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+{"text":"\\section{Syst\\`emes int\\'egrables quantiques}\n\nLe {\\bf mod\\`ele \\`a 6 sommets} est un c\\'el\\`ebre mod\\`ele de {\\bf physique statistique} introduit par Pauling en 1935, qui permet notamment \nde d\\'ecrire le cristal de la glace  (voir \\cite{BaxterBook}). Il est r\\'ealis\\'e sur un r\\'eseau dont chaque\nsommet est reli\\'e \\`a 4 autres sommets. Un \\'etat du syst\\`eme est une orientation\ndes ar\\^etes telle qu'\\`a chaque sommet arrivent exactement 2 fl\\^eches (Figure 1). \nLes fl\\^eches repr\\'esentent l'orientation des mol\\'ecules d'eau du cristal\nles unes par rapport aux autres. \nIl y a 6 configurations\npossibles \\`a chaque sommet (Figure 2), ce qui justifie l'appellation de ce mod\\`ele.\n~\\begin{center}\n\\begin{figure}\\label{orientation}\n \\hspace{4.5cm}  \\epsfig{file=orientation.eps,width=0.3\n   \\linewidth}\n      \\caption{Une orientation d'un r\\'eseau (mod\\`ele \\`a 6 sommets).}\n\\end{figure}\n\\end{center}\n\\begin{center}\n\\begin{figure}\\label{6vertex}\n  \\hspace{1cm} \\epsfig{file=6vertex.eps,width=0.8\\linewidth}\n      \\caption{6 configurations possibles \\`a chaque sommet.}\n\\end{figure}\n\\end{center}\n\n\nL'\\'etude du mod\\`ele de la glace est fortement li\\'ee \\`a celle d'un autre mod\\`ele, cette fois-ci en {\\bf physique statistique quantique}, appel\\'e {\\bf mod\\`ele $XXZ$} de Spin $1\/2$, dit de Heisenberg quantique (1928). Il s'agit d'une variante en physique quantique du mod\\`ele d'Ising (1925) (voir \\cite{JM}), qui mod\\'elise des cha\\^ines de spins magn\\'etiques quantiques\nayant deux \\'etats classiques, haut ou bas (Figure 3).\n\\begin{center}\n\\begin{figure}\\label{spin}\n \\hspace{4.5cm}  \\epsfig{file=spin2.eps,width=0.3\n   \\linewidth\n   }\n      \\caption{Etats d'un Spin $1\/2$ (haut ou bas).}\n\\end{figure}\n\\end{center}\n\nCes deux mod\\`eles, mod\\`ele \\`a 6 sommets et mod\\`ele $XXZ$, figurent parmi les plus \\'etudi\\'es\nen physiques statistique et quantique. Les structures math\\'ematiques qui les sous-tendent sont tr\\`es proches.\nEn d\\'epit de leur formulation assez \\'el\\'ementaire,\nils sont extr\\^emement riches et leur analyse a une tr\\`es longue histoire.\n\nEn physique statistique (quantique), le comportement du syst\\`eme est contr\\^ol\\'e par la {\\bf fonction de partition} $\\mathcal{Z}$ \\footnote{En physique statistique, la fonction de partition s'exprime comme la somme $\\sum_j \\text{exp}(-E_j\/(k_BT))$ sur tous les \\'etats $j$ du syst\\`eme, o\\`u $E_j$ est l'\\'energie de l'\\'etat $j$, $T$ est la temp\\'erature du syst\\`eme et $k_B$ la constante de Boltzmann. En physique quantique, la somme est remplac\\'ee par une trace $\\text{Tr}_W(\\text{exp}(-E\/(k_BT)))$ o\\`u $E$ est l'op\\'erateur \\og hamiltonien\\fg{}  qui agit sur l'espace $W$ des \\'etats quantiques du syst\\`eme.}, qui permet d'obtenir les grandeurs mesurables\\footnote{Une grandeur mesurable $Q$ est obtenue comme moyenne pond\\'er\\'ee sur les \\'etats $\\frac{\\sum_j  \\text{exp}(-E_j\/(k_BT)) Q_j}{\\mathcal{Z}}$ des valeurs $Q_j$ sur chaque \\'etat $j$.}. Cette fonction\n$\\mathcal{Z}$ est tr\\`es difficile \\`a calculer en g\\'en\\'eral. La m\\'ethode de la {\\bf matrice de transfert}\nest un proc\\'ed\\'e pour tenter de la d\\'eterminer : il s'agit d'\\'ecrire $\\mathcal{Z}$ comme trace\nd'un op\\'erateur $\\mathcal{T}$ (la matrice de transfert) agissant sur l'{\\bf espace des \\'etats} $W$ : \n$$\\mathcal{Z} = \\on{Tr}_W (\\mathcal{T}^M).$$\nIci $M$ est un entier associ\\'e \\`a la taille du r\\'eseau du mod\\`ele.\nAinsi, pour trouver $\\mathcal{Z}$, il suffit d'obtenir les valeurs propres $\\lambda_j$ de $\\mathcal{T}$ :\n$$\\mathcal{Z} = \\sum_j \\lambda_j^M.$$\nLe spectre $\\{ \\lambda_j \\}_j$ de $\\mathcal{T}$ est appel\\'e le {\\bf spectre du syst\\`eme quantique}.\n\nDans un c\\'el\\`ebre article s\\'eminal de 1971, inspir\\'e notamment par les travaux de Bethe (1931), Baxter \\cite{Baxter} a compl\\`etement r\\'esolu ce probl\\`eme\\footnote{Baxter a introduit la m\\'ethode puissante des \\og Q-op\\'erateurs\\fg{} qui lui a \\'egalement permis de r\\'esoudre le mod\\`ele \\og \\`a 8 sommets\\fg, plus complexe. Le mod\\`ele \\`a 6 sommets avait aussi \\'et\\'e r\\'esolu par d'autres m\\'ethodes, notamment dans les travaux de Lieb et Sutherland (1967).}. Gr\\^ace \\`a une \\'etude\ntr\\`es pr\\'ecise il a notamment montr\\'e\nque les valeurs propres $\\lambda_j$ de $\\mathcal{T}$ ont une structure tout \\`a fait remarquable : elles\ns'expriment sous la forme\n\\begin{equation}    \\label{relB}\n\\lambda_j = A(z) \\frac{Q_j(zq^2)}{Q_j(z)} + D(z)  \\frac{Q_j(zq^{-2})}{Q_j(z)},\n\\end{equation}\no\\`u $z,q\\in\\CC^*$ sont des param\\`etres du mod\\`ele (respectivement spectral et quantique),\n$A(z)$ et $D(z)$ sont des fonctions \\og universelles\\fg{} (au sens o\\`u elles ne d\\'ependent pas de la valeur propre\n$\\lambda_j$). La fonction $Q_j(z)$ d\\'epend de la valeur propre, mais c'est un polyn\\^ome.\nLa relation (\\ref{relB}) est la fameuse {\\bf relation de Baxter} (ou \\og relation $TQ$ de Baxter\\fg).\nLes polyn\\^omes $Q_j$ sont appel\\'es {\\bf polyn\\^omes de Baxter}.\n\n\\medskip\n\nEn r\\'esultent alors naturellement les questions suivantes :\n\n- Y a-t-il une explication pour l'existence de la relation de Baxter  ?\n\n- Une expression analogue avec des polyn\\^omes permet-elle de d\\'ecrire le spectre d'autres syst\\`emes quantiques ?\n\n\\medskip\n\nUne conjecture formul\\'ee en 1998 par Frenkel-Reshetikhin \\cite{Fre} affirme que la deuxi\\`eme question doit avoir une r\\'eponse positive. Comme on ne peut esp\\'erer effectuer en g\\'en\\'eral le calcul d\\'etaill\\'e de Baxter qui est connu pour le mod\\`ele $XXZ$, c'est en r\\'epondant \\`a la premi\\`ere question que nous pouvons d\\'emontrer cette conjecture. Pour ce faire, \\'etudions les structures math\\'ematiques, alg\\'ebriques, sous-jacentes \\`a la th\\'eorie.\n\n\\section{Groupes quantiques et leurs repr\\'esentations}\n\nLes {\\bf groupes quantiques} sont graduellement apparus au cours des ann\\'ees 1970, en particulier dans les travaux de l'\\'ecole de Leningrad, comme le cadre naturel math\\'ematique pour \\'etudier les matrices de transfert. Drinfeld \\cite{Dri} et Jimbo \\cite{J} ont ind\\'ependamment d\\'ecouvert une formulation alg\\'ebrique uniforme sous forme d'{\\bf alg\\`ebres de Hopf}. Il s'agit d'un des r\\'esultats cit\\'es pour la m\\'edaille Fields de Drinfeld en 1990.\n\nPour introduire les groupes quantiques de Drinfeld-Jimbo, consid\\'erons d'abord un objet tr\\`es classique, une {\\bf alg\\`ebre de Lie} (simple) complexe de dimension finie. Il s'agit d'un espace vectoriel de dimension finie $\\mathfrak{g}$ muni d'un {\\bf crochet de Lie}, c'est-\\`a-dire d'une application bilin\\'eaire antisym\\'etrique \n$$[,]:\\mathfrak{g}\\times \\mathfrak{g}\\rightarrow \\mathfrak{g}$$ \nsatisfaisant la formule de Jacobi\n$$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0\\text{ pour tous }x,y,z\\in\\mathfrak{g}.$$\nL'exemple le plus simple, mais n\\'eanmoins non trivial car il correspond au mod\\`ele $XXZ$, est celui de l'alg\\`ebre de Lie $\\mathfrak{g} = sl_2$ : c'est l'espace des matrices complexes $2\\times 2$ de trace nulle, muni du crochet naturel \n$$[M,N] = MN - NM$$ \npour lequel il est clairement stable. Pour les g\\'en\\'erateurs lin\\'eaires\n$$E = \\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\text{ , }F = \\begin{pmatrix}0&0\\\\1&0\\end{pmatrix}\n\\text{ , }H = \\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix},$$\non a par exemple la relation\n\\begin{equation}\\label{crochet}[E,F] = H.\\end{equation}\n\nCes alg\\`ebres de Lie ont des analogues naturelles de dimension infinie, les {\\bf alg\\`ebres de lacets} \n$$\\hat{\\Glie} = \\Glie \\otimes \\CC[t^{\\pm 1}],$$\navec le crochet de Lie d\\'efini par\n$$[x\\otimes f(t),y\\otimes g(t)] = [x,y]\\otimes (fg)(t)\\text{ pour $x,y\\in\\Glie$ et $f(t),g(t)\\in \\CC[t^{\\pm 1}]$},$$ \nce qui revient \\`a remplacer le corps $\\CC$ par l'anneau des polyn\\^omes de Laurent complexes \n$$\\CC[t^{\\pm 1}] = \\left\\{\\sum_{N\\leq i\\leq M} a_i t^i| N,M\\in\\ZZ, a_i\\in\\CC\\right\\}.$$ \nCes alg\\`ebres sont des quotients d'{\\bf alg\\`ebres de Kac-Moody affines}, qui ont des propri\\'et\\'es alg\\'ebriques semblables \\`a celles des alg\\`ebres de Lie simples de dimension finie (notamment une pr\\'esentation analogue \\`a celle de Serre pour $\\Glie$, comme l'ont montr\\'e Kac (1968) et Moody (1969), voir \\cite{ka}). Elles ont \\'et\\'e \\'etudi\\'ees intensivement pour leurs diverses applications en math\\'ematiques et physique math\\'ematique.\n\nMaintenant, pour \\'etudier les syst\\`emes quantiques qui nous int\\'eressent, ces alg\\`ebres de Lie classiques doivent \\^etre {\\bf quantifi\\'ees}, c'est-\\`a-dire d\\'eform\\'ees en tenant compte du param\\`etre quantique \n$$q = \\text{exp}(h)\\in\\CC^*,$$ \no\\`u $h$ est un analogue de la grandeur de Planck ($q$ sera bien identifi\\'e au param\\`etre quantique de la relation (\\ref{relB})). \nOn retrouve les structures classiques pour $h\\rightarrow 0$, donc $q\\rightarrow 1$. {\\it On supposera dans la suite que $q$ n'est pas une racine de l'unit\\'e.}\n\nBien qu'une telle quantification des alg\\`ebres de Lie $\\Glie$ ou $\\hat{\\Glie}$ elles-m\\^emes ne soit pas connue, Drinfeld et Jimbo ont d\\'ecouvert qu'il existe une quantification naturelle de leurs {\\bf alg\\`ebres enveloppantes} respectives $\\mathcal{U}(\\Glie)$ et $\\mathcal{U}(\\hat{\\Glie})$ (alg\\`ebres universelles d\\'efinies \\`a partir des alg\\`ebres de Lie, par exemple en rempla\\c cant dans la pr\\'esentation de Serre les crochets $[x,y]$ par des expressions alg\\'ebriques $xy - yx$ dans l'alg\\`ebre). On obtient alors les groupes quantiques $\\mathcal{U}_q(\\Glie)$, $\\mathcal{U}_q(\\hat{\\Glie})$ qui d\\'ependent\\footnote{Elles peuvent \\^etre d\\'efinies comme des alg\\`ebres sur $\\CC[[h]]$.} du param\\`etre quantique $q$, voir \\cite{CP}. \n\nPar exemple dans $\\mathcal{U}_q(sl_2)$ la relation (\\ref{crochet}) devient\n$$[E,F] = \\frac{e^{hH} - e^{-hH}}{q - q^{-1}}$$\nqui tend bien vers $H$ quand $h$ tend vers $0$.\n\nLe cas des {\\bf alg\\`ebres affines quantiques} $\\mathcal{U}_q(\\hat{\\Glie})$ est particuli\\`erement remarquable car Drinfeld \\cite{Dri2} a d\\'emontr\\'e\\footnote{La preuve a \\'et\\'e pr\\'ecis\\'ee par la suite par Beck puis par Damiani.} qu'elles peuvent non seulement \\^etre obtenues comme quantification de $\\mathcal{U}(\\hat{\\Glie})$, mais \\'egalement, par un autre proc\\'ed\\'e, comme {\\bf affinisation} du groupe quantique $\\mathcal{U}_q(\\Glie)$. C'est la {\\bf r\\'ealisation de Drinfeld} des alg\\`ebres affines quantiques. Ceci peut \\^etre \\'enonc\\'e dans le diagramme \\og commutatif\\fg{} suivant :\n$$\\xymatrix{ &\\hat{\\Glie}\\ar@{-->}[dr]^{\\text{Quantification}}&   \n\\\\ \\Glie \\ar@{-->}[ur]^{\\text{Affinisation}}\\ar@{-->}[dr]_{\\text{Quantification}}& &  \\U_q(\\hat{\\Glie})\n\\\\ & \\U_q(\\Glie)\\ar@{-->}[ur]_{\\text{Affinisation quantique}}& }$$\nCe th\\'eor\\`eme, qui revient \\`a donner deux pr\\'esentations isomorphes de $\\mathcal{U}_q(\\hat{\\Glie})$, est un analogue quantique du th\\'eor\\`eme classique de Kac et Moody. Il s'agit d'une bonne indication de l'importance de ces alg\\`ebres d'un point de vue alg\\'ebrique.