\pi_n)^T$ such that \n\\begin{equation}\\label{fperm}\n\\pi_i\\preccurlyeq\\pi_j\\preccurlyeq\\pi_k \\quad \\iff \\quad \nf(\\pi_i,\\pi_j)\\geq f(\\pi_i,\\pi_k) \\quad \\text{and} \\quad\nf(\\pi_j,\\pi_k)\\geq f(\\pi_i,\\pi_k).\n\\end{equation}\nIt is natural to associate to such correlation function a real symmetric matrix\n$F$, whose entries are defined by $f_{ij}=f(i,j)$.\nThis matrix plays exactly the role of the similarity matrix $S$ discussed in\nSection~\\ref{sec:mathback}, as the following theorem states.\n\n\\begin{theorem}\\label{theo:fs}\nA matrix $F$ is an R-matrix if and only if \\eqref{fperm} holds.\n\\end{theorem}\n\n\\begin{proof}\nLet us assume that the permutation ${\\boldsymbol{\\pi}}$ which realizes \\eqref{fperm} has\nalready been applied to the units. \nThen, since a permutation of the units corresponds to a simultaneous\npermutation of the rows and columns of the matrix $F$, we obtain\n$$\ni\\leq j\\leq k \\quad \\iff \\quad \nf_{ij}\\geq f_{ik} \\quad \\text{and} \\quad\nf_{jk}\\geq f_{ik}.\n$$\nThe first inequality $f_{ij}\\geq f_{ik}$ is exactly \\eqref{rmatrix2}.\nKeeping into account the symmetry of $F$ and cyclically permuting the indexes,\nfrom the second inequality we get\n$$\nj\\leq k\\leq i \\quad \\iff \\quad \nf_{ij}\\leq f_{ik},\n$$\nwhich corresponds to \\eqref{rmatrix1}.\n\\end{proof}\n\nIf a seriation data set is described by an adjacency (or abundance) matrix $A$,\nwe will set $F=AA^T$.\nIf $F$ is pre-$R$ (see Section~\\ref{sec:mathback}), there exists a rows\/columns\npermutation that takes it in $R$-form. Unfortunately, this property cannot be\nstated in advance, in general. This property can be ascertained, e.g., after\napplying the algorithm discussed in this section; see below.\n\nThe authors approach in~\\cite{atkins1998spectral}, see\nalso~\\cite{estrada2010network}, is to consider the minimization of the\nfollowing penalty function \n$$\nh(\\bm{x}) = \\frac{1}{2}\\sum_{i,j=1}^{n} f_{ij}(x_i-x_j)^2, \n\\quad \\bm{x}\\in {\\mathbb{R}}^n,\n$$ \nwhose value is small for a vector $\\bm{x}$ such that each pair $(i,j)$ of\nhighly correlated units is associated to components $x_i$ and $x_j$ with close\nvalues.\nOnce the minimizing vector $\\bm{x}_{\\min}$ is computed, it is sorted in\neither nonincreasing or nondecreasing value order, yielding\n$\\bm{x}_{\\boldsymbol{\\pi}}=(x_{\\pi_1},\\ldots,x_{\\pi_n})^T$.\nThe permutation of the units ${\\boldsymbol{\\pi}}$ realizes \\eqref{fperm}.\n\nNote that $h$ does not have a unique minimizer, since its value does not change\nif a constant is added to each of the components $x_i$ of the vector\n$\\bm{x}$.\nIn order to ensure uniqueness and to rule out the trivial solution, it is\nnecessary to impose two suitable constraints on the components of the vector\n$\\mathbf{x}$.\nThe resulting minimization problem is:\n$$\n\\begin{aligned}\n&\\text{minimize} & &\nh(\\bm{x}) = \\frac{1}{2}\\sum_{i,j=1}^{n} f_{ij}(x_i-x_j)^2 \\\\\n&\\text{subject to} & & \\sum_i x_i = 0 \\quad \\text{and} \\quad \\sum_i x_i^2 = 1.\n\\end{aligned}\n$$\n\nThe solution to this approximated problem may be obtained from the Fiedler\nvector of the Laplacian $L$ of the correlation matrix $F$. \nLetting $D = \\diag(d_i)$ be the degree matrix, $d_i=\\sum_{j=1}^n f_{ij}$, it is\nimmediate to observe that\n$$\nh(\\bm{x}) = \\frac{1}{2}\\sum_{i,j=1}^{n} f_{ij}(x_i^2+x_j^2-2x_ix_j)\n= \\bm{x}^T D \\bm{x} - \\bm{x}^T F \\bm{x}.\n$$ \nThis shows that the previous minimization problem can be rewritten as \n\\begin{eqnarray*}\n\\min_{\\|\\mathbf{x}\\|=1,\\ \\bm{x}^T\\bm{e} = 0}\n\\mathbf{x}^TL\\mathbf{x}\n\\end{eqnarray*}\nwhere $L=D-F$.\nThe constraints require $\\mathbf{x}$ to be a unit vector orthogonal to\n$\\mathbf{e}$.\nBeing $L$ symmetric, all the eigenvectors except $\\bm{e}$ satisfy the\nconstraints.\nConsequently, a Fiedler vector is a solution to the constrained minimization\nproblem.\n\nIn fact, Theorem~3.2 from~\\cite{atkins1998spectral} proves that an R-matrix has\na monotone Fiedler vector, while Theorem~3.3, under suitable assumptions,\nimplies that a reordering of the Fiedler vector takes a pre-R matrix to R-form.\nThis confirms that the problem is well posed only when $F$ is pre-R.\nNevertheless, real data sets may be inconsistent, in the sense that do not\nnecessarily lead to pre-R similarity matrices.\nIn such cases, it may be useful to construct an approximate solution to the\nseriation problem, and sorting the entries of the Fiedler vector generates an\nordering that tries to keep highly correlated elements close to each other.\nThis is relevant because techniques based on Fiedler vectors are being used for\nthe solution of different sequencing\nproblems~\\cite{barnard1995spectral,greenberg1995physical,higham2007,juvan1992optimal}. \nIn particular, they are employed in complex network analysis, e.g., for\ncommunity detection and partitioning of graphs~\\cite{estrada2012,estrada2015}.\n\nThe algorithm proposed in~\\cite{atkins1998spectral} is based upon the above\nidea, and uses a PQ-tree to store the permutations of the units that produce a\nsolution to the seriation problem; its implementation is described in\nAlgorithm~\\ref{alg:spectrsort}.