\n\nLes alg\\`ebres affines quantiques $\\U_q(\\hat{\\Glie})$ ont en fait une structure beaucoup plus riche, ce sont des alg\\`ebres de Hopf. Elles sont notamment munies d'une {\\bf comultiplication} (qui est une op\\'eration duale de la multiplication), c'est-\\`a-dire d'un morphisme d'alg\\`ebre\n\\begin{equation}\\label{coproduit}\\Delta : \\U_q(\\hat{\\Glie}) \\rightarrow \\U_q(\\hat{\\Glie})\\otimes \\U_q(\\hat{\\Glie}).\\end{equation}\nMais surtout, $\\U_q(\\hat{\\Glie})$ poss\\`ede une {\\bf $R$-matrice universelle}, c'est-\\`a-dire un \\'el\\'ement (canonique) dans le carr\\'e tensoriel\\footnote{En fait, dans une l\\'eg\\`ere compl\\'etion du carr\\'e tensoriel.}\n$$\\mathcal{R}(z) \\in (\\U_q(\\hat{\\Glie})\\otimes \\U_q(\\hat{\\Glie}))[[z]]$$\nqui est notamment une solution de l'{\\bf \\'equation de Yang-Baxter quantique} :\n$$\\mathcal{R}_{12}(z)\\mathcal{R}_{13}(zw)\\mathcal{R}_{23}(w) = \\mathcal{R}_{23}(w)\n\\mathcal{R}_{13}(zw) \\mathcal{R}_{12}(z).$$\nLes param\\`etres formels $z$, $w$ sont appel\\'es {\\bf param\\`etres spectraux}. Cette \\'equation est \\`a valeurs dans le cube tensoriel \n$$(\\U_q(\\hat{\\Glie}))^{\\otimes 3}[[z, w]].$$\nLes indices dans les facteurs indiquent l'emplacement des termes de la $R$-matrice universelle : \n$$\\mathcal{R}_{12}(z) = \\mathcal{R}(z)\\otimes 1\\text{ , }\\mathcal{R}_{23}(z) = 1 \\otimes \\mathcal{R}(z)...$$\nIl s'agit d'une \\'equation hautement non triviale, li\\'ee aux mouvements de tresses. En effet, dans la figure 4 on retrouve l'\\'equation en lisant de bas en haut et en multipliant\npar un facteur $\\mathcal{R}_{\\alpha\\beta}$ d'indice $(\\alpha,\\beta)$ lorsque le brin $\\alpha$ croise le brin $\\beta$. C'est pour cette raison que la th\\'eorie des repr\\'esentations des groupes quantiques permet de construire des invariants en topologie de basse dimension (notamment les polyn\\^omes de Jones des n\\oe uds). Il s'agit historiquement, avec la construction par Lusztig et Kashiwara de bases canoniques de repr\\'esentations des alg\\`ebres de Lie classiques, d'un des premiers grands succ\\`es de la th\\'eorie des groupes quantiques. Nous n'aborderons pas ces sujets ici pour nous concentrer sur les applications aux syst\\`emes quantiques.\n \\begin{center}\n\\begin{figure}\\label{tresse}\n \\hspace{4cm}  \n {\\epsfig{file=tresse.eps,width=.4\\linewidth}\n}\n      \\caption{Equation de Yang-Baxter}\n\\end{figure}\n\\end{center}\n\nPour d\\'ecrire des solutions de l'\\'equation de Yang-Baxter quantique, on peut sp\\'ecialiser sur des {\\bf repr\\'esentations} de dimension finie de $\\U_q(\\hat{\\Glie})$.\nUne repr\\'esentation (lin\\'eaire) de $\\U_q(\\hat{\\Glie})$ est un espace vectoriel $V$ (ici complexe) muni d'un morphisme d'alg\\`ebre\n$$\\rho_V : \\U_q(\\hat{\\Glie}) \\rightarrow \\text{End}(V).$$\nAutrement dit, l'alg\\`ebre $\\U_q(\\hat{\\Glie})$ agit sur l'espace $V$ par op\\'erateurs lin\\'eaires. \n\nL'\\'etude des repr\\'esentations est un vaste domaine, central en math\\'ematiques, appel\\'e {\\bf th\\'eorie des repr\\'esentations}. En arithm\\'etique par exemple, les repr\\'esentations \nde groupes de Galois jouent un r\\^ole crucial. Elles sont \\'egalement essentielles dans la formulation m\\^eme des principes de la physique quantique car ils font intervenir des repr\\'esentations\nde l'alg\\`ebre des observables.\n\nOn d\\'efinit naturellement la {\\bf somme directe de repr\\'esentations} $(V,\\rho_V)$ et $(V',\\rho_{V'})$ avec l'application $\\rho_{V\\oplus V'} = \\rho_V + \\rho_{V'}$ \\`a valeurs\ndans $\\text{End}(V \\oplus V')$.\n\nLes {\\bf repr\\'esentations simples}, c'est-\\`a-dire qui n'ont pas de sous-repr\\'esentation (sous-espace stable pour l'action de l'alg\\`ebre) propre, sont particuli\\`erement importantes, comme nous allons le voir dans notre \\'etude.\nElles constituent les \\og briques \\'el\\'ementaires\\fg{} de la th\\'eorie des repr\\'esentations. Par exemple, toute repr\\'esentation de dimension finie de $\\U_q(\\Glie)$ est {\\bf semi-simple}, c'est-\\`a-dire isomorphe \\`a une somme\ndirecte de repr\\'esentations simples\\footnote{Ce r\\'esultat d\\'emontr\\'e par M. Rosso et G. Lusztig est un analogue quantique du th\\'eor\\`eme classique de Weyl qui assure que toute repr\\'esentation de dimension finie de $\\U(\\Glie)$ est semi-simple.}. Ce n'est pas le cas\\footnote{Cependant, toute repr\\'esentation $V$ de dimension finie de $\\U_q(\\hat{\\Glie})$ admet une filtration de Jordan-H\\\"older par des sous-repr\\'esentations $V_0 = V\\supset V_1\\supset V_2 \\cdots \\supset V_N = \\{0\\}$  avec les  $V_i\/V_{i+1}$ simples.} pour l'alg\\`ebre affine quantique $\\U_q(\\hat{\\Glie})$.\n\n\nComme $\\U_q(\\hat{\\Glie})$ est munie d'un coproduit (\\ref{coproduit}), pour deux repr\\'esentations $(V,\\rho_V)$ et $(V',\\rho_{V'})$, le produit tensoriel $V\\otimes V'$\nest aussi une repr\\'esentation en utilisant\n$$\\rho_{V\\otimes V'} = (\\rho_V\\otimes \\rho_{V'})\\circ \\Delta : \\U_q(\\hat{\\Glie})\\rightarrow \\text{End}(V)\\otimes \\text{End}(V') = \\text{End}(V\\otimes V').$$\nCette action sur un {\\bf produit tensoriel de repr\\'esentation} sera utile par la suite. Mais ind\\'ependamment on peut faire aussi agir directement la $R$-matrice\nuniverselle sur un carr\\'e tensoriel $V\\otimes V$ pour $V$ une repr\\'esentation de dimension finie de $\\U_q(\\hat{\\Glie})$ : on peut en effet consid\\'erer l'image de la $R$-matrice universelle dans $\\text{End}(V^{\\otimes 2})(z)$\n$$\\mathcal{R}_{V,V}(z) = (\\rho_V\\otimes \\rho_V)(\\mathcal{R}(z))\\in \\text{End}(V)^{\\otimes 2}[[z]] = \\text{End}(V^{\\otimes 2})[[z]].$$\nOn obtient aussi une solution de l'\\'equation de Yang-Baxter quantique, dite {\\bf $R$-matrice}, mais dans l'alg\\`ebre de dimension finie $\\text{End}(V^{\\otimes 2})[[z]]$.\n\nPar exemple, dans le cas $\\Glie = sl_2$, l'alg\\`ebre affine quantique $\\U_q(\\hat{sl_2})$ poss\\`ede une repr\\'esentation de dimension $2$ dite {\\bf repr\\'esentation fondamentale}\net not\\'ee $V_1$. Par le proc\\'ed\\'e d\\'ecrit ci-dessus, elle produit la $R$-matrice suivante\n\\footnote{La solution explicite de l'\\'equation de Yang-Baxter donn\\'ee ici est la $R$-matrice \\og normalis\\'ee\\fg, obtenue en multipliant $\\mathcal{R}_{V_1, V_1}(z)$ par une certaine fonction scalaire de $z$. On peut constater que ses coefficients sont des fonctions m\\'eromorphes en $z$. C'est un ph\\'enom\\`ene g\\'en\\'eral, voir \\cite{efk}.} dans $\\text{End}(V_1^{\\otimes 2})[[z]]$ avec $V_1^{\\otimes 2}$ qui est de dimension $4$ :\n$$\\begin{pmatrix}1&0&0&0\\\\ 0&\\frac{q^{-1}(z-1)}{z-q^{-2}}&\\frac{z(1 - q^{-2})}{z-q^{-2}}&0\\\\0&\\frac{1-q^{-2}}{z - q^{-2}}&\\frac{q^{-1}(z - 1)}{z - q^{-2}}&0\\\\0&0&0&1\\end{pmatrix}.$$\nC'est la $R$-matrice associ\\'ee au mod\\`ele $XXZ$. Mais la th\\'eorie des groupes quantiques en produit beaucoup d'autres, selon qu'on change l'alg\\`ebre de Lie $\\Glie$ ou la repr\\'esentation $V$. \nElles correspondent \\`a autant de syst\\`emes quantiques. \n\nLa {\\bf matrice de transfert} $\\mathcal{T}_V(z)$ est alors d\\'efinie en prenant la trace partielle sur la repr\\'esentation, c'est-\\`a-dire \n\\begin{equation}\\label{transfer}\n\\mathcal{T}_V(z) = ((\\on{Tr}_V \\circ \\rho_V) \\otimes \\on{id})({\\mathcal{R}(z)})\\in \\U_q(\\hat{\\Glie})[[z]].\n\\end{equation}\nLa repr\\'esentation $V$ qui sert \\`a construire la matrice de transfert $\\mathcal{T}_V(z)$ est appel\\'ee {\\bf espace auxiliaire}.\nComme cons\\'equence de l'\\'equation de Baxter, les matrices de transfert commutent, c'est-\\`a-dire que pour une autre repr\\'esentation $V'$ on a\n$$\\mathcal{T}_V(z)\\mathcal{T}_V(z') = \\mathcal{T}_V(z')\\mathcal{T}_V(z)\\text{ dans }\\U_q(\\hat{\\Glie})[[z,z']].$$\nAinsi, les coefficients $\\mathcal{T}_V[N]$ des matrices de transfert, d\\'efinis par\n$$\\mathcal{T}_V(z) = \\sum_{N\\geq 0}z^N \\mathcal{T}_V[N],$$\nengendrent une sous-alg\\`ebre commutative de $\\U_q(\\hat{\\Glie})$. \n\nDonnons-nous une autre repr\\'esentation de dimension finie $W$ de $\\U_q(\\hat{\\Glie})$, dite {\\bf espace des \\'etats}. Les coefficients $\\mathcal{T}_V[N]$ des matrices de transfert agissent donc sur $W$ en une grande famille commutative d'op\\'erateurs. Ainsi, il fait sens de parler des valeurs propres des matrices de transfert $\\mathcal{T}_V(z)$ sur $W$.\n\nDans le cas particulier du mod\\`ele $XXZ$, on rappelle que ${\\mathfrak g} = sl_2$ et $V = V_1$ est une repr\\'esentation fondamentale de dimension $2$.\nL'espace des \\'etats $W$ est un produit tensoriel de repr\\'esentations fondamentales de dimension $2$ et l'image de l'op\\'erateur $\\mathcal{T}_{V_1}(z)$\ndans $\\text{End}(W)[[z]]$ est bien la matrice de transfert de Baxter. Les r\\'esultats de Baxter donnent donc la structure du spectre de $\\mathcal{T}_{V_1}(z)$\nsur $W$ dans ce cas. \n\n\\medskip\n\nQue dire en g\\'en\\'eral ?\n\n\\section{La conjecture du spectre quantique}\n\nEn 1998 \\cite{Fre}, E. Frenkel et N. Reshetikhin ont propos\\'e une nouvelle approche dans le but de g\\'en\\'eraliser les formules de Baxter.\n\n\\`A cette fin, ils ont introduit le {\\bf $q$-caract\\`ere} $\\chi_q(V)$ d'une repr\\'esentation $V$ de dimension finie de $\\U_q(\\hat{\\Glie})$. Il s'agit d'un polyn\\^ome\nde Laurent \\`a coefficients entiers en des ind\\'etermin\\'ees $Y_{i,a}$ ($1\\leq i\\leq n$, $a\\in\\CC^*$) \n$$\\chi_q(V) \\in \\ZZ[Y_{i,a}^{\\pm 1}]_{1\\leq i\\leq n, a\\in\\CC^*}.$$\nL'entier $n$ est ici le {\\bf rang} de l'alg\\`ebre de Lie $\\Glie$, qui par exemple vaut bien $n$ pour $\\Glie = sl_{n+1}$. La d\\'efinition du $q$-caract\\`ere de $V$ repose\nsur une d\\'ecomposition de $V$ en sous-espaces de Jordan\\footnote{Pour une famille commutative d'op\\'erateurs sur $W$, obtenus \\`a partir de la r\\'ealisation de Drinfeld de $\\U_q(\\hat{\\mathfrak{g}})$ et distincts en g\\'en\\'eral des coefficients des matrices de transfert.} $V_m$ param\\'etr\\'es par des mon\\^ome $m$ en les variables $Y_{i,a}^{\\pm 1}$ :\n$$V = \\bigoplus_m V_m.$$\nLe $q$-caract\\`ere encode les dimensions de cette d\\'ecomposition. Il est d\\'efini par\n$$\\chi_q(V) = \\sum_m \\text{dim}(V_m) m.$$\nAinsi, les coefficients de $\\chi_q(V)$ sont en fait positifs et leur somme est la dimension $V$. \n\nPar exemple, pour $\\Glie = sl_2$ et $V = V_1$ la repr\\'esentation fondamentale de dimension $2$,  \n\\begin{equation}\\label{qcar}\n\\chi_q(V) = Y_{1,q^{-1}} + Y_{1,q}^{-1}.\n\\end{equation}\nOn a donc dans ce cas deux sous-espaces de Jordan de dimension $1$ associ\\'es aux mon\\^omes respectifs $Y_{1,q^{-1}}$ et $Y_{1,q}^{-1}$ :\n$$V = V_{Y_{1,q^{-1}}} \\oplus V_{Y_{1,q}^{-1}}.