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $T = \\spectrsort(F,U)$\n\\STATE $n=$ row size of $F$\n\\STATE $\\alpha = \\min_{i,j} f_{i,j}$, \\If $\\alpha\\neq 0$, \n\t$\\bm{e}=(1,\\ldots,1)^T$, $F=F-\\alpha \\bm{e}\\bm{e}^{T}$, \\End\n\\label{line2}\n\\STATE call $\\getconcomp$ to construct the connected components \n$\\{F_1,\\dots,F_k\\}$ of $F$\n\\label{line3}\n\\STATE \\phantom{XXX} and the corresponding index sets $U=\\{U_1,\\dots,U_k\\}$ \n\\label{line4}\n\\IF {$k>1$}\n\t\\FOR {$j=1,\\dots,k$}\n\\label{line6}\n\t\t\\STATE $v(j) = \\spectrsort(F_j,U_j)$\n\t\\ENDFOR\n\t\\STATE $T = \\pnode(v)$\n\\label{line9}\n\\ELSE \n\t\\IF {$n=1$}\n\\label{line11}\n\t\t\\STATE $T=\\lnode(U)$\n\t\\ELSEIF {$n=2$}\n\t\t\\STATE $T=\\pnode(U)$\n\t\\ELSE\n\\label{line15}\n\t\t\\STATE $L=$ Laplacian matrix of $F$\n\\label{line16}\n\t\t\\STATE compute (part of) the eigenvalues and eigenvectors of $L$\n\\label{line17}\n\t\t\\STATE determine multiplicity $n_F$ of the Fiedler value\n\t\t\taccording to a tolerance $\\tau$\n\t\t\\IF {$n_F = 1$}\n\t\t\t\\STATE $\\bm{x}=$ sorted Fiedler vector\n\t\t\t\\STATE $t$ number of distinct values in $\\bm{x}$\n\t\t\t\taccording to a tolerance $\\tau$\n\\label{line19}\n\t\t\t\\FOR {$j=1,\\dots,t$}\n\t\t\t\t\\STATE $u_j$ indices of elements in $\\bm{x}$ \n\t\t\t\t\twith value $x_j$\n\t\t\t\t\\IF $u_j$ has just one element\n\t\t\t\t\t\\STATE $v_j=\\lnode(u_j)$\n\\label{line26}\n\t\t\t\t\\ELSE \n\t\t\t\t\t\\STATE $v(j) = \\spectrsort(F(u_j,u_j),U(u_j,u_j))$\n\\label{line28}\n\t\t\t\t\\ENDIF\n\t\t\t\\ENDFOR\n\t\t\t\\STATE $T = \\qnode(v)$\n\t\t\\ELSE\n\t\t\t\\STATE $T=\\mnode(U)$\n\\label{line33}\n\t\t\\ENDIF\n\t\\ENDIF\n\\ENDIF\n\\end{algo}\n\\caption{Spectral sort algorithm.}\n\\label{alg:spectrsort}\n\\end{algorithm}\n\nThe algorithm starts translating all the entries of the correlation matrix so\nthat the smallest is 0 , i.e., \n\\begin{equation}\\label{trans}\n\\tilde{F}=F-\\alpha \\bm{e}\\bm{e}^{T}, \\qquad \\alpha = \\min_{i,j} f_{ij};\n\\end{equation}\nsee line~\\ref{line2} of Algorithm~\\ref{alg:spectrsort}.\nThis is justified by the fact that $F$ and $\\tilde{F}$ have the same Fiedler\nvectors and that if $F$ is an irreducible R-matrix such translation ensures\nthat the Fiedler value is a simple eigenvalue of\n$L$~\\cite[Lemma~4.1~and~Theorem~4.6]{atkins1998spectral}.\nOur software allows the user to disable this procedure (see\nTable~\\ref{tab:opts} below)\nas he may decide to\nsuitably preprocess the similarity matrix in order to reduce the\ncomputational load.\nIndeed, the translation procedure is repeated each time the algorithm calls\nitself recursively.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $U = \\getconcomp(F)$\n\\STATE preallocate the cell-array $U$, $chlist=$empty vector\n\\STATE $root=$\\{node 1\\}, $list=root$, $n=$row size of $F$\n\\STATE $i=0$, $flag=true$ (logical variable)\n\\WHILE $flag$\n\\STATE $i =i+1$\n\\STATE $list =\\graphvisit(root,list)$\n\\STATE $U\\{i\\}=list$\n\\STATE update $chlist$ adding the nodes in $list$ and sort the vector\n\\STATE $flag=true$ if the number of elements in $chlist$ is different from $n$\n\\STATE \\phantom{XXX} otherwise $flag=false$\n\\IF $flag$\n\t\\STATE choose the $root$ for a new connected component\n\t\\IF there are no connected components left\n\t\t\\STATE exit\n\t\\ENDIF\n\t\\STATE $list=root$\n\\ENDIF\n\\ENDWHILE\n\\end{algo}\n\\caption{Detect the connected components of a graph.}\n\\label{alg:getconcomp}\n\\end{algorithm}\n\nIf the matrix $F$ is reducible, then the seriation problem can be\ndecoupled~\\cite[Lemma~4.2]{atkins1998spectral}.\nLines~\\ref{line3}--\\ref{line4} of the algorithm detect the irreducible blocks\nof the correlation matrix by using the function \\textsf{getconcomp.m}, which\nalso identifies the corresponding index sets.\nThe function, described in Algorithm~\\ref{alg:getconcomp}, constructs a\n\\emph{cell array} containing the indices which identify each connected\ncomponent of a graph.\nIt calls the function \\textsf{graphvisit.m}, which visits a graph starting from\na chosen node; see Algorithm~\\ref{alg:graphvisit}.\nNote that these two functions, in order to reduce the stack consumption due to\nrecursion, use a global variable to store the correlation matrix.\n\n\\begin{algorithm}[!ht]\n\\begin{algo}\n\\STATE \\Function $list = \\graphvisit(root,list)$\n\\STATE construct the list $l$ of the indices of the nodes connected to the root\n\\STATE initialize an empty list $nlist$\n\\STATE find the elements of $l$ which are not in $list$\n\\STATE add the new elements to $list$ and to $nlist$\n\\IF $nlist$ is not empty\n\t\\STATE sort $list$\n\t\\FOR each node $i$ in $nlist$\n\t\t\\STATE $list=\\graphvisit(nlist(i),list)$\n\t\\ENDFOR\n\\ENDIF\n\\end{algo}\n\\caption{Visit a graph starting from a node.}\n\\label{alg:graphvisit}\n\\end{algorithm}\n\nIf more than one connected component is found, then the function calls itself\non each component, and stores the returned lists of nodes as children of\na P-node (lines \\ref{line6}--\\ref{line9}).\nIf the matrix is irreducible, the dimension $n$ of the matrix is\nconsidered (lines~\\ref{line11}--\\ref{line15}).\nThe cases $n=1,2$ are trivial.\nIf $n>2$, the Laplacian matrix $L$ is computed, as well as the Fiedler value\nand vector (lines~\\ref{line16}--\\ref{line17}). \nDepending on the matrix being ``small'' or ``large'' different algorithms are\nused.\nFor a small scale problem the full spectral decomposition of the Laplacian is\ncomputed by the \\ttt{eig} function of Matlab.\nFor a large scale problem only a small subset of the eigenvalues and\neigenvectors are evaluated using the \\ttt{eigs} function, which is based on a\nKrylov space projection method.\nThe \\texttt{PQser} toolbox computes by default the eigenpairs corresponding to the\nthree eigenvalues of smallest magnitude, since they are sufficient to\nunderstand if the Fiedler value is simple or multiple, but the default value\ncan be modified.\nThe choice between the two approaches is automatically performed and it may be\ninfluenced by the user; see Table~\\ref{tab:opts} in Section~\\ref{sec:implser}.\n\nThen, the algorithm determines the multiplicity of the Fiedler value according\nto a given tolerance.\nIf the Fiedler value is a simple eigenvalue of $L$, the algorithm sorts the\nelements of the current list according to the reordering of the Fiedler vector\nand stores them as the children of a Q-node.\nIf two or more values of the Fiedler vector are repeated the function invokes\nitself recursively (line~\\ref{line28}), in accordance\nwith~\\cite[Theorem~4.7]{atkins1998spectral}; on the contrary, the corresponding\nnode becomes a leaf (line~\\ref{line26}).