$$\n\nLa {\\bf conjecture du spectre quantique} de Frenkel et Reshetikhin \\cite{Fre} pr\\'edit\\footnote{Dans des cas particuliers, une conjecture analogue avait \\'et\\'e formul\\'ee par N. Reshetikhin \\cite{R3}; V. Bazhanov et N. Reshetikhin\n\\cite{BR}; et A. Kuniba et J. Suzuki \\cite{KS}.} que pour une repr\\'esentation de dimension finie donn\\'ee $V$,\nles valeurs propres $\\lambda_j$ de $\\mathcal{T}_V(z)$ sur une repr\\'esentation simple\\footnote{\nPlus g\\'en\\'eralement, $W$ peut \\^etre un produit tensoriel de repr\\'esentations simples.} $W$ sont obtenues de la mani\\`ere suivante :\ndans le $q$-caract\\`ere $\\chi_q(V)$ de $V$, on remplace chaque variable formelle $Y_{i,a}$ par\\footnote{Pour simplifier l'exposition, on supposera dans la suite de $\\Glie$\nest simplement lac\\'ee (c'est le cas notamment des alg\\`ebres de Lie $sl_{n+1}$).}\n$$F_{i}(az) q^{\\text{deg}(Q_{i,j})} \\frac{Q_{i,j}(zaq^{-1})}{Q_{i,j}(zaq)},$$\no\\`u $F_{i}(z)$ est une fonction universelle, au sens o\\`u elle ne d\\'epend pas de la valeur propre $\\lambda_j$, et $Q_{i,j}(z)$ d\\'epend de la valeur propre $\\lambda_j$ mais est un polyn\\^ome. C'est l'analogue du polyn\\^ome de Baxter.\n\nNotons que c'est bien le $q$-carat\\`ere {\\it de l'espace auxiliaire} $V$ qui est utilis\\'e pour \\'ecrire la formule du spectre de la matrice de transfert {\\it sur l'espace des \\'etats} $W$.\n\nDans le cas particulier du mod\\`ele $XXZ$, on obtient  \\`a partir de (\\ref{qcar}) la formule\n$$\\lambda_j = F_{1}(zq^{-1}) q^{\\text{deg}(Q_{1,j})} \\frac{Q_{1,j}(zq^{-2})}{Q_{1,j}(z)} + (F_1(zq))^{-1} q^{-\\text{deg}(Q_{1,j})} \\frac{Q_{1,j}(zq^2)}{Q_{1,j}(z)}.$$\nAinsi, la conjecture est bien compatible avec la formule de Baxter (\\ref{relB}) en identifiant \n$$A(z) = (D(zq^2))^{-1} = (F_1(zq))^{-1}q^{-\\text{deg}(Q_{1,j})}.$$ \nOn peut d\\'etailler par exemple le cas o\\`u l'espace des \\'etats $W\\simeq V_1$ est de dimension $2$. On a alors $2$ valeurs propres $\\lambda_0$ et $\\lambda_1$. La fonction universelle est\n$$F_1(z) = q^{1\/2}\\text{exp}\\left(  \\sum_{r > 0}  \\frac{z^r (q^{-r} - q^r)}{r(q^r + q^{-r})}\\right),$$\net les polyn\\^omes de Baxter sont\n$$Q_{1,0}(z) = 1\\text{ et }Q_{1,1}(z) = 1 - z(1 + q + q^2).$$\nOn obtient donc le spectre\n$$\\lambda_0 = F_1(zq^{-1})\\left(1 + q^{-3}\\frac{1 - z^{-1}}{1 - z^{-1}q^{-2}}\\right),$$\n\\begin{equation}\\label{bethe}\\lambda_1 = F_1(zq^{-1}) \\left(q\\frac{1-z(1+q^{-1} + q^{-2})}{1 - z(1 + q + q^2)} + q^{-4}\\frac{(1 - z^{-1})(1 - z(q^2 + q^3+ q^4))}{(1 - z^{-1}q^{-2})(1 - z(1 + q + q^2))}\\right).\\end{equation}\n\nEn g\\'en\\'eral la formule peut avoir plus de deux termes.\nPar exemple, dans le cas d'une certaine repr\\'esentation fondamentale $V$ de dimension $3$ de $\\U_q(\\hat{sl_3})$, le $q$-caract\\`ere est\n\\begin{equation}\\label{sl3}\\chi_q(V) = Y_{1,q^{-1}} + Y_{1,q}^{-1}Y_{2,1} + Y_{2,q^2}^{-1},\\end{equation}\net la formule pour le spectre est\n$$F_1(zq^{-1}) q^{\\text{deg}(Q_{1,j})}\\frac{Q_{1,j}(zq^{-2})}{Q_{1,j}(z)} +\\frac{F_2(z) q^{\\text{deg}(Q_{2,j})}}{F_1(zq)q^{\\text{deg}(Q_{1,j})}} \\frac{Q_{1,j}(zq^2)Q_{2,j}(zq^{-1})}{Q_{1,j}(z)Q_{2,j}(zq)} + \\frac{q^{-\\text{deg}(Q_{2,j})}}{(F_2(zq^2))^{-1}} \\frac{Q_{2,j}(zq^3)}{Q_{2,j}(zq)}.$$\n\nNotons qu'en g\\'en\\'eral les repr\\'esentations simples $V$ de dimension finie peuvent avoir une dimension \\og tr\\`es grande\\fg. Par exemple, H. Nakajima a obtenu (\\`a l'aide d'un super-calculateur et en s'appuyant sur \\cite{Nak}) que dans le cas de l'alg\\`ebre de Lie exceptionelle de type $E_8$, une des repr\\'esentations fondamentales a un $q$-caract\\`ere avec $6899079264$ mon\\^omes qui n\\'ecessite un fichier de taille m\\'emoire $180$ Go pour \\^etre \\'ecrit. Il y a donc autant de termes dans la formule de Baxter correspondante. Et les repr\\'esentations fondamentales sont les repr\\'esentations simples de dimensions les plus basses. \n\nIl est donc hors de question d'aborder cette conjecture par un calcul explicite en g\\'en\\'eral. D'ailleurs, m\\^eme si les repr\\'esentations simples de dimension finie de $\\U_q(\\hat{\\Glie})$ ont \\'et\\'e intensivement \\'etudi\\'ees ces vingt-cinq derni\\`eres ann\\'ees, on ne conna\\^it pas en g\\'en\\'eral de formule pour leur $q$-caract\\`ere, ni m\\^eme en fait pour leur dimension.\n\nAinsi, il faut de nouvelles structures pour aborder la conjecture du spectre quantique.\n\nNotre d\\'emonstration avec E. Frenkel \\cite{FH} de la conjecture du spectre quantique repose ainsi sur \nde nouveaux ingr\\'edients dont nous donnons un bref aper\\c cu dans les sections suivantes.\n\n\\section{Repr\\'esentations pr\\'efondamentales}\n\nL'id\\'ee g\\'en\\'erale de la preuve est d'interpr\\'eter les $Q_i$ eux-m\\^emes comme des valeurs\npropres de nouvelles matrices de transfert, construites non pas \\`a partir de repr\\'esentations de dimension\nfinie $V$, mais de repr\\'esentations de dimension infinie dite {\\bf repr\\'esentations pr\\'efondamentales}\n$L_{i,a}^+$ o\\`u $1\\leq i\\leq n$ et $a\\in\\CC^*$.\n\nNous avions construit pr\\'ealablement ces repr\\'esentations pr\\'efondamentales avec M. Jimbo \\cite{HJ} dans un contexte un peu diff\\'erent. Ce ne sont pas des repr\\'esentations de l'alg\\`ebre enti\\`ere $\\U_q({\\hat{\\mathfrak g}})$, mais d'une certaine sous-alg\\`ebre, la {\\bf sous-alg\\`ebre de Borel} \n$$\\U_q(\\hat{\\mathfrak{b}})\\subset \\U_q({\\hat{\\mathfrak g}}).$$ \nCela ne pose cependant pas de probl\\`eme pour construire la matrice de transfert $\\mathcal{T}_{i,a}(z)$ associ\\'ee \\`a la repr\\'esentation pr\\'efondamentale $L_{i,a}$ par la formule (\\ref{transfer}), car il est justement connu que la partie \\og gauche\\fg{} de la $R$-matrice universelle (celle \\`a qui on applique $\\rho_{L_{i,a}}$) est dans la sous-alg\\`ebre de Borel\\footnote{On ne peut cependant pas appliquer la trace \\`a un espace de dimension infinie. On utilise une graduation naturelle de $L_i$ par des espaces de dimension finie (les espaces de poids). Ainsi dans la suite, les traces, matrices de transfert, etc. sont \\og tordues\\fg{} par cette graduation.} :\n$$\\mathcal{T}_{i,a}(z) =  ((\\on{Tr}_{L_{i,a}} \\circ \\rho_{L_{i,a}}) \\otimes \\on{id})({\\mathcal{R}(z)})\\in \\U_q(\\hat{\\Glie})[[z]].$$\nIl n'est alors pas difficile de montrer qu'en utilisant un certain automorphisme de $\\U_q(\\hat{\\mathfrak{b}})$ on a \n$$\\mathcal{T}_{i,a}(z) = \\mathcal{T}_i(az)\\text{ o\\`u }\\mathcal{T}_i(z) = \\mathcal{T}_{i,1}(z).$$\n\nPour le cas du mod\\`ele $XXZ$, c'est-\\`a-dire pour $\\Glie = sl_2$, V. Bazhanov, S. Lukyanov, et A. Zamolodchikov avaient d\\'ej\\`a construit \\og \\`a la main\\fg{} une repr\\'esentation pr\\'efondamentale (appel\\'ee repr\\'esentation de $q$-oscillation) et la matrice de transfert associ\\'ee (appel\\'ee $Q$-op\\'erateur de Baxter) dans l'article important \\cite{BLZ}. \n\nPour obtenir l'existence des repr\\'esentations pr\\'efondamentales en g\\'en\\'eral \\cite{HJ}, on ne peut encore une fois pas faire de calculs explicites : le point crucial est de consid\\'erer des syst\\`emes inductifs\\footnote{Les inclusions $L_k\\subset L_{k+1}$, construites \\`a l'aide de produits tensoriels de sous-espaces \\cite{h3}, ne sont pas compatibles avec l'action de $\\U_q(\\hat{\\Glie})$ enti\\`ere mais avec celle d'une sous-alg\\`ebre $\\U_q^+(\\hat{\\mathfrak{b}})$ de $\\U_q(\\hat{\\mathfrak{b}})$.} de repr\\'esentations simples $L_k$ (les repr\\'esentations de Kirillov-Reshetikhin) de dimension finie strictement croissante avec $k\\geq 0$ et de d\\'eterminer en quel sens l'action de la sous-alg\\`ebre de Borel $\\U_q(\\hat{\\mathfrak{b}})$ \\og converge\\fg{} sur la limite inductive $L_\\infty$, qui elle est de dimension infinie :\n$$L_0\\subset L_1\\subset L_2\\subset \\cdots \\subset L_k\\subset L_{k+1}\\subset \\cdots  \\subset L_{\\infty}.$$\nIl s'agit ainsi d'une construction asymptotique des repr\\'esentations pr\\'efondamentales.\n\nEn utilisant certaines filtrations de la repr\\'esentation pr\\'efondamentale $L_{i,a}$, nous \\'etablissons qu'effectivement, \\`a un facteur scalaire universel $f_i(z)$ pr\\`es, la matrice de transfert associ\\'ee $\\mathcal{T}_i(z)$ agit sur l'espace des \\'etats $W$ par un op\\'erateur polyn\\^omial :\n$$\\rho_W(\\mathcal{T}_i(z)) \\in f_i(z) \\times (\\text{End}(W))[z].$$\nIl n'est pas difficile d'\\'ecrire une formule explicite pour la fonction universelle scalaire $f_i(z)\\in\\CC[[z]]$ (elle ne d\\'epend que de $V$ et de $W$). Il est beaucoup plus d\\'elicat d'obtenir des informations sur la partie lin\\'eaire polyn\\^omiale\n$$(f_i(z))^{-1}\\rho_W(\\mathcal{T}_i(z)) \\in (\\text{End}(W))[z].$$\n\nDe m\\^eme que les matrices de transfert usuelles commutent, on a \n$$\\mathcal{T}_i(z)\\mathcal{T}_i(z') = \\mathcal{T}_i(z')\\mathcal{T}_i(z),$$\net donc on obtient une famille commutative $\\mathcal{T}_i[m]$ si on \\'ecrit\n$$\\mathcal{T}_i(z) = \\sum_{m\\geq 0}\\mathcal{T}_i[m]z^m.$$\nEn utilisant la trigonalisation simultan\\'ee, cette commutativit\\'e implique que les valeurs propres sur $W$ de $(F_i(z))^{-1}\\mathcal{T}_i(z)$ elles-m\\^emes sont \\'egalement des polyn\\^omes.\n\n\\section{Anneau de Grothendieck et relations de Baxter}\n\nIl faut enfin d\\'emontrer que les valeurs propres de la matrice de transfert $\\mathcal{T}_V(z)$\ns'expriment, comme pr\\'evu dans la conjecture, en terme des valeurs propres des $\\mathcal{T}_i(z)$\nselon le $q$-caract\\`ere de $V$. Autrement dit, en rempla\\c cant dans $\\chi_q(V)$ chaque\nvariable $Y_{i,a}$ par le quotient\\footnote{Ce quotient doit en fait \\^etre multipli\\'e par une matrice de transfert d'une repr\\'esentation de dimension $1$ que nous omettons dans la suite pour simplifier l'exposition.} \n$$\\mathcal{T}_i(azq^{-1})\/\\mathcal{T}_i(azq),$$ \nobtient-on la matrice de transfert $\\mathcal{T}_V(z)$ ?\n\nDans le cas ${\\mathfrak g} = \\wh{sl}_2$ et $V$ de dimension du mod\\`ele $XXZ$, un calcul \\cite{BLZ} donne le r\\'esultat. On a bien : \n$$\\mathcal{T}_V(z) = \\frac{\\mathcal{T}_1(zq^{-1})}{\\mathcal{T}_1(zq)} + \\frac{\\mathcal{T}_1(zq^3)}{\\mathcal{T}_1(zq)}.$$\n\nEn g\\'en\\'eral, nous proposons d'utiliser la {\\bf cat\\'egorie $\\mathcal{O}$} que nous avons d\\'efinie avec M. Jimbo \\cite{HJ}. Il s'agit d'une {\\bf cat\\'egorie mono\\\" idale} (stable par produits tensoriels) de repr\\'esentations de l'alg\\`ebre de Borel $\\U_q(\\hat{\\mathfrak{b}})$, contenant les repr\\'esentations de dimension finie ainsi que les repr\\'esentations pr\\'efondamentales. \nNous {\\bf cat\\'egorifions} les relations de Baxter g\\'en\\'eralis\\'ees, c'est-\\`a-dire que nous les exprimons en termes de la cat\\'egorie $\\mathcal{O}$. Pour ce faire, on peut d\\'efinir l'{\\bf anneau de Grothendieck} $K(\\mathcal{O})$ de cette cat\\'egorie. En tant que groupe, il s'agit du groupe libre engendr\\'e par les classes d'isomorphismes de repr\\'esentations simples :\n$$K(\\mathcal{O}) = \\bigoplus_{[V]\\text{ Classe d'un simple dans }\\mathcal{O}.} \\ZZ [V].$$\nAlors tout objet (non n\\'ecessairement simple) de $\\mathcal{O}$ a une image dans $K(\\mathcal{O})$ en imposant la relation\n$$[V''] = [V] + [V']$$\nsi on a une suite exacte dans la cat\\'egorie\n$$0\\rightarrow V \\rightarrow V''\\rightarrow V'\\rightarrow 0.$$\nOn peut alors munir $K(\\mathcal{O})$ d'une structure d'anneau par la relation\n$$[V\\otimes V'] = [V][V']$$\npour des objets $V$, $V'$ de la cat\\'egorie $\\mathcal{O}$.\n\nUn des th\\'eor\\`emes principaux de \\cite{FH} est qu'en rempla\\c cant dans $\\chi_q(V)$ chaque\nvariable $Y_{i,a}$ par le quotient \n$$\\frac{[L_{i,aq^{-1}}]}{[L_{i,aq}]},$$ \nen repla\\c cant $\\chi_q(V)$ par $[V]$ puis en \\og chassant\\fg{} les d\\'enominateurs, on obtient une relation dans l'anneau de Grothendieck $K(\\mathcal{O})$.\n\nPar exemple, dans notre cas favori du mod\\`ele $XXZ$, on obtient\n$$[V] = \\frac{[L_{1,q^{-1}}]}{[L_{1,q}]} + \\frac{[L_{1,q^3}]}{[L_{1,q}]}$$\nqui donne la relation de Baxter cat\\'egorifi\\'ee dans l'anneau de Grothendieck\n$$[V][L_{1,q}] = [V\\otimes L_{1,q}] = [L_{1,q^{-1}}] + [L_{1,q^3}].$$\nEn g\\'en\\'eral on obtient des relations avec plus de termes, comme dans l'exemple pour $\\Glie = sl_3$ ci-dessus pour lequel la formule (\\ref{sl3}) donne\n$$[V\\otimes L_{1,1}\\otimes L_{2,q}] = [L_{1,q^{-2}}\\otimes L_{2,q}] + [L_{1,q^2}\\otimes L_{2,q^{-1}}] + [L_{2,q^3}\\otimes L_{1,1}].$$\nMaintenant, \\og prendre la matrice de transfert\\fg{} est additif et multiplicatif, c'est \\`a dire qu'on a un morphisme d'anneau\\footnote{On peut montrer que ce morphisme d'anneau est injectif et donc que l'anneau de Grothendieck $K(\\mathcal{O})$ est commutatif (bien que la cat\\'egorie ne soit pas tress\\'ee, c'est-\\`a-dire que $V\\otimes V'$ et $V'\\otimes V$ ne sont pas isomorphes en g\\'en\\'eral). En fait, l'application de $q$-caract\\`eres $[V]\\mapsto \\chi_q(V)$ elle-m\\^eme peut \\^etre prolong\\'ee en un morphisme d'anneau injectif sur $K(\\mathcal{O})$.}\n$$\\mathcal{T} : K(\\mathcal{O})\\rightarrow \\mathcal{U}_q(\\hat{\\Glie})[[z]]\\text{ , }[V]\\mapsto \\mathcal{T}_V(z).$$\nAinsi, les relations de Baxter g\\'en\\'eralis\\'ees dans l'anneau de Grothendieck $K(\\mathcal{O})$ impliquent les relations voulues entre les matrices de transfert. La conjecture du spectre quantique est donc d\\'emontr\\'ee.\n\n\\begin{center}*\\end{center}\n\nPour conclure, les formules pour les valeurs propres des matrices de transfert en terme\ndes polyn\\^omes $Q_{i,j}$ impliquent des \\'equations entre les racines de ces polyn\\^omes pour garantir que les p\\^oles\napparents se simplifient (par exemple dans l'\\'equation (\\ref{bethe}), $(1 + q + q^2)^{-1}$ n'est en fait pas un p\\^ole de $\\lambda_1$). Dans le cas du mod\\`ele $XXZ$ ce sont les fameuses \\'equations de l'Ansatz de Bethe.\nCes consid\\'erations ont men\\'e N. Reshetikhin \\cite{R3} \\`a formuler ces \\'equations dans le cas g\\'en\\'eral\n  (voir aussi \\cite{BR,KS, F:icmp}). La preuve de la conjecture du spectre quantique permet de donner une explication et une approche uniforme \\`a ces formules. On a maintenant une autre conjecture importante et ouverte : l'existence d'une bijection entre toutes les valeurs propres et les solutions des \\'equations de l'Ansatz de Bethe (conjecture de compl\\'etude).\n\n\\medskip\n\n{\\bf Remerciements} : je souhaite adresser mes remerciements \\`a E. Ghys pour m'avoir encourag\\'e \\`a \\'ecrire cet article, \\`a E. Frenkel et M. Jimbo pour notre collaboration et enfin \\`a J. Dumont, C. Hernandez, P. Zinn-Justin et l'\\'equipe d'Images de Math\\'ematiques pour leurs remarques sur une version pr\\'eliminaire de ce texte.\n\n\n\\backmatter\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nBuilding large-scale quantum computers is still a challenging task due to a plethora of engineering obstacles \\cite{eng}. One prominent challenge is the intrinsic noise. In fact, implementing scalable and reliable quantum computers requires implementing quantum gates with sufficiently low error rates. There has been substantial progress in characterizing noise in a quantum system \\cite{noise3,noise2,noise1} and in building error correcting schemes that can detect and correct certain types of errors \\cite{qec1,qec2,qec3}.\\\\\nNumerous protocols have been constructed to characterize the noise in quantum devices. Many of these protocols fail in achieving one of the following desirables: scalability to large-scale quantum computers and efficient characterization of the noise. Quantum Process Tomography \\cite{QPT} is a protocol that can give a complete description of the dynamics of a quantum black box, however, it's not scalable to large-scale quantum systems. Randomized Benchmarking (RB) is another protocol that's typically used to estimate the error rate of some set of quantum gates \\cite{ ScalableNoise, RBQG}. Although RB is a scalable protocol in principle, it can only measure a single error rate that's used to approximate the average gate infidelity thus providing an incomplete description of noise. Various other protocols based on RB protocol are able to characterize the correlations of noise between the different qubits, however, these protocols lack scalability \\cite{ScalableNoise,SimultaneousRB,ThreeRB}.\nQuantum Error Mitigation \\cite{QEM} (QEM) is a recently emerging field that aims to improve the accuracy of near-term quantum computational tasks. Whereas Quantum Error Correction (QEC) \\cite{AQEC1,AQEC2} necessitates additional qubits to encode a quantum state in a multiqubit entangled state, QEM does not demand any additional quantum resources. It is considered an excellent alternative for enhancing the performance of Noisy Intermediate-Scale Quantum (NISQ) computing \\cite{nisq}. QEM protocols include zero-noise Richardson extrapolation of results from a collection of experiments of varying noise \\cite{Extrapolation}, probabilistic error cancellation through sampling from a set of quantum circuits generated from a target circuit mixed with Pauli gates from an optimized distribution \n\\cite{pem,ProbabilisticMitigation}, and exploiting state-dependent bias through invert-and-measure techniques to map the predicted state to the strongest state \\cite{bias}.\nMeasurement Error Mitigation (MEM) is another QEM protocol that models the noise in a quantum circuit as a measurement noise matrix $\\bm{E}_{meas}$ applied to the ideal output of the circuit. The columns of $\\bm{E}_{meas}$ are the probability distributions obtained through preparing and immediately measuring all possible $2^n$ basis input states \\cite{QiskitTextbook}.\\\\\nRecently, the authors in \\cite{Nature} developed a protocol based on the RB that relies on the concept of a Gibbs Random Field (GRF) to completely and efficiently estimate the error rates of the Pauli Channel and detect correlated errors between the qubits in a quantum computer. Their effort paves the way to enable quantum error correction and\/or mitigation schemes. Herein, we refer to their efficient learning protocol as the \\{EL protocol\\}. \nIn this paper, we build upon the EL protocol and decompose the average noise of a quantum circuit of specific depth into State Preparation and Measurement (SPAM) error and average gate error.\nWe propose a linear algebraic based protocol and proof to efficiently construct and model the average behavior of noise in a quantum system for any desired circuit depth without having to run a large number of quantum circuits on the quantum computer or simulator. We then rely on this model to mitigate the noisy output of the quantum device. For an n-qubit quantum system, the average behavior of the noise can be well approximated as a special form of a Pauli Channel \\cite{Knill2005,Wallman_2016,Ware_2021}. \nA Pauli channel $\\varepsilon$ acts on a qubit state $\\boldsymbol{\\rho}$ to produce\n\\begin{equation}\n\\varepsilon(\\bm{\\rho})=\\sum_{i}p_i\\bm{P}_{i}\\bm{\\rho}\\bm{P}_{i}\n\\label{equation 1}\n\\end{equation}\nwhere $p_i$ is an error rate associated with the Pauli operator $\\bm{P}_{i}$. The $p_i$'s form a probability distribution $(\\sum_{i}p_{i}=1)$, and are related to the eigenvalues, $\\boldsymbol{\\lambda}$, of the Pauli Channel defined as \n\\begin{equation}\n\\lambda_i=2^{-n}Tr(\\bm{P}_{i}\\varepsilon(\\bm{P}_{i}))\n\\label{Equation 2}\n\\end{equation}\nThus, when a state $\\bm{\\rho}$ is subjected to the noisy\nchannel $\\varepsilon$, $p_i$ describes the probability of a multiqubit Pauli error $\\bm{P}_{i}$ affecting the system, while $\\lambda_i$ describes how faithfully a given multispin Pauli operator is transmitted. $\\bm{p}$ and $\\bm{\\lambda}$ are related by Walsh-Hadamard transform where \n\n\n\\begin{equation}\n\\bm{\\lambda}=\\bm{Wp}\n\\label{Equation 3}\n\\end{equation}\n\nWhile RB only estimates the average value of all $\\lambda_i$ of the Pauli Channel, the EL protocol estimates the individual $\\lambda_i$. A complete characterization of the Pauli channel requires learning more than the eigenvalues or error rates associated with single-qubit Pauli operators such as $\\bm{\\sigma}_{z}^{(1)}$ or  $\\bm{\\sigma}_{x}^{(3)}$; it requires learning all of the noise correlations in the system, that is, also learning the eigenvalues and error rates associated with multiqubit Pauli operators such as $\\bm{\\sigma}_{z}^{(1)} \\otimes \\mathbf{1}^{(2)} \\otimes \\bm{\\sigma}_{x}^{(3)}$ and how they vary compared to the ones obtained under independent local noise. Estimating these correlations is essential for performing optimal QEC and\/or QEM. However, these  correlations increase exponentially as the number of qubits increases, so having an efficient noise characterization protocol is crucial to direct the error mitigation efforts to capture the critical noise correlations.\\\\\nOur method relies on the error rates vector $\\bm{p}$ of the Pauli-Channel to decompose the average behavior of noise for circuits of depth $m$ into two noise components: a SPAM error matrix denoted by the matrix $\\bm{N}$ and a depth dependent component comprising an average gate error matrix denoted by the matrix $\\bm{M}$. We evaluate our model for the average noise by predicting the average probability distribution for circuits of depth $m$ and computing the distance between this predicted distribution and the empirically obtained one. Finally, we use our proposed decomposition to mitigate noisy outputs of random circuits and compare our mitigation protocol with the MEM protocol \\cite{QiskitTextbook}. We applied our noise characterization and mitigation protocols on the following IBM Q 5-qubit quantum computers: Manila, Lima, and Belem\\cite{IBM}.\n\\section*{Results}\n\\subsection*{Proposed Protocol Theory}\nThe ideal output probability distribution of an $n$-qubit quantum circuit with depth $m$ is perturbed by the SPAM and the average gate errors. Our aim is to construct a comprehensive linear algebraic model that takes into account both these errors for an arbitrary depth $m$. Matrix algebra can then be employed to mitigate the noise as follows: \n\\begin{equation}\n    \\bm{C}_{ideal}= \\bm{Q}_{m}^{-1}\\bm{C}_{noisy}\n\\end{equation}\nwhere $\\bm{Q}_{m}$ is the characterized noise matrix for circuits of depth $m$, $\\bm{C}_{ideal}$ and $\\bm{C}_{noisy}$ are the ideal and noisy outputs of a given circuit of depth $m$, respectively.