\nIn our implementation we introduce a tolerance $\\tau$ to distinguish ``equal''\nand ``different'' numbers: $a$ and $b$ are considered ``equal'' if\n$|a-b|<\\tau$. The default value for $\\tau$ is $10^{-8}$.\n\nIn the case of a multiple Fiedler value, the algorithm conventionally\nconstructs an ``M-node'' (line~\\ref{line33}).\nThis new type of node has been introduced in order to flag this particular\nsituation, which will be further discussed in Section~\\ref{sec:mult}.\n\nAlgorithm~\\ref{alg:spectrsort} produces a PQ-tree whether $F$ is a pre-R matrix\nor not.\nIf all the Fiedler values computed are simple, the starting matrix is pre-R and\nany permutation encoded in the PQ-tree will take it to R-form.\nIn the presence of a multiple Fiedler vector the problem is not well posed and\nan approximate solution is computed.\n\nThe number $N$ of all the admissible permutations generated by the algorithm\ncan be obtained by counting all the admissible boundaries of the tree.\nIn the case of a PQ-tree consisting of a single Q-node $N$ is equal to 2,\nbecause only the left-to-right order of the children leaves and its reverse\nare possible. \nFor a single P-node, the number of all the permutations is the factorial of the\nnumber of the children.\nAn M-node is temporarily treated as a P-node, although we experimentally\nobserved that not all the permutations are admissible; this aspect is discussed\nin Section~\\ref{sec:mult}.\n\n\n\\subsection{Implementation of spectral seriation}\\label{sec:implser}\n\nThe functions included in the \\texttt{PQser} toolbox are listed in\nTable~\\ref{tab:serfuncs}.\nBesides the function \\ttt{spectrsort}, which implements \nAlgorithm~\\ref{alg:spectrsort}, there is a parallel version of the same method,\n\\ttt{pspectrsort}, which distributes the \\ttt{for} loop at\nline~\\ref{line6} of the algorithm among the available processing units.\nIn order to execute the function \\ttt{pspectrsort}, the Parallel Computing\nToolbox must be present in the current Matlab installation.\n\n\\begin{table}[htb]\n\\footnotesize\n\\centering\n\\begin{tabular}{ll}\n\\hline\n\\ttt{spectrsort} & spectral sort for the seriation problem \\\\\n\\ttt{pspectrsort} & parallel version of \\ttt{spectrsort} \\\\\n\\ttt{fiedvecs} & compute the Fiedler vectors and values of a Laplacian \\\\\n\\ttt{getconcomp} & determine the connected components of a graph \\\\\n\\ttt{graphvisit} & visit a graph starting from a node \\\\\n\\ttt{distinct} & sort and level the elements of a vector \\\\\n\\ttt{lapl} & construct the graph Laplacian of a matrix \\\\\n\\ttt{testmatr} & test matrices for PQser \\\\\n\\hline\n\\end{tabular}\n\\caption{Functions in the \\texttt{PQser} toolbox devoted to the solution of the\nseriation problem.}\n\\label{tab:serfuncs}\n\\end{table}\n\nThe function \\ttt{testmatr} allows one to create some simple test problems.\nThe remaining functions of Table~\\ref{tab:serfuncs} are not likely to be used\nin the common use of the toolbox. \nThey are made available to the expert user, who may decide to call them\ndirectly or to modify their content.\n\n\\begin{table}[htb]\n\\footnotesize\n\\centering\n\\begin{tabular}{ll}\n\\hline\n\\ttt{tau} & tolerance used to distinguish between ``equal'' and ``different'' \\\\\n\t & values (\\ttt{spectrsort} and \\ttt{fiedvecs}, def. $10^{-8}$) \\\\\n\\ttt{translate} & applies translation \\eqref{trans} (\\ttt{spectrsort}, def. 1) \\\\\n\\ttt{lrg} & used to select small scale or large scale algorithm (\\ttt{fiedvecs}, \\\\\n\t& true if the input matrix is sparse) \\\\\n\\ttt{nlarge} & if matrix size is below this value, the small scale algorithm \\\\\n\t & is used (\\ttt{fiedvecs}, def. 1000) \\\\\n\\ttt{neig} & number of eigenpairs to be computed when the large scale \\\\\n\t & algorithm is used (\\ttt{fiedvecs}, def. 3) \\\\\n\\ttt{maxncomp} & maximum number of connected components (\\ttt{getconcomp}, def. 100) \\\\\n\\ttt{bw} & half bandwidth of test matrix (\\ttt{testmatr}, type 2 example, def. 2) \\\\\n\\ttt{spar} & construct a sparse test matrix (\\ttt{testmatr}, type 2 example, def. 1) \\\\\n\\hline\n\\end{tabular}\n\\caption{Tuning parameters for the \\texttt{PQser} toolbox; the functions affected are\nreported in parentheses, together with the default value of each parameter.}\n\\label{tab:opts}\n\\end{table}\n\nThe toolbox has some tuning parameters, which are set to a default value, but\ncan be modified by the user. This can be done by passing to a function, as an\noptional argument, a variable of type \\ttt{struct} with fields chosen among the\nones listed in Table~\\ref{tab:opts}.\nFor example:\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nopts.translate = 0;\nT = spectrsort(F,opts);\n\\end{verbatim}\n\\end{quote}\napplies Algorithm~\\ref{alg:spectrsort} to a similarity matrix $F$ omitting\nthe translation process described in \\eqref{trans}.\n\nTo illustrate the use of the toolbox, we consider a similarity matrix $R$\nsatisfying the Robinson criterion \n$$\nR = \\begin{bmatrix}\n200 & 150 & 120 & 80 & 40 & 0 & 0 & 0 & 0 & 0\\\\\n150 & 200 & 160 & 120 & 80 & 40 & 0 & 0 & 0 & 0\\\\\n120 & 160 & 200 & 160 & 120 & 80 & 40 & 0 & 0 & 0\\\\\n80 & 120 & 160 & 200 & 160 & 120 & 80 & 40 & 0 & 0\\\\\n40 & 80 & 120 & 160 & 200 & 160 & 120 & 80 & 40 & 0\\\\\n0 & 40 & 80 & 120 & 160 & 200 & 160 & 120 & 80 & 40\\\\\n0 & 0 & 40 & 80 & 120 & 160 & 200 & 160 & 120 & 80\\\\\n0 & 0 & 0 & 40 & 80 & 120 & 160 & 200 & 160 & 120\\\\\n0 & 0 & 0 & 0 & 40 & 80 & 120 & 160 & 200 & 150\\\\\n0 & 0 & 0 & 0 & 0 & 40 & 80 & 120 & 150 & 200\n\\end{bmatrix}\n$$\nand the pre-R matrix obtained by applying to the rows and columns of $R$ a\nrandom permutation \n$$\nF = \\begin{bmatrix} \n200 & 0 & 0 & 150 & 120 & 0 & 160 & 40 & 0 & 80\\\\\n0 & 200 & 150 & 0 & 0 & 120 & 0 & 80 & 160 & 40\\\\ \n0 & 150 & 200 & 0 & 0 & 80 & 0 & 40 & 120 & 0\\\\ \n150 & 0 & 0 & 200 & 80 & 0 & 120 & 0 & 0 & 40\\\\ \n120 & 0 & 0 & 80 & 200 & 80 & 160 & 120 & 40 & 160\\\\ \n0 & 120 & 80 & 0 & 80 & 200 & 40 & 160 & 160 & 120\\\\ \n160 & 0 & 0 & 120 & 160 & 40 & 200 & 80 & 0 & 120\\\\ \n40 & 80 & 40 & 0 & 120 & 160 & 80 & 200 & 120 & 160\\\\ \n0 & 160 & 120 & 0 & 40 & 160 & 0 & 120 & 200 & 80\\\\ \n80 & 40 & 0 & 40 & 160 & 120 & 120 & 160 & 80 & 200 \n\\end{bmatrix}.