\nThe straight-forward approach would be to construct $\\bm{Q}_m$ from empirical simulations in a similar fashion to the $\\bm{E}_{meas}$ noise matrix that was characterized in the MEM scheme. The columns of $\\bm{Q}_{m}$ comprise the emperical average probability distributions for basis input states $\\ket{\\bm{in}}\\in \\{ \\ket{\\bm{0}},\\,\\ket{\\bm{1}},\\,\\dots,\\,\\ket{\\bm{2^n-1}}\\}$, denoted by $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$, where $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ are obtained through sampling a number of depth $m$ circuits to incorporate the average gate and SPAM errors.\n\\begin{equation}\n\\bm{Q}_m=\n\\begin{bmatrix}\n \\hat{\\bm{q}}(m,\\ket{\\bm{0}}) & \\hat{\\bm{q}}(m,\\ket{\\bm{1}}) & \\hdots &\\hat{\\bm{q}}(m,\\ket{\\bm{2^n-1}})\n\\end{bmatrix}\n\\label{eq:Q}\n\\end{equation}\nBuilding $\\bm{Q}_m$, however, through empirical simulations can be expensive especially when the circuit depth is large. Herein, we propose a method for an efficient estimation of $\\bm{Q}_m$ where the individual probability distributions $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ are estimated as follows:\n\\begin{equation}\n    \\bm{q}'(m, \\ket{\\bm{in}}) = \\bm{N}_{in}\\bm{M}_{in}^m\\ket{\\bm{in}}\n    \\label{eq:qhat}\n\\end{equation}\nwhere $\\bm{N}_{in}$ and $\\bm{M}_{in}$ are input-specific matrices that represent the SPAM error matrix and average gate error for input $\\ket{\\bm{in}}$, respectively. Both $\\bm{M}_{in}$ and $\\bm{N}_{in}$ are extracted empirically using random circuits from a set of small circuit depths $T$ and then used in mitigating the outputs for circuits with higher depths. We first show the construction of $\\bm{N}_{0}$ and $\\bm{M}_{0}$.\\\\\nThe construction of $\\bm{N}_{0}$ and $\\bm{M}_{0}$ proceeds by estimating the error rates vector $\\bm{p}$ associated with the Pauli Channel based on the assumption in Equation \\ref{equation 1} for the average behavior of the noisy quantum device at hand using the EL protocol. The protocol proceeds by constructing $K$ random identity circuits of depth $m \\in T$ \\cite{SimultaneousRB, Nature}. Each circuit is constructed by initializing the qubits to the all-zeros state $\\ket{\\bm{0}}$ followed by choosing a random sequence $s \\in S_{m}$, the set of all length $m$ sequences of one-qubit Clifford gates applied independently on each qubit, followed by an inverse gate for the chosen sequence to ensure an identity circuit. It then estimates the resulting empirical probability distribution $\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$ by averaging over all the empirical probability distributions $\\hat{\\bm{q}}(m,s,\\ket{\\bm{0}})$ for the constructed random identity circuits of depth $m$, that is, \n\\begin{equation}\n    \\hat{\\bm{q}}(m,\\ket{\\bm{0}})= \\frac{1}{K}\\sum \\hat{\\bm{q}}(m,s,\\ket{\\bm{0}})\n    \\label{Equation 4}\n\\end{equation}\n$\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$ is a vector with $2^n$ entries each corresponding to the possible observed outcome. A Walsh-Hadamard transform is then applied on each $\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$ to obtain\n\\begin{equation}\n    \\bm{\\Lambda}(m)=\\bm{W}\\hat{\\bm{q}}(m,\\ket{\\bm{0})}\n    \\label{Equation 5}\n\\end{equation}\nEach parameter $\\Lambda_{i}(m)$ in $\\bm{\\Lambda}(m)$ is fitted to the model \n\\begin{equation}\n    \\Lambda_{i}(m)=A_{i}\\lambda_{i}^{m}\n    \\label{Equation 6}\n\\end{equation}\nwhere $A_i$ is a constant that absorbs SPAM errors and \nthe vector $\\bm{\\lambda}$ of all fitted parameters $\\lambda_i$ is a SPAM-free estimate to the eigenvalues of the Pauli Channel defined in Equation \\ref{Equation 2}. Notice that we can rewrite Equation \\ref{Equation 6} as\n\\begin{equation}\n    \\bm{\\Lambda}(m)=\\bm{A\\lambda}^{m}\n    \\label{Equation 7}\n\\end{equation}\nwhere $\\bm{A}$ is a diagonal matrix where the diagonal entries are $A_i$ and $\\bm{\\lambda}^m$ is an element-wise exponentiation of a vector. An inverse Walsh-Hadamard Transform is then applied on $\\bm{\\lambda}$ to get the error rate vector $\\bm{p}$ of the Pauli Channel as\n\\begin{equation}\n    \\bm{p}=\\bm{W}^{-1}\\bm{\\lambda}\n    \\label{Equation 8}\n\\end{equation}\n$\\bm{p}$ is then projected onto a probability simplex to ensure $\\sum_{i}p_i=1$. Introducing the GRF model by the EL protocol allows the scalability of estimating $\\bm{p}$ with the increase in the number of qubits. The GRF model assumes the noise correlations are bounded between a number of neighboring qubits depending on the architecture of the quantum computer at hand. Thus, decreasing the number of noise correlations to be estimated.\\\\\nThe final outcome $\\bm{p}$ of the EL protocol represents the SPAM-free probability distribution of the average noise in the quantum computer. Each element $p_i \\in \\bm{p}$ corresponds to the probability of an error of the form $binary(i)$ on an input state $\\ket{\\bm{0}}$. For example, for a 5-qubit quantum computer, $p_0$ corresponds to the probability of no bit flips on the input state, i.e., error of the form  $IIIII$, $p_1$ to the error of the form $IIIIX$, $p_2$ to the error of the form $IIIXI$, etc\u2026\\\\\nIn order to proceed with the proof for our proposed decomposition of Equation (\\ref{eq:qhat}) for input state $\\ket{\\bm{0}}$, we first state the following lemma (the detailed proof of the lemma can be found Section I in the supplementary):\n\\begin{lemma} \nLet $\\bm{\\lambda}$ and $\\bm{p}$ be the respective eigenvalues and error rates of a Pauli Channel with $n$ qubits, then $\\bm{\\lambda}^m=\\bm{WM}^m\\ket{\\bm{0}}$\nwhere $\\bm{M}$ is a $2^n \\times 2^n$ matrix such that $M_{ij}=p_{i\\oplus j}$ ($i \\oplus j$ is the bitwise exclusive-OR operator).\n\\label{Lemma}\n\\end{lemma}\n\\noindent Using Lemma \\ref{Lemma} and Equations \\ref{Equation 5} and \\ref{Equation 7}, $\\hat{q}(m,\\ket{0})$ can be estimated as\n\\begin{equation}\n    \\bm{q}'(m,\\ket{\\bm{0}})=\\bm{W}^{-1}\\bm{AWM}^{m}\\ket{\\bm{0}}\n    \\label{Equation 11}\n\\end{equation}\nThe transition matrix $\\bm{M}=\\bm{M}_{0}$ represents the average error per gate while the $\\bm{N}=\\bm{W}^{-1}\\bm{AW}=\\bm{N}_{0}$ matrix represents the SPAM errors for an input state $\\ket{\\bm{0}}$. Notice that the average noise for depth $m$ circuits on an input state $\\ket{\\bm{0}}$ behaves as a sequence of $m$ average noise gates $\\bm{M}_{0}$ followed by SPAM errors $\\bm{N}_{0}$.\\\\\nThe construction of $\\bm{N}_{in}$ and $\\bm{M}_{in}$ for input state $\\ket{\\bm{in}}$ proceeds similar to the procedure of constructing $\\bm{N}_{0}$ and $\\bm{M}_{0}$, however, a permutation of $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ is required before applying a Walsh-Hadamard transform to ensure that each element $p_{i}(\\ket{\\bm{in}})$ in the input-specific error rate vector $\\bm{p}(\\ket{\\bm{in}})$ corresponds to the probability of an error of the form $binary(i)$ on an input state $\\ket{\\bm{in}}$. This permutation is done by applying an input-specific permutation matrix $\\bm{\\pi}_{in}$ on $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ $\\forall m$ where $\\pi_{in_{ij}}=1$ if $i\\oplus j=in$ and $0$ otherwise. \n\n\\subsection*{Experiments}\nIn this section, we evaluate the accuracy of the model in Equation $\\ref{Equation 11}$ in predicting the average probability output, $\\hat{\\bm{q}}(m,\\ket{\\bm{0}})$, for identity circuits of higher depths by estimating $\\bm{A}_0$ and $\\bm{p}(\\ket{\\bm{0}})$ using only simulations of lower depths identity circuits. Denote by $\\bm{q}'(m,\\ket{\\bm{0}})$ the predicted average probability distribution obtained using Equation \\ref{Equation 11}. We select a \\textit{training set of depths} $T=\\{1,\\,2,\\,\\dots,\\,m_{max}\\}$ to estimate $\\bm{A}_0$ and $\\bm{p}$ using the EL protocol followed by the construction of the average gate error matrix $\\bm{M}_{0}$ and SPAM error matrix $\\bm{N}_{0}$ where $M_{0_{ij}}=p_{i \\oplus j}(\\ket{\\bm{0}})$ and $\\bm{N}_{0}=\\bm{W}^{-1}\\bm{A}_{0}\\bm{W}$. A new \\textit{testing set of depths} $T'=\\{m_{max}+1,\\,m_{max}+2,\\,\\dots,\\,100\\}$ is then selected where we compute the \\textit{Jensen-Shannon Divergence} ($JSD$) between $\\hat{\\bm{q}}(m',\\ket{\\bm{0}})$ and $\\bm{q}'(m',\\ket{\\bm{0}})$ $\\forall m'\\in T'$. The $JSD$ measures the similarity between the two probability distributions \\cite{JSD}. The lower the $JSD$, the closer the two distributions are. More information about the $JSD$ can be found in Section II in the supplementary. Figure \\ref{VaryingTrainingDepth} presents the computed $JSD$ for different quantum computers while varying $m_{max}$. Figure \\ref{TestErrorBars} presents the average and standard deviation for the test $JSD$ values for the different quantum computers. The average test $JSD$ varies between $0.024$ and $0.056$ for the different $m_{max}$ values with lower average $JSD$ values noted for high $m$ for $m_{max}=80$ as indicated in Figure \\ref{TestErrorBars}b.\n\\begin{figure}[h]\n\\begin{subfigure}[t]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Manila}\n  \\caption{IBM Q Manila}\n  \\label{Manila}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Lima}\n  \\caption{IBM Q Lima}\n  \\label{Lima}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Belem}\n  \\caption{IBM Q Belem}\n  \\label{Belem}\n\\end{subfigure}\n\\caption{$JSD(\\hat{\\bm{q}}(m,\\ket{\\bm{0}}),\\,\\bm{q}'(m,\\ket{\\bm{0}}))$ for training sets of depths $T$ and testing sets of depths $T'$ with variable maximum training depth $m_{max}\\in\\{20,\\,50,\\,80\\}$ on different IBM Q 5-qubit quantum computers.}\n\\label{VaryingTrainingDepth}\n\\end{figure}\n\\FloatBarrier\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n      \\centering\n      \\includegraphics[width=\\columnwidth]{TestErrorBars.png}\n      \\caption{}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{0.48\\textwidth}\n      \\centering\n      \\includegraphics[width=\\columnwidth]{TestErrorBars80_100.png}\n      \\caption{}\n    \\end{subfigure}\n    \\caption{The average and standard deviation of $JSD(\\hat{\\bm{q}}(m,\\ket{\\bm{0}}),\\,\\bm{q}'(m,\\ket{\\bm{0}}))$; (a) over all depths $m \\in [m_{max}+1,100]$ and  (b) over depths $m \\in [80,100]$ while varying the maximum training depth $m_{max}$ on different IBM Q 5-qubit quantum computers.