\n$$\n\nThe PQ-tree $T$ containing the solution of the reordering problem is\nconstructed by calling the function \\ttt{spectrsort}, which returns the\nresulting data structure:\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nT = spectrsort(F,opts)\nT = \n struct with fields:\n type: 'Q'\n value: [1x10 struct]\n\\end{verbatim}\n\\end{quote}\nUsing the function \\ttt{pqtreeplot} \n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\npqtreeplot(T)\n\\end{verbatim}\n\\end{quote}\nwe obtain the representation of the PQ-tree displayed in Figure~\\ref{pqtree2}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.52\\textwidth]{pqtree2}\n\\caption{A PQ-tree corresponding to a pre-R matrix of dimension 10.}\n\\label{pqtree2}\n\\end{center}\n\\end{figure}\n\nIn this particular case, the PQ-tree $T$ consists of just a Q-node as a root,\nso only two permutations of the leaves are allowed.\nThey can be extracted from the tree using the function \\ttt{pqtreeperms}, whose\noutput is\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nperms_matrix = pqtreeperms(T)\nperms_matrix =\n 4 1 7 5 10 8 6 9 2 3\n 3 2 9 6 8 10 5 7 1 4\n\\end{verbatim}\n\\end{quote} \nIn some occasions, a PQ-tree may contain a very large number of permutations.\nIn such cases, the function \\ttt{pqtree1perm} extracts just one of the\npermutations, in order to apply it to the rows and columns of the matrix $F$:\n\\begin{quote}\n\\footnotesize\n\\begin{verbatim}\nseq = pqtree1perm(T);\nAR = F(seq,seq);\n\\end{verbatim}\n\\end{quote} \nSince $F$ is pre-R, we clearly reconstruct the starting similarity matrix $R$.\n\nThis experiment is contained in the script \\texttt{graf2.m}, in the\n\\texttt{demo} sub-directory.\nThe script \\texttt{tutorial.m}, in the same directory, illustrates the use of\nthe toolbox on other numerical examples.\n\n\n\\section{The case of a multiple Fiedler value}\\label{sec:mult}\n\nIn this section we discuss the case where the Fiedler value is a multiple root\nof the characteristic polynomial of the Laplacian $L$.\nWhen this happens, the eigenspace corresponding to the smallest nonzero\neigenvalue of $L$ has dimension larger than one, so there is no uniqueness in\nthe choice of the Fiedler vector.\n\nWe conjecture that sorting the entries of a Fiedler vector, that is, of any\nvector in the eigenspace of the Fiedler value, does not necessarily lead to all\npossible indexes permutations, i.e., the factorial of the number $n$ of units.\nWe observed that there may be some constraints that limit the number of\npermutations deriving from the Fiedler vector, and this number does not appear\nto be related to the multiplicity of the Fiedler value by a simple formula.\nWe will illustrate this issue by a numerical experiment.\n\nAs we did not find any reference to this problem in the literature, we plan\nto study it in a subsequent paper.\nThis is the reason why the \\texttt{PQser} toolbox conventionally associates an\n\\emph{M-node}, to the presence of a multiple Fiedler value.\nIn the software, the new type of node is temporarily treated as a P-node.\nThis leaves the possibility to implement the correct treatment of the case,\nonce the problem will be understood.\n\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\begin{tikzpicture}[->,>=stealth',auto,node distance=1cm,\n\ton grid,semithick, every state\/.style={fill=cyan!20!white,draw=none,\n\tcircular drop shadow,text=black,inner sep=0pt}]\n\t\\tiny\n\\node[state](A){1};\n\\node[state](B)[below=of A]{2};\n\\node[state](C)[below=of B]{3};\n\\node[state](D)[below=of C]{4};\n\\node[state](E)[below=of D]{5};\n\\node[state,fill=red!20!white,node distance=4cm](L)[right=of A]{1};\n\\node[state,fill=red!20!white,node distance=4cm](M)[right=of B]{2};\n\\node[state,fill=red!20!white,node distance=4cm](N)[right=of C]{3};\n\\node[state,fill=red!20!white,node distance=4cm](O)[right=of D]{4};\n\\node[state,fill=red!20!white,node distance=4cm](P)[right=of E]{5};\n\\path (A) edge (L);\n\\path (A) edge (M);\n\\path (B) edge (M);\n\\path (B) edge (N);\n\\path (C) edge (N);\n\\path (C) edge (O);\n\\path (D) edge (O);\n\\path (D) edge (P);\n\\path (E) edge (P);\n\\path (E) edge (L);\n\\end{tikzpicture}\n\\caption{The \\emph{cycle} seriation problem; the units are on the left, the\ntypes on the right.}\n\\label{fig:cycle}\n\\end{center}\n\\end{figure}\n\nHere we present a simple example to justify our conjecture.\nLet us consider the seriation problem described by the bipartite graph depicted\nin Figure~\\ref{fig:cycle}.\nThe nodes on the left are the units, e.g., the excavation sites; on the right\nthere are the types, which may be seen as the archeological findings. The\nrelationships between units and types are represented by edges connecting the\nnodes.\n\nThe problem is clearly unsolvable, as the associated graph describes a\n\\emph{cycle}: each unit is related to surrounding units by a connection to a\ncommon type, and the two extremal units are related to each other in the same\nway.\n\nAt the same time, not all the units permutations are admissible.\nFor example, one may argue that the permutation ${\\boldsymbol{\\pi}}_1=(3,4,5,1,2)^T$ should\nbe considered partially feasible, as it breaks only one of the constraints\ncontained in the bipartite graph, while the ordering ${\\boldsymbol{\\pi}}_2=(1,4,2,5,3)^T$ has\nnothing to do with the problem considered.\n\nThe adjacency matrix associated to the graph in Figure~\\ref{fig:cycle} is the\nfollowing\n\\begin{equation}\\label{incidCycle}\nA = \\begin{bmatrix}\n1 & 1 & 0 & 0 & 0\\\\\n0 & 1 & 1 & 0 & 0\\\\\n0 & 0 & 1 & 1 & 0\\\\\n0 & 0 & 0 & 1 & 1\\\\\n1 & 0 & 0 & 0 & 1\n\\end{bmatrix}.\n\\end{equation}\nWe can associate to \\eqref{incidCycle} the similarity matrix\n\\begin{equation}\\label{cycleF}\nF = AA^T = \\begin{bmatrix}\n2 & 1 & 0 & 0 & 1\\\\\n1 & 2 & 1 & 0 & 0\\\\\n0 & 1 & 2 & 1 & 0\\\\\n0 & 0 & 1 & 2 & 1\\\\\n1 & 0 & 0 & 1 & 2\n\\end{bmatrix},\n\\end{equation}\nwhose Laplacian is\n$$\nL = D - F = \\begin{bmatrix}\n2 & -1 & 0 & 0 & -1\\\\\n-1 & 2 & -1 & 0 & 0\\\\\n0 & -1 & 2 & -1 & 0\\\\\n0 & 0 & -1 & 2 & -1\\\\\n-1 & 0 & 0 & -1 & 2\n\\end{bmatrix}.\n$$\n\nThe matrix $L$ is circulant, that is, it is fully specified by its first\ncolumn, while the other columns are cyclic permutations of the first one\nwith an offset equal to the column index. \nA complete treatment of circulant matrices can be found\nin~\\cite{davis1979circulant}, while~\\cite{redivo2012smt} implements a Matlab\nclass for optimized circulant matrix computations.\n\nOne of the basic properties of circulant matrices is that their spectrum is\nanalytically known. In particular, the eigenvalues of $L$ are given by\n$$\n\\{\\widehat{L}(1), \\widehat{L}(\\omega), \\widehat{L}(\\omega^2),\n\\widehat{L}(\\omega^3), \\widehat{L}(\\omega^4) \\},\n$$\nwhere $\\widehat{L}(\\zeta)$ is the discrete Fourier transform of the first\ncolumn of $L$\n$$\n\\hat{L}(\\zeta) = 2 -\\zeta ^{-1}-\\zeta ^{-4},\n$$\nand $\\omega = {\\mathrm{e}}^{\\frac{2\\pi{\\mathrm{i}}}{5}}$ is the minimal phase $5$th root of\nunity; see~\\cite{davis1979circulant}.\nA simple computation shows that\n$$\n\\widehat{L}(1)=0, \\quad\n\\widehat{L}(\\omega)=\\widehat{L}(\\omega^4)=2-2\\cos\\frac{2\\pi}{5}, \\quad\n\\widehat{L}(\\omega^2)=\\widehat{L}(\\omega^3)=2-2\\cos\\frac{4\\pi}{5}, \n$$\nso that the Fiedler value $\\widehat{L}(\\omega)$ has multiplicity 2.\n\nTo explore this situation we performed the following numerical experiment.\nWe considered 10000 random linear combinations of an orthonormal basis for the\neigenspace corresponding to the Fiedler value. This produces a set of random\nvectors, belonging to a plane immersed in ${\\mathbb{R}}^5$, which can all be considered\nas legitimate ``Fiedler vectors''.\n\nEach vector was sorted, and the corresponding permutations of indexes were\nstored in the columns of a matrix.\nIn the end, all the repeated permutations were removed.\nWe obtained 10 permutations, reported in the columns of the following matrix\n\\begin{equation}\\label{permat}\n\\begin{bmatrix}\n3 & 2 & 5 & 3 & 2 & 4 & 4 & 1 & 5 & 1 \\\\\n2 & 1 & 4 & 4 & 3 & 3 & 5 & 5 & 1 & 2 \\\\\n4 & 3 & 1 & 2 & 1 & 5 & 3 & 2 & 4 & 5 \\\\\n1 & 5 & 3 & 5 & 4 & 2 & 1 & 4 & 2 & 3 \\\\\n5 & 4 & 2 & 1 & 5 & 1 & 2 & 3 & 3 & 4\n\\end{bmatrix}.\n\\end{equation}\nThey are much less than the $5!=120$ possible permutations, and reduce to 5 if\nwe remove the columns which are the reverse of another column.\nThis confirms our conjecture: when a Fiedler value is multiple some constraints\nare imposed on the admissible permutations of the units.\n\nIt is relevant to notice that matrix \\eqref{permat} does not contain the cyclic\npermutations of the units depicted in Figure~\\ref{fig:cycle}.\nIndeed, the spectral algorithm aims at moving the nonzero components close to\nthe main diagonal, and this contrasts with the presence of nonzeros in the\nelements $f_{51}$ and $f_{15}$ of matrix \\eqref{cycleF}.\nAll permutations contained in \\eqref{permat}, when applied to the similarity\nmatrix $F$, produce the same matrix\n$$\n\\tilde{F} = \\begin{bmatrix}\n2 & 1 & 1 & 0 & 0 \\\\\n1 & 2 & 0 & 1 & 0 \\\\\n1 & 0 & 2 & 0 & 1 \\\\\n0 & 1 & 0 & 2 & 1 \\\\\n0 & 0 & 1 & 1 & 2\n\\end{bmatrix},\n$$\nwhich exhibits a smaller bandwidth than \\eqref{cycleF}.\n\nThis experiment can be reproduced by executing the script \\texttt{mfiedval.m},\nwhich is found in the \\texttt{demo} sub-directory.\n\n\n\\section{Numerical experiments}\\label{sec:numexp}\n\nIn this section we illustrate the application of the \\texttt{PQser} toolbox to some\nnumerical examples.\nThe experiments can be repeated by running the related Matlab scripts located\nin the \\texttt{demo} sub-directory of the toolbox.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width=.30\\textwidth]{bornmat}\\hfil \n\\includegraphics[width=.30\\textwidth]{bornmix}\\hfil \n\\includegraphics[width=.30\\textwidth]{bornord}\n\\caption{Processing of the Bornholm data set: the spy plot on the left shows\nthe original matrix, a permuted version is reported in the central graph, on\nthe right we display the matrix reordered by the spectral algorithm.}\n\\label{bornholm}\n\\end{center}\n\\end{figure}\n\nThe first example is the numerical processing of the Bornholm data set,\npresented in Table~\\ref{tab:adjacency}.\nWe randomly permute the rows of the adjacency matrix and apply the\nspectral algorithm to the similarity matrix associated to the permuted matrix. \nThe resulting PQ-tree contains just a Q-node, so there is only one solution\n(actually, this is a proof that the matrix is pre-R) which we use to reorder\nthe permuted matrix.\nThe computational code is contained in the file \\ttt{exper1.m}.\n\nFigure~\\ref{bornholm} reports the spy plots which represent the nonzero\nentries of the initial matrix, its permuted version, and the final reordering.\nIt is immediate to observe that the lower band of the reordered matrix is\nslightly narrower than the initial matrix, showing that the spectral algorithm\nwas able to improve the results obtained empirically by archaeologists.\n\nThe second example concerns the comparison of the spectral algorithm with its\nparallel version in the solution of a large scale problem.\nThe experiments were performed on a dual Xeon CPU E5-2620 system (12 cores),\nrunning the Debian GNU\/Linux operating system and Matlab 9.2.\n\nThe function \\ttt{testmatr} of the toolbox allows the user to create a block\ndiagonal matrix, formed by $m$ banded blocks whose size is chosen using a\nsecond input parameter.\nThe matrix is randomly permuted in order to hide its reducible structure.