}\n    \\label{TestErrorBars}\n\\end{figure}\n\\FloatBarrier\n\\noindent We rely on $\\bm{q}'(m, \\ket{\\bm{in}})$ to construct and evaluate the mitigation power of $\\bm{Q}_m$ for different depths. We first  select a \\textit{training set of depths} $T=\\{1,\\,20,\\,40,\\,60,\\,80,\\,100\\}$ to estimate $\\bm{A}_{in}$ and $\\bm{p}(\\ket{\\bm{in}})$ for each input state $\\ket{\\bm{in}}$ using the EL protocol followed by the construction of $\\bm{M}_{in}$ using $\\bm{p}(\\ket{\\bm{in}})$ and $\\bm{N}_{in}=\\bm{W}^{-1}\\bm{A}_{in}\\bm{W}$. We then estimate $\\hat{\\bm{q}}(m,\\ket{\\bm{in}})$ as $\\bm{q}'(m,\\ket{\\bm{in}})$ for all inputs using Equation \\ref{eq:qhat} in order to construct $\\bm{Q}_{m}$ using Equation \\ref{eq:Q}. We then choose a new \\textit{testing set of depths} $T'=\\{10,\\,30,\\,50,\\,70,\\,90\\}$ so that $\\bm{Q}_m$ is used in mitigating the outputs for circuits of depth $m \\in T'$ where for a given identity circuit of depth $m$ with input $\\ket{\\bm{in}}$ and sequence $s$ of gates, the mitigated output $\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}$ in obtained as \n\\begin{equation}\n    \\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}=\\bm{Q}_{m}^{-1}\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})\n\\end{equation}\n$\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}$ is projected onto a probability simplex to ensure a probability distribution. The $JSD$ between $\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})_{mit}$ and the ideal output $\\ket{\\bm{in}}$ is computed and then averaged over all input states and all random circuits of depth $m$. We also compare our proposed mitigation protocol using $\\bm{Q}_m$ with the MEM scheme (Figure \\ref{Mitigation}). We report upto 88\\% improvement in the $JSD$ value for the proposed approach compared to the unmitigated approach, and upto 69\\% improvement compared to MEM approach. Note that for the results presented here, we rely on the average SPAM free error rate, $\\bm{p}_{avg}=\\frac{1}{2^n}\\sum_{in=0}^{2^n-1}{\\bm{p}(\\ket{\\bm{in}})}$ to construct $\\bm{M}_{in}=\\bm{M}_{avg}$ for all inputs. We compare the results using $\\bm{p}_{avg}$ and $\\bm{p}(\\ket{\\bm{in}})$ in the supplementary Section V. $\\bm{N}_{in}$ remains input specific. Further elaborations on the results are presented in supplementary Section VI. \n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[h]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{AveManilaAllInputs.png}\n  \\caption{IBM Q Manila}\n  \\label{LimaMitigation}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{AveLimaAllInputs.png}\n  \\caption{IBM Q Lima}\n  \\label{AthensMitigation}\n\\end{subfigure}\n\\begin{subfigure}[h]{0.32\\textwidth}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{AveBelemAllInputs.png}\n  \\caption{IBM Q Belem}\n  \\label{belem Mitigation}\n\\end{subfigure}\n\\caption{Average $JSD$ between the ideal output $\\ket{\\bm{in}}$ and each of the unmitigated output $\\hat{\\bm{q}}(m,s,\\ket{\\bm{in}})$, mitigated output by the MEM protocol, and mitigated output by our proposed noise model for each depth $m$ on IBM Q 5-qubit quantum computers.}\n\\label{Mitigation}\n\\end{figure}\n\\FloatBarrier\n\\subsection*{Complexity}\nSo far in the estimation of $\\bm{M}_{in}$ and $\\bm{N}_{in}$ for each input state $\\ket{\\bm{in}}$ using the EL protocol, $K$ random circuits are generated for each depth $1\\le m \\le m_{max}$ where the EL protocol requires $O(2^{2n})$ for the Walsh-Hadamard transform which can be reduced into $O(n^2)$ using fast Walsh-Hadamard transform. Thus, the overall complexity of the construction of $\\bm{M}_{in}$ and $\\bm{N}_{in}$ for all input states is $O(m_{max}Kn^22^{n})$. Furthermore,  the GRF model factors the error rates vector into a product of $f\\sim O(n)$ factors, depending on the architecture of the quantum computer, where each factor depends on a subset of adjacent qubits of cardinality $N<<n$ (typically $N=4$). Thus, the complexity is reduced further into $O(nm_{max}KN^{2}2^{n})$. The construction of $\\bm{Q}_m$ would be based on Equation \\ref{eq:qhat} for each input state $\\ket{\\bm{in}}$ where $\\bm{M}_{in}^m$ can be computed efficiently using the Singular Value Decomposition (SVD) of $\\bm{M}_{in}$, thus the construction of $\\bm{Q}_m$ is $O(2^{3n})$. For the MEM scheme, the construction of $\\bm{E}_{meas}$ requires only generating $K$ circuits with no gates for each input state $\\ket{\\bm{in}}$, thus the complexity is $O(K2^{n})$. For mitigation, both protocols are based on matrix inversion, thus the complexity for mitigation is $O(2^{3n})$.\n\n\n\n\\section*{Discussion}\nThe proposed mitigation protocol builds upon the SPAM-free noise characterization protocols for low circuit depths to generate a SPAM-error matrix $\\bm{N}_{in}$ and an average gate error matrix $\\bm{M}_{in}$ for each input state $\\ket{\\bm{in}}$. It then constructs a noise mitigation matrix $\\bm{Q}_{m}$ for arbitrary circuit depths $m$ where the columns of $\\bm{Q}_{m}$ are the estimated average probability distributions $\\bm{q'}(m,\\ket{\\bm{in}})=\\bm{N}_{in}\\bm{M}_{in}^m$.\nThe mitigated output $\\hat{\\bm{q}}(m,\\,s)_{mit}$ of a given circuit of depth $m$ with sequence $s$ of gates is obtained by applying ${\\bm{Q}_{m}}^{-1}$ on the empirical circuit output $\\hat{\\bm{q}}(m,\\,s)$.\n\n\n\nWe evaluated the accuracy of our model in estimating the average probability distributions for high depth circuits and evaluated our mitigation protocol on the IBM Q 5-qubit quantum devices: Belem, Lima, and Manila.\nFor the model accuracy evaluations, for the different $m_{max}$ values, we reported on average a test  $JSD(\\hat{\\bm{q}}(m,\\ket{\\bm{0}}),\\bm{q}'(m,\\ket{\\bm{0}}))$ value around 0.022-0.028 for Manilla, 0.03-0.055 for Lima, and 0.028-0.048 for Belem. For $m_{max}=20$, the test JSD values varied between 0.005 and 0.05 for Lima computer, 0.01 and 0.06 for Manila computer, and 0.02 and 0.09 for Belem computer. We note that for $m_{max}=20$ the test spans $m \\in [21-100]$. For higher depths $m \\in [80-100]$, on average $m_{max}=80$ resulted in better model error than $m_{max}=50$ and $m_{max}=20$. Results for IBM Q Athens are presented in the supplementary.\n\nFinally, we report upto 88\\% JSD improvement for the proposed approach compared to the unmitigated approach with significant mitigation improvement compared to MEM at mid to higher depths. Specifically, for $m=90$, we reported 58\\%, 66\\% and 85\\% JSD improvement for the proposed approach compared to the unmitigated on Belem, Manilla, and Lima respectively. This is compared 12\\%, 17\\% and 51\\% respectively for the MEM. On average for all the test depths across the different machines, we report 68.4\\% JSD improvement for the proposed versus versus 38.2\\% for the MEM improvement compared to the unmitigated approach.\n\\section*{Methods}\nIn evaluating the accuracy of the model, we run $K=1000$ random identity circuits with each submission requesting 1024 shots for each depth $m \\in \\{1,\\,2,\\,\\dots,\\,100\\}$. In evaluating the mitigation power of $\\bm{Q}_m$, we run $K=1000$ random identity circuits for depths $m \\in \\{1,\\,10,\\,20,\\,30,\\,40,\\,50,\\,60,\\,70,\\,80,\\,90,\\,100\\}$ for each basis input state $\\ket{\\bm{in}}$ with each circuit requesting 1024 shots. The constructed circuits contain single qubit Clifford gates only. We run the circuits on the following IBM Q 5-qubit quantum computers: Manila, Lima, and Belem. Theoretical derivations and numerical details essential to the study are presented in the Results section. More details can be found in the Supplementary Information. For the configurations and noise profiles of the IBM quantum machines, please go to IBM Quantum Experience at \\url{http:\/\/www.research.ibm.com\/quantum}.\n\n\n\\subsection*{Data availability}\nThe data that supports the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\n\\section{Introduction}\n\\label{intro}\nBuilding large-scale quantum computers is still an unachievable task due to a plethora of engineering challenges \\cite{eng}. One prominent challenge is that of noise. There has been substantial progress in analyzing the noise in a quantum system \\cite{noise3,noise2,noise1} and in building error correcting schemes that can detect and correct some types of errors \\cite{qec1,qec2,qec3}. However, no known protocol can completely and efficiently characterize quantum noise. Implementing scalable and reliable quantum computers can be done by implementing quantum gates with sufficiently low error rates. Most protocols for determining the error rates are not scalable. These protocols also suffer from state preparation and measurement errors (SPAM).\n\nWhile numerous protocols have been constructed to characterize the noise in quantum devices, many of these protocols fail in achieving one of the following desirables: scalability to large-scale quantum computers and efficient characterization of the noise.\n\nQuantum Process Tomography \\cite{QPT} is a protocol that can give a complete description of the dynamics of a quantum black box, however, it's not scalable to large-scale quantum systems. \n\nRandomized Benchmarking (RB) is another protocol that's typically used to estimate the error rate of some set of quantum gates \\cite{RB,ScalableNoise,RBQG}. Although RB is a scalable protocol in principle, it can only measure a single error rate that's used to approximate the average gate infidelity thus providing an incomplete description of noise. Recent work has improved the scalability of the RB protocol and has broadened its applicability. Various other protocols extend RB protocol to be able to characterize the correlations of noise between different qubits, however, these protocols are no longer scalable \\cite{RB,ScalableNoise}. \n\nFurthermore, \\cite{Nature} develops a protocol based on the RB that can completely and efficiently estimate the error rates of noise and detect all correlated errors between the qubits in a quantum computer.\n\nHere we extend the protocol developed in \\cite{Nature} to model the average behavior of noise in a quantum system and predict the average output for any desired circuit depth without having to run a large number of quantum circuits on the quantum computer or simulator. \n\n\\section{Background Review}\n\\subsection{Depolarizing Channel}\nA depolarizing channel $\\varepsilon$ is a simple model that describes noise in quantum system \\cite{nielson} . For a single qubit, the depolarizing channel replaces the qubit state with the completely mixed state $\\frac{I}{2}$ with probability $\\alpha$ and keeps it untouched with probability $(1-\\alpha)$. Thus, the state of the qubit after being submitted to depolarizing noise becomes\n\n\\begin{equation}\n    \\varepsilon(\\rho)=(1-\\alpha)\\rho+\\alpha\\frac{I}{2}\n\\end{equation}\n\nThe depolarizing channel is thus characterized by a single parameter $\\alpha$. RB and other protocols aim to estimate this $\\alpha$ for multi-qubit quantum computers.\n\n\\subsection{Quantum Process Tomography (QPT)}\nQuantum Process Tomography (QPT) is a procedure that can characterize the complete characteristics of a quantum black box \\cite{QPT}.\n\nA quantum channel is a Complete Positive Trace Preserving (CTPT) linear map $\\varepsilon$ that describes the dynamics of a quantum system.\n\n\\begin{equation}\n    \\rho \\rightarrow \\rho'=\\varepsilon(\\rho)\n\\end{equation}\n\n\nKraus theorem indicates that $\\varepsilon(\\rho)$ can be described \\begin{equation*}\n    \\varepsilon(\\rho)=\\sum_{i} A_i \\rho A_{i}^\\dagger\n\\end{equation*} where $A_i$ are called Kraus Operators and they satisfy the completeness relation \n\\begin{equation*}\n    \\sum_{i}A_i A_{i}^{\\dagger} = \\mathbb{I}\n\\end{equation*}\n\nThe $A_i$ describes the dynamics of the system alone. These dynamics include quantum unitary operations, measurements, and any noise affecting the system (decoherence).\n\nTypically, $A_i$ operators are derived from the theoretical model of the system and its environment. QPT can be used to experimentally derive the $A_i$ operators thus completely describing the dynamics of a quantum black box. \n\nFor a system of $n$ qubits, QPT proceeds by experimentally preparing $N=2^{2n}$ orthonormal pure states $\\rho_j = \\ket{\\psi_j}\\bra{\\psi_j},j=1,...,N$ and then measuring the output $\\rho^{'}=\\varepsilon(\\rho)$ for each input. The inputs and their corresponding outputs are then used to determine the unknown operators $A_i$. The quantum operation $\\varepsilon$ on any input state $\\rho$ can thus be determined by linearly extending $\\varepsilon$ to all states.\n\n\\cite{QPT} shows the mathematical steps for determining the Kraus operators for a quantum black box of one and two qubit quantum system. However, it becomes cumbersome for larger systems.\n\nOnce the operator-sum representation is determined, many important quantities including entanglement fidelity (measure how closely the dynamics of the quantum system under consideration approximates that of some ideal quantum system), the minimum fidelity, and the channel capacity.\n\n\\subsection{Randomized Benchmarking (RB)}\nRB protocol is used to obtain information about the average error rate over the Clifford group \\cite{RB}.The RB protocol proceeds as follows. \n\n\\begin{algorithm}\n\\caption{Randomized Benchmarking Protocol}\n\\begin{algorithmic}[1]\n\\State Choose a circuit depth $m$.\n\\State Choose a random sequence $s\\in S_m$ of Clifford Gates followed by an inverse gate for this sequence.\n\\State Obtain an estimate $\\hat{q}(m,s)$ of the expectation value of an observable $E$ after preparing state $\\rho$ and applying the gates in $s$.\n\\State Repeat steps 2-3 $K_m$ times to obtain an estimate $\\hat{q}(m)$ of \\begin{equation*}\n    \\bar{q}(m)=\\frac{1}{|S_m|}\\sum_{s \\in S_m}q(m,s)\n\\end{equation*}\n\n\\State Repeat steps 1-4 to fit the model \n\\begin{equation*}\n    \\hat{q}(m)=A\\alpha^m+B\n\\end{equation*}\n\n\\end{algorithmic}\n\\end{algorithm}\n\nThe average gate error $r$ can be extracted from $\\alpha$ as follows:\n\n\\begin{equation*}\n    r=\\frac{(2^n-1)(1-\\alpha)}{2^n}  \n\\end{equation*}\nwhere $n$ is the number of qubits.\n\nWhile RB is a scalable protocol and robust to SPAM errors, it only estimates a single parameter of interest; hence,  it doesn't give a complete description of the dynamics of the quantum system (including correlation of errors between qubits).\n\n\\subsection{Gibbs Random Field Protocol}\n\\cite{Nature} improves the RB protocol to an efficient and reliable protocol that can give a complete description of the noise in the quantum computer.\n\nAlthough not every noisy channel is a Pauli Channel, the average noise in any noisy quantum system can be well approximated by a Pauli Channel. A Pauli Channel $\\varepsilon$ acting on a state $\\rho$ is of the form \\begin{equation*}\n    \\varepsilon(\\rho)=\\sum_{j}p(j)P_{j}\\rho P_{j}\n\\end{equation*} where $p(j)$ is the error rate associated with the Pauli operator $P_{j}$. They are closely related to, but distinct from, the eigenvalues of the Pauli channel, which are defined to be \n\\begin{equation*}\n\\lambda(j)=2^{-n}Tr(P_{j}\\varepsilon(P_{j}))\n\\end{equation*}\n\nThus, when a state $\\rho$ is subjected to the noisy channel $\\varepsilon$, $p(j)$ describes the probability of a multiqubit Pauli error $P_{j}$ affecting the system, while the respective eigenvalue describes how faithfully a given multispin Pauli operator is transmitted. The $p(j)$ and $\\lambda(j)$ are related by a Walsh\u2013Hadamard transform where\n\\begin{equation*}\n    \\mathbf{p}=W\\mathbf{\\lambda}\n\\end{equation*}\n\nThus estimating $p(j)$ would give a complete description of the average noise in the quantum system.\n\nThe protocol in \\cite{Nature} proceeds as follows:\n\\begin{algorithm}\n\\caption{Gibbs Random Field Protocol}\n\\begin{algorithmic}[1]\n\\State Choose a circuit depth $m$.\n\\State Choose a random sequence $s\\in S_m$ of Clifford Gates followed by an inverse gate for this sequence.\n\\State Obtain an estimate $\\mathbf{\\hat{q}(m,s)}$ of the probability distribution over the $2^n$ different measurement outcomes.\n\\State Repeat steps 2-3 $K_m$ times to obtain an estimate $\\mathbf{\\hat{q}(m)}$ of \n\\begin{equation*}\n    \\mathbf{\\bar{q}(m)}=\\frac{1}{|S_m|}\\sum_{s\\in S_m}\\mathbf{q(m,s)}\n\\end{equation*}\n\n\\State Apply Walsh-Hadamard transform $W$ on $\\mathbf{\\hat{q}(m)}$ to obtain\n\\begin{equation*}\n    \\mathbf{\\lambda(m)}=W\\mathbf{\\hat{q}(m)}\n\\end{equation*}\n\n\\State Repeat steps 1-5 for different circuit depths.\n\n\\State For each parameter in $\\mathbf{\\lambda(m)}$, fit to the model\n\\begin{equation*}\n    \\lambda_{i}(m)=A\\lambda_{i}^m\n\\end{equation*}\n\n\\State Apply an Inverse Walsh-Hadamard transform on $\\mathbf{\\lambda}$ to obtain\n\\begin{equation*}\n    \\mathbf{p}=W^{-1}\\mathbf{\\lambda}\n\\end{equation*}\n\n\\end{algorithmic}\n\\end{algorithm}\n\nThe effective error rates $p$ can still describe important body correlations between different sets of qubits.\n\nWhile the nature protocol is not scalable, it introduces the Gibbs Random Field (GRF) model for a scalable estimation of $p(j)$ on a quantum computer. GRF is a strictly positive probability distribution that obeys certain conditional independence properties known as Markov conditions (Methods). These conditions restrict the range of possible correlations enough to make the problem of noise characterization tractable, but they allow sufficient expressive power that a GRF can accurately model noise in real devices. The underlying (but testable) assumption is that realistic devices will only have correlations between a bounded number of qubits. \nHowever, this model depends on the architecture of the quantum processor.\n\n\\section{Proposed Approach}\n\\subsection{Backward Propagation Approach (BPA) for Efficient Noise Mitigation}\nAlthough the noise in a given quantum system behaves randomly, we can analyze the average behavior of the noise in the system. We investigate the protocol used in \\cite{Nature} to develop a model of the average noise in a quantum system.\n\nThe final outcome $p$ of the protocol used in \\cite{Nature} is the SPAM free probability distribution of the average noise in the quantum computer. Each element $p_i$ in $p$ corresponds to the probability of an error of the form $binary(i)$ on an input state $\\ket{0}^{\\otimes n}$(the state where all qubits are initialized to $\\ket{0}$). For example, for a 5-qubit quantum computer, $p_0$ corresponds to the probability of no \"bit flips\" on the input state, $p_1$ to the error of the form 00001 (last qubit is flipped), etc\u2026\n\nUsing $p$, we construct a Markovian transition matrix $M$ that can be used to estimate the average noisy output probability distribution of an identity circuit of depth $m$. $M$ is constructed as follows: $M_{ij}=p_{i\\oplus j}$ where $\\oplus$ is the bitwise $xor$ operator. In other words, each column of $M$ corresponds to a different shuffling of $p$. We show here the construction $M$ for a 5-qubit quantum computer.\n\n\\begin{center}\n$M=\n\\begin{pmatrix}\n &p_0     &p_1     &p_2    &\\hdots &p_{31}\\\\ \n &p_1     &p_0     &p_3    &\\hdots &p_{30}\\\\ \n &p_2     &p_3     &p_0    &\\hdots &p_{29} \\\\ \n &\\vdots  &\\vdots  &\\vdots &\\ddots &\\vdots\\\\ \n &p_{31}  &p_{30}  &p_{29}    &\\hdots &p_0\n\\end{pmatrix}\n$\n\\end{center}\n\nAfter each \"average\" gate, the output for the identity circuit would be multiplied by this transition matrix. Thus, the average \"noisy\" probability distribution of an identity circuit of sequence length $m$ would be equal to $M^m\\ket{0}^{\\otimes n}$where $\\ket{0}^{\\otimes n}=[1,0, ... ,0]^T$. Notice for $m=1$, the probability distribution of the average output:\n\n\\begin{center}\n$\nM\\ket{0}^{\\otimes n}=\n\\begin{pmatrix}\np_0\\\\ \np_1\\\\ \np_2\\\\ \n\\vdots\\\\ \np_{31}\n\\end{pmatrix}\n=p\n$\n\\end{center}\n\nThe transition matrix $M$ doesn't include the SPAM errors, thus it doesn't give a good description of the behavior of the quantum circuit on average. However, we know that\n\\begin{equation}\n    M^m\\ket{0}^{\\otimes n}=W^{-1}\\lambda^m\n\\end{equation}\nAnd we also know that after fitting, we get \n\\begin{equation}\n    \\lambda^m=A^{-1}W\\hat{q}(m)\n\\end{equation}\nwhere $A$ is a diagonal matrix obtained from fitting.\nUsing (1) and (2), we get:\n\\begin{equation}\n    M^m\\ket{0}^{\\otimes n}=W^{-1}A^{-1}W\\hat{q}(m)\n\\end{equation}\nOr in other words:\n\\begin{equation}\n    W^{-1}AWM^m\\ket{0}^{\\otimes n}=\\hat{q}(m)\n\\end{equation}\n\nThe detailed proof of equation 6 can be found in the appendix.\n\n\\begin{algorithm}\n\\caption{Proposed Protocol}\n\\begin{algorithmic}[1]\n\\State Choose a training set of sequence lengths $T=[m_{1},m_{2},...]$.\n\\State Use the protocol in \\cite{Nature} to construct $A$ and $\\lambda$.\n\\State Construct $p=W^{-1}\\lambda$.\n\\State Construct the matrix $M$ where $M_{ij}=p_{i \\oplus j}$ ($\\oplus$ is the bitwise $xor$).\n\\State Choose a new testing set of sequence lengths $T'=[m'_{1},m'_{2}...]$.\n\\State for each $m'\\in T'$, construct the average probability distribution $\\hat{q}(m')=W^{-1}AWM^{m'}\\ket{0}^{\\otimes n}$\n\\end{algorithmic}\n\\end{algorithm}\n\nThus, the average noise of random circuits of fixed length $m$ can be described as a Markovian matrix applied to itself $m$ times followed by a matrix $B=W^{-1}AW$ that takes into account the SPAM errors.\n\nUsing this model, we can predict the average output for an input of $\\ket{0}^{\\otimes n}$ for any sequence length $m'$ not included in our fitting set of sequence lengths. Moreover, we can predict the average output for any input state\n\nOur modeling allows the extraction of data in a short period of time without having to run circuits on a real quantum computer.\n\n\\section{Results}\n\\subsection{Data Generation}\nFor each depth $m \\in$ [1,...,100], we run $1000$ random identity circuits with each submission requesting 1024  (Figure \\ref{DataGeneration}). The constructed circuits contain single qubit gates only. We run the circuits on the following 5-qubit IBM quantum computers: Athens, Lima, and Belem.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{DataGeneration}\n\\caption{Generating data for circuit depth $m$, $G_1,G_2,...,G_m$ are random gates selected from the Clifford Group, $G^{-1}$ ensures we have an identity circuit}\n\\label{DataGeneration}\n\\end{figure}\n\\FloatBarrier\n\n\\subsection{Results of our Proposed Protocol}\nAfter generating the data, the protocol developed in \\cite{Nature} was used to obtain the vector $p$ that we used to estimate the average noisy probability output $\\hat{q}(m)$ for depth $m$ (Figure \\ref{Protocol}).