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width=.49\\textwidth]{exper2time}\\hfil \n\\includegraphics[width=.47\\textwidth]{exper2spup}\n\\caption{Comparison between the sequential and the parallel versions of\nAlgorithm~\\ref{alg:spectrsort}: on the left the execution time in seconds, on\nthe right the parallel speedup, defined as the ratio between the sequential and\nthe parallel timings. The test matrix is of dimension $2^{15}=32768$, the size\nof each reducible block is $2^j$, where $j$ is the index reported in the\nhorizontal axis.}\n\\label{exper2}\n\\end{center}\n\\end{figure}\n\nWe let the size of the problem be $n=2^{15}=32768$ and, for $j=1,2\\ldots,15$,\nwe generate a sequence of test matrices containing $n\\cdot 2^{-j}$ blocks, each\nof size $2^j$.\n\nWe apply the function \\ttt{spectrsort} that implements\nAlgorithm~\\ref{alg:spectrsort} to the above problems, as well as its parallel\nversion \\ttt{pspectrsort}, and record the execution time; see the file\n\\ttt{exper2.m}.\nThe number of processors available on our computer was 12.\n\nThe graph on the left of Figure~\\ref{exper2} shows that there is a significant\nadvantage from running the toolbox on a parallel computing system when the\nnetwork associated to the problem is composed by a small number of large\nconnected components.\nThis is confirmed by the plot of the parallel speedup, that is, the ratio\nbetween the timings of the sequential and the parallel implementations,\ndisplayed in the graph on the right in the same figure.\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[width=.30\\textwidth]{exper3data}\\hfil \n\\includegraphics[width=.30\\textwidth]{exper3spec}\\hfil \n\\includegraphics[width=.30\\textwidth]{exper3rcm}\n\\caption{Bandwidth reduction of a sparse matrix of size 1024: the three spy\nplots display the initial matrix, the reordered matrix resulting from the\nspectral algorithm, and the one produced by the \\ttt{symrcm} function of\nMatlab.}\n\\label{exper3}\n\\end{center}\n\\end{figure}\n\nTo conclude, we consider another important application of the reordering\ndefined by the Fiedler vector of the Laplacian, namely, the reduction of the\nbandwidth for a sparse matrix; see~\\cite{barnard1995spectral}.\n\nWe generate a sparse symmetric matrix of size $n=1024$, having approximately\n0.2\\% nonzero elements, and reorder its rows and columns by\nAlgorithm~\\ref{alg:spectrsort}.\nNotice that in this case the spectral algorithm must be applied to a matrix\nwhose elements are taken in absolute value.\nThe computation is described in the script \\ttt{exper3.m}.\n\nThe resulting matrix is depicted by displaying its nonzero pattern in\nFigure~\\ref{exper3}, where it is compared to the reverse Cuthill-McKee\nordering, as implemented in the \\ttt{symrcm} function of Matlab.\nThe spectral algorithm appears to be less effective than \\ttt{symrcm},\nleading to a reordered matrix with a wider band.\nThis is due to the fact that \\ttt{spectrsort} aims at placing the largest\nentries close to the diagonal, and this does not necessarily produce the\nmaximal bandwidth reduction.\nExperimenting with sparser matrices we observed that often the two methods\nproduce similar results.\n\nWe remark that, also in this application, the presence of a multiple Fiedler\nvalue may constitute a problem.\nFor example, we were not able to correctly process the Matlab test matrix\n\\ttt{bucky} (the connectivity graph of the Buckminster Fuller geodesic dome),\nbecause the associated Laplacian possesses a triple Fiedler value.\n\n\n\\section{Conclusions}\\label{sec:last}\n\nIn this paper we present a new Matlab toolbox principally aimed to the\nsolution of the seriation problem, but which can be applied to other related\nproblems.\n\nIt is based on a spectral algorithm introduced in~\\cite{atkins1998spectral},\nand contains an implementation of PQ-trees as well as some tools for their\nmanipulation, including an interactive visualization tool.\nThe implemented algorithm includes the possibility to choose between a small\nscale and a large scale algorithm for the computation of the Fiedler vector,\nand to detect equal components in the same vector according to a chosen\ntolerance. Further, a parallel version of the method is provided.\n\nWe also point out the importance of the presence of multiple Fiedler values, a\nproblem which has not been considered before in the literature and which has a\nsignificant influence on the computation of an approximate solution to the\nseriation problem.\n\nThe use of the toolbox is illustrated by a few practical examples, and its\nperformance is investigated through a set of numerical experiments, both of\nsmall and large scale.\n\n\n\\section{Acknowledgements}\n\nWe would like to thank Matteo Sommacal for pointing our attention to the\nproblem of seriation and to its application in archaeology.\nThe paper~\\cite{ps05}, which he coauthored, was our principal source of\ninformation when the research which lead to this paper started.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\n Herbig-Haro (HH) jets are the optical manifestations of\ncollimated outflows from young stellar objects (YSOs) in their early\nstages of evolution and, along with typical HH objects, are apparent\ntracers of active star formation. YSOs, unfortunately, are usually deeply\nembedded in dense, extended envelopes and\/or opaque molecular cloud\ncores that strongly impede efforts for investigations in the optical \nregime and make our view of the early stages of star formation still \nquite a puzzle. In some cases, however, YSOs are located within\nthe confines of HII regions (Reipurth et al. 1998) and their power sources\nsuffer very low extinction and are therefore optically visible. Similar\njets immersed in a photoionized medium were identified in the Orion\nnebula and near the late B stars in NGC~1333 (Bally et al. 2000;\nBally \\& Reipurth 2001).\nHowever, some properties of photoionized jets in HII regions are\nfound to be highly dependent on their environment. Those immersed in the \noutskirts of HII regions, or near soft-UV radiation sources such as late B stars,\ntend to emerge as bipolar jets which usually share a ``C\" shaped appearance\n(Bally et al. 2000; Bally \\& Reipurth 2001), whereas jets deeply embedded \nin HII regions \nendure a more disruptive environment and seem to favor highly asymmetric\nor even monopolar jet formation (Reipurth et al. 1998).\n\nThe Rosette Nebula is a spectacular HII region excavated by the strong\nstellar winds from O stars in the center of the young open\ncluster NGC~2244, which has an age of about 3 x $10^6$ yr (Ogura \\& Ishida 1981).\nThis on-going star forming region (Meaburn \\& Walsh 1986;\nClayton et al. 1998; Phelps \\& Lada 1997; Li et al. 2002; Li 2003) is located\nat a distance of $\\sim$1500 pc at the tip of a giant molecular cloud\ncomplex with an extent of around 100 pc (Dorland \\& Montmerle 1987).\nIn this paper, we present the discovery of an optical jet system embedded \nin the Rosette Nebula with a fascinating morphology that displays exceptional \nnew features not known before.\n\n\\section{Observations and Data Reduction}\n\n\\subsection{Narrowband Imaging}\n\nNarrow band images of the Rosette Nebula were obtained 3 March 1999 with the\nKitt Peak National Observatory 0.9-meter telescope and the 8k$\\times$8k MOSAIC I\ncamera (Muller et al. 1998). Images were taken with the H${\\alpha}$,\n[OIII] and [SII] filters, whose central wavelengths\/FWHMs are 6569\\AA\/80\\AA, 5021\\AA\/55\\AA,\nand 6730\\AA\/80\\AA, respectively. For each filter, five exposures of 600~seconds\nwere taken, each image slightly offset to fill in physical gaps between the\nMOSAIC CCDs. The pixel scale is 0.423\\arcsec~pixel$^{-1}$, resulting in a 59\\arcmin~x~59\\arcmin\\\nfield of view. An astrometric solution for these images was determined with\nthe use of stars in the Guide Star Catalog 1.2 (Roser et al. 1998). The astrometric\naccuracy in the optical images presented is $\\sim$1\\arcsec.\n\n\\subsection{Optical Spectroscopy}\n\nOptical spectroscopy of the jet system, with a\n200\\AA~mm$^{-1}$, 4.8\\AA~pixel$^{-1}$ dispersion, and a 2.5\\arcsec slit,\nwas carried out with the 2.16m telescope of the National Astronomical\nObservatories of China on 10 and 16 January 2003, with slit positions\nalong and orthogonal to the jet direction. A possible\nphysical companion of the apparent power source was also observed with the\northogonal slit position to avoid contamination from the jet system.\nAn OMR (Optomechanics Research Inc.) spectrograph and a Tektronix 1024${\\times}$1024 CCD\ndetector were used in this run of observations.\n\nThe spectral data were reduced following\nstandard procedures in the NOAO Image Reduction and Analysis Facility\n(IRAF, version 2.11) software package. The CCD reductions included bias\nand flat-field correction, successful nebular background subtraction, and\ncosmic rays removal. Wavelength calibration was performed based on helium-argon\nlamps exposed at both the beginning and the end of the observations every night.\nFlux density calibration of each spectrum was conducted based on observations of at\nleast two of the KPNO spectral standards (Massey et al. 1988) per night.\n\n\\section{Results and Discussion}\n\n\\subsection{The Jet System}\n\nAn optical jet system with a striking morphology was identified\nwhen scrutinizing the narrow band images of the Rosette Nebula (Fig. 1). A\nprominent jet with a position angle of 312\\arcdeg\\ is clearly seen traced back\nto a faint visible star, indicating rather low extinction along the\nline of sight. The circumstellar envelope of the YSO\nmight have been stripped away, leaving a photoablating disk exposed\nto the strong UV radiation field of the exciting sources in the HII region.\nThe optical jet remains highly collimated for a projected distance\nof $>$8000~AU from the energy source if a mean distance\nof 1500 pc to Rosette is adopted. At the end of the collimated jet is a \nside-impacted shock structure, probably due to severe radiation pressure from the\nstrong UV field present and strong stellar winds encountered. \nA prominent knot or compact mass bullet resembling a point source is immediately\nnoticed in the highly collimated part of the jet with a projected separation of\n7\\arcsec\\ from the source. Another one or \ntwo knots can be marginally noticed in the bow shaped extension of the jet, \nindicating episodic eruption events from the driving source.\n\n We infer the possible existence of \na counterjet from the presence of a promising bow shock\nstructure on the opposite side of the jet. Successive episodes\nof counterjet ejection may have ceased or may be too faint. \nAlternatively, this structure may simply be shocked\ninterstellar material that was swept up and photoionized by the strong\nradiation field, which happen to be projected onto the vicinity of\nthe jet exciting source. Another possible explanation is that it is\ndisrupted jet material from a possible T Tauri companion (please refer to\nFig.~1) or from nearby young stars. If it is indeed a \ndegenerated counterjet, however, it may be the only existing observational \nevidence for how bipolar jets evolve into monopolar or \nhighly asymmetric jets.\nWe further argue that the long-puzzling high-velocity components from\nthe photoionized gas fronts aggregated in the Rosette (Meaburn \\& Walsh 1986; Clayton \\& Meaburn 1995; Clayton, Meaburn \\& Lopez et al. 1998) could\nactually be from rapidly disrupted young disk-jet systems in the inner\npart of the HII region. Diffuse [OIII] emission is prominent from \nall parts of the jet system, which is likely due to its photo-dissipating nature. \n[SII] emission from the highly-collimated part\nof the jet decreases rapidly from the base of the jet and is undetected\nin the shocked structures, indicating a steep decline of internal\nshock effects or rather overwhelming external ionization.