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=0.3\\columnwidth,width=\\columnwidth]{Protocol}\n\\caption{Generating data for circuit depth $m$, $G_1,G_2,...,G_m$ are random gates selected from the Clifford Group, $G^{-1}$ ensures we have an identity circuit}\n\\label{Protocol}\n\\end{figure}\n\\FloatBarrier\n\nWe fix the maximum depth of out training set of depths to $m_{max}$, and then try to predict the average noisy output for our testing depths $m_{max}+1,...,100$. We computed the Jenson-Shannon divergence (JSD) between $BM^m\\ket{0}^{\\otimes n}$ and $\\hat{q}(m)$ for each quantum computer with $m_{max}=50$ fixed. The JSD is a measure of the similarity be two probability distributions defined as: \n\n\\begin{equation*}\n    JSD(p,q)=\\frac{1}{2}D(p||m)+\\frac{1}{2}D(q||m) \n\\end{equation*}\nwhere $m=\\frac{1}{2}(p+q)$ and \n\\begin{equation*}\n    D(P,Q)=\\sum_{x}P(x) log(\\frac{P(x)}{Q(x)})\n\\end{equation*}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{JSDQuantumComputers}\n\\caption{$JSD$ between $BM^m \\ket{0}^{\\otimes n}$ and $\\hat{q}(m)$ for different quantum computers. The dashed lines correspond to the $JSD$ for the training depths with $m_{max}=50$ and the solid lines correspond to the $JSD$ for the testing depths}\n\\label{JSDQuantumComputers}\n\\end{figure}\n\\FloatBarrier\n\nFigure \\ref{JSDQuantumComputers} shows that the JSD fluctuates between 0 and reaches a maximum of 0.10 on the Lima quantum computer, indicating a higher similarity between the constructed average $BM^m \\ket{0}^{\\otimes n}$ and the experimentally obtained average $\\hat{q}(m)$ for each sequence length. Ofcourse, the maximum $JSD$ is reduced if we remove the outliers for each quantum computer.\n\nFurthermore, for each quantum computer, we vary $m_{max}$ and measure the corresponding $JSD$ fore each quantum computer (Figures \\ref{VaryingTrainingDepth}).\n\n\\begin{figure}[h]\n\\begin{subfigure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Belem}\n  \\caption{Belem Quantum Computer}\n\\end{subfigure}\n\\begin{subfigure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Lima}\n  \\caption{Lima Quantum Computer}\n\\end{subfigure}\n\\begin{subfigure}\n  \\centering\n  \\includegraphics[width=\\columnwidth]{Athens}\n  \\caption{Athens Quantum Computer}\n\\end{subfigure}\n\\caption{$JSD(\\hat{q}(m),BM^{m}\\ket{0})$ for different maximum training depth $m_{max}$}\n\\label{VaryingTrainingDepth}\n\\end{figure}\n\\FloatBarrier\n\nFigure \\ref{VaryingTrainingDepth} shows that fitting with a smaller training depth would result in the same testing pattern as fitting with a larger one. It also shows that a $m_{max}=50$ gives the lowest $JSD$ thus results in the best prediction.\n\nWe also tried the same protocol for different basis inputs state, i.e. $\\ket{in} \\in \\{\\ket{0},\\ket{1},...,\\ket{2^n-1}\\}$ where we estimated $\\hat{q}_{in}=W^{-1}A_{in}WM_{in}|in>$ where $A_{in}$ and $M_{in}$ corresponds to the matrices constructed when running the Gibbs Random Field Protocol on each input. Notice that the protocol can be done with some permutations performed on any $\\ket{in}$ to the $\\ket{0}$.\n\nFigure \\ref{lambdas} and \\ref{As} shows the variation of the entries of the $\\lambda$ and $A$ vector as function of the inputs, respectively.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=10cm,width=\\columnwidth]{lambdas}\n\\caption{Variation of the entries of the $\\lambda$ vector as function of the inputs on Athens Quantum Computer}\n\\label{lambdas}\n\\end{figure}\n\\FloatBarrier\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=10cm,width=\\columnwidth]{As}\n\\caption{Variation of the entries of the $A$ vector as function of the inputs on Athens Quantum Compute}\n\\label{As}\n\\end{figure}\n\\FloatBarrier\n\n\\section{Conclusion}\nabc.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe understanding of the turbulence is one of the main unsolved problems\nof classical physics, in spite of the more than 250 years of\nstrong investigations initiated by D.Bernoulli and L.Euler.\n\nIn the stochastic approach to turbulence~\\cite{Monin},~\\cite{Frish}\nthe turbulent cascade is\nconsidered as a stochastic process, described by the probability\ndistribution $P(\\lambda,v)$, where $\\lambda$ and $v$ are the\nappropriate scaled length and velocity increment respectively.\nRecently~\\cite{Friedrich} R.Friedrich and J.Peinke presented\nexperimental evidence that\nthe probability\ndensity function  $P(\\lambda,v)$ obeys a Fokker-Planck\nequation (FPE)~\\cite{Risken} (see fig.1 and fig.2\nin~\\cite{Friedrich}):\n\\begin{equation}\n\\frac{\\partial P(\\lambda,v)}{\\partial \\lambda} =\n\\left[ -\\frac{\\partial}{\\partial v} D^1(\\lambda,v)\n+ \\frac{\\partial^2}{\\partial v^2} D^2(\\lambda,v) \\right]\nP(\\lambda,v),\n\\label{FPP}\n\\end{equation}\nwhere the drift and duffusion coefficients\n$D^1$ and $D^2$ respectively are derived by analysis of experimental\ndata of a fluid dynamical experiment (see fig.3 in~\\cite{Friedrich}).\n\nIn their paper Friedrich and Peinke consider the application\nof the FPE to obtain the Kolmogorov scaling  with simplified\nassumptions that $D^1$ and $D^2$ are $\\lambda$-independent,\n$D^1$ is linear in $v$ and $D^2$ is quadratic in $v$:\n$$\nD^1= -a\\, v, \\qquad a>0; \\qquad\nD^2 = c\\, v^2,\\qquad c>0 \\,.\n$$\n( In the notations of~\\cite{Friedrich} :\n$ a \\equiv \\gamma $ and $ c \\equiv Q $.)\n\nHere we will consider a more realistic situation\n(see fig.3 in~\\cite{Friedrich})\nof $\\lambda$-dependent $D^1$ and $D^2$ :\n\\begin{equation}\nD^1= -a(\\lambda)\\, v, \\qquad a(\\lambda)>0 ;\\qquad\nD^2 = c(\\lambda)\\, v^2,\\qquad c(\\lambda)>0 .\n\\label{D}\n\\end{equation}\nThus the FPE ~(\\ref{FPP}) will take the form:\n\\begin{equation}\n\\frac{\\partial P}{\\partial \\lambda} = b_0(\\lambda)P(\\lambda,v)+\nb_1(\\lambda) v \\frac{\\partial P}{\\partial v} +\nc(\\lambda)\\left( v \\frac{\\partial}{\\partial v}\\right)^2 P(\\lambda,v),\n\\label{P}\n\\end{equation}\nwhere\n\\begin{equation}\nb_0(\\lambda)= a(\\lambda)+2 c(\\lambda),\n\\quad b_1(\\lambda) = a(\\lambda) + 3 c(\\lambda) .\n\\label{b}\n\\end{equation}\n\n\\begin{center}\n\\section{Exact Solution of the Cauchy Problem for the\nEq.~(\\ref{P})}\n\\end{center}\nIn this section we will find the solution\n$P(\\lambda,v)$\n of the Cauchy problem\nfor the Eq.~(\\ref{P}) with the initial condition\n\\begin{equation}\nP(0,v) = \\varphi (v).\n\\label{In}\n\\end{equation}\nAccording to \\cite{Monin}$-\\!$~\\cite{Friedrich}, when the probability\ndensity function is known, one may derive all properties\nof the turbulent cascade considered as a stochastic process.\n\nFor the solution of the problem (\\ref{P}),~(\\ref{In}) we\nshall use the approach\nof M.Suzuki~\\cite{Suzuki} to the FPE (see also ~\\cite {Donkov} ),\nbased on the disentangling techniques of R.Feynman~\\cite{Feynman}\n and the operational methods developed in the functional\nanalysis, in particular in the theory of pseudodifferential equations\nwith partial derivatives ~\\cite{Hoermander}$-\\!$~\\cite{Maslov}\n\nIn the spirit of the operational methods using the\npseudodifferential operators  we can write the solution\nof the Cauchy problem (\\ref{P}),~(\\ref{In}) in the form\n\\begin{equation}\nP(\\lambda, v) =\n\\left(exp_+ \\int_0^{\\lambda} \\left[ b_0(s)+b_1(s)v\n\\frac{\\partial}{\\partial v}\n+ c(s)\\left(v \\frac{\\partial}{\\partial v}\\right)^2 \\right] {\\rm d}s\n\\right) \\varphi(v) ,\n\\label{form}\n\\end{equation}\nwhere the symbol $\\;\\;exp_+\\;\\;$ designates the V.Volterra ordered exponential\n\\begin{equation}\nexp_+ \\int_0^{\\lambda} \\hat C(s) {\\rm d}s =\n\\hat 1 + \\lim_{n\\to\\infty} \\sum_{k=1}^n\\int_0^{\\lambda}{\\rm d}\\lambda_1\n\\int_0^{\\lambda_1}{\\rm d}\\lambda_2 \\dots \\int_0^{\\lambda_{k-1}}\n{\\rm d}\\lambda_{k}\n\\hat C(\\lambda_1) \\hat C(\\lambda_2) \\dots \\hat C(\\lambda_{k}).\n\\label{exp}\n\\end{equation}\n\nThe linearity of the integral and the explicit form of the operators\nin~(\\ref{form}) permit to write the solution $P(\\lambda,v)$ in terms\nof usual, not ordered, operator valued exponent\n\\begin{equation}\nP(\\lambda,v) = {\\rm e}^{\\beta_0 (\\lambda)}\\,\n{\\rm e}^{\\beta_1(\\lambda)v\\frac{\\partial}{\\partial v} +\n\\gamma (\\lambda)\\left(v\\frac{\\partial}{\\partial v}\\right)^2}\n\\varphi(v) ,\n\\label{expP}\n\\end{equation}\nwhere for convenience we have denoted\n\\begin{equation}\n\\beta_j(\\lambda) = \\int_0^{\\lambda}b_j(s){\\rm d}s,\\;\\; (j=0,1);\n\\qquad \\gamma(\\lambda) = \\int_0^{\\lambda}c(s) {\\rm d}s .\n\\label{beta}\n\\end{equation}\nConsequently (from now on \"$'$\" means $\\frac{\\rm d}{{\\rm d}t} $ ) :\n\\begin{equation}\n\\beta_j(0)=0,\\;\\;\\; {\\beta}'_j(\\lambda) =b_j(\\lambda),\\;\\;\\; (j=0,1);\n\\qquad \\gamma(0)=0,\\;\\;\\; {\\gamma}'(\\lambda)=c(\\lambda).\n\\label{betaPR}\n\\end{equation}\n\nSince the operators \\qquad\n$\\hat A \\equiv \\beta_1(\\lambda) v \\frac{\\partial}{\\partial v}$ \\quad and \\quad\n$\\hat B \\equiv \\gamma (\\lambda)\\left(v \\frac{\\partial}{\\partial v}\\right)^2$\\qquad\ncommute : $[\\hat A , \\,\\hat B] = 0$ ,\nfrom Eq.~(\\ref{expP}) we have\n\n\\begin{equation}\nP(\\lambda,v)=\n{\\rm e}^{\\beta_0(\\lambda)}\\,\n{\\rm e}^{\\beta_1(\\lambda) v\\frac{\\partial}{\\partial v}} \\,\n{\\rm e}^{\\gamma(\\lambda)\\left(v\\frac{\\partial}{\\partial v}\\right)^2}\n\\varphi(v).\n\\label{Form}\n\\end{equation}\n\nTherefore, taking into account the formulae\n\\begin{equation}\n{\\rm e}^{\\beta_1(\\lambda)v\\frac{\\partial}{\\partial v}}f(v)\n= f\\left(v{\\rm e}^{\\beta_1(\\lambda)}\\right)\n\\label{F1}\n\\end{equation}\nand\n$\n\\,\\,{\\rm e}^{\\gamma(\\lambda)\\left(v\\frac{\\partial}{\\partial v}\\right)^2}g(v)\n=\\frac{1}{\\sqrt{4\\pi\\gamma(\\lambda)}}\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{s^2}{4\\gamma(\\lambda)}} g\\left(v{\\rm e}^\n{-s}\\right){\\rm d}s\n$$\n\\begin{equation}\n\\qquad\\qquad\\qquad=\\frac{1}{\\sqrt{4\\pi\\gamma(\\lambda)}}\n\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{\\left(\\ln v-y\\right)^2}{4\\gamma(\\lambda)}}\ng\\left({\\rm e}^\n{y}\\right){\\rm d}y,\n\\label{F2}\n\\end{equation}\nwe obtain the following expression for the exact solution of the\nCauchy problem (\\ref{P}),(\\ref{In})\n$$\nP(\\lambda,v)\n=\\frac{{\\rm e}^{\\beta_0(\\lambda)}}\n{\\sqrt{4\\pi\\gamma(\\lambda)}}\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{s^2}{4\\gamma(\\lambda)}} \\varphi\\left(v{\\rm e}^\n{\\beta_1(\\lambda)-s}\\right){\\rm d}s\n$$\n\\begin{equation}\n\\label{end}\n\\qquad\\qquad=\\frac{{\\rm e}^{\\beta_0(\\lambda)}}\n{\\sqrt{4\\pi\\gamma(\\lambda)}}\\int_{-\\infty}^{\\infty}\n{\\rm e}^{-\\frac{\\left(\\ln v+\\beta_1(\\lambda)-y\\right)^2}\n{4\\gamma(\\lambda)}}g\\left({\\rm e}^{y}\\right){\\rm d}y,\n\\end{equation}\nwhere $\\beta_0(\\lambda), \\beta_1(\\lambda)$  and $\\gamma(\\lambda)$\nare defined in~(\\ref{beta}).\n\nSubstituting the expression~(\\