\n\nSpectroscopic observation along the jet direction reveals prominent H$\\alpha$ \n($W_{\\lambda} \\sim$~35\\AA) as well as [SII] emission. The presence of\nsignificant [OIII] $\\lambda\\lambda$4959, 5007 line emission \nconfirms our suspicion that the jet is in a high-excitation state.\nFig. 2 clearly shows the existence of a weak counterjet as discussed above.\nNote that enhanced [NII] emission is detected along the jet, which is not\nshown here.\n\n\\subsection{The exciting source}\n\nSpectroscopic observations of the exciting source reveal primarily the \nspectrum of a normal F8Ve star with weak H$\\alpha$ ($W_{\\lambda} \\sim$~6.7\\AA)\nand prominent [OIII] emission. However, the spectrum also shows a \nsignificant red-displaced absorption component in H$\\alpha$, with \na receding velocity of 500$\\pm$50~km~s$^{-1}$ (Fig. 3). Two weak\nbut significant emission features with unclear nature are also present immediately\nto the red of the rest H$\\alpha$ emission and its red-displaced absorption. One is\ncentered at 6582.5 \\AA~ with an accuracy of $\\pm$ 1 \\AA~, and can be \nthe pronounced [NII] $\\lambda$ 6583.6 emission detected also along the jet. \nThis gives a high [NII] to H$\\alpha$ flux ratio of about 1:3.\nAlbeit the other with a central wavelength of 6613.3 \\AA~ cannot be identified \nwith any known emission lines with rest velocities. A possible interpretation\nis a red-displaced emission component of H$\\alpha$. This however would suggest\nan extraordinarily high receding velocity of 2300$\\pm$50~km~s$^{-1}$ and should\nbe treated with great caution. The possible ejection of high velocity, compact \nmass bullets from the energy source, if true, can be partially supported by the \nclear existence of a compact knot in the highly collimated part of the jet\nas mentioned in the previous subsection, and may offer a big challenge to \ncurrently available disk-jet models. This scenario, however, seems to match well\nwith Meaburn's (2003) recent speculations on the formation of ablated jets\nin HII regions.\n\n Spectral energy distribution (Kurucz 1991) fitting of the energy source\nbased on spectroscopic observations and data extracted from 2MASS indicated a relic\ndisk mass of only 0.006 M$_{\\odot}$, consistent with a photoablating\norigin of the system. It is noteworthy that Inverse P Cygni (IPC) profiles\nseldom appear in low Balmer\nemission lines such as H$\\alpha$. The most well known case is the YY Orionis\ntype, weak-lined T Tauri star T Cha (Alcala, Covino \\& Franchini et al. 1993), \nfrom which, at most, a relic circumstellar\ndisk can be expected. An IPC profile in H$\\alpha$ is usually attributed to its\noptically thin nature, which normally leads to the conclusion of a high\ninclination angle of the relic disk. Furthermore, this must happen under\ntight conditions such as low-density in the residual circumstellar\nmaterial and weak H$\\alpha$ emission (Alcala, Covino \\& Franchini et al. 1993).\nBoth are consistent with properties of this jet system.\n\n\\subsection{Implications}\n\nA similar red-displaced absorption profile is also extracted \naround the power source from the spectra taken along the jet direction, \nwhich helps to elucidate the reliability of the spectral reduction and time \nelapsed mass accretion of the central YSO. The simultaneous existence of \nsignatures of mass inflow and outflow, along with the absence of apparent \nveiling or blue excess, the absence of signatures of chromospheric \nactivity, and its weak or absent Balmer emission in combination is strong evidence \nthat the forming star is highly affected by its location in the strong UV \nradiation fields. Indeed, Reipurth et al. (1998) state that external ionization may \nhelp on the feeding and\/or launching of the jet, which makes jet formation in such\nsystems survive. We further speculate that the aforementioned effects eventually \nchange the configuration of the stellar-disk magnetosphere \nsuch that photodissipated material in the relic disk is \neasily loaded onto the magnetic channels, especially on the side facing the\nstrong radiation fields. It is conceivable that most or all of this material could \nhave been ejected in the form of a jet on the opposite side of the \nmass loading, instead of eventually ramming into the\ncontracting infant star embarrassed with starvation. \nIt's therefore hard for the energy source to further grow in mass, its accretion\ndisk could be dissipated on a comparatively short time scale and its evolution\ntoward the main sequence is either highly accelerated or in some cases even led \nto the formation of a failed star that would have evolved into a higher mass\nstar under normal conditions.\nMany young, very low mass stars and brown dwarfs \nare found to show evidence of mass accretion from its circumstellar environment \n(Fernandez \\& Comeron 2001; Luhman, Briceno \\& Stauffer 2003; Jayawardhana, \nMohanty \\& Basri 2003; White \\& Basri, 2003). This study presents\nfurther implications on how incipience of \nmassive stars in giant molecular clouds inhibits further generations of low-mass \nstar formation. It could also well serve as an inspiring case of how isolated \nsubstellar\/planetary mass objects in regions of massive star formation, especially \nthose found in HII regions (Zapatero Osorio et al. 2000), were prohibited from \ngrowing in mass, ceased their accretion process and came into being.\n\n{\\flushleft \\bf Acknowledgments~}\n\nWe are grateful to an anonymous referee for the \nmany helpful comments on the paper. Thanks prof. You-Hua Chu very much\nfor her valuable comments and suggestions. Appreciations \nto prof. W. P. Chen and W. H. Ip for their kind accommodations and help \nduring my two years stay at NCU. This work has made use of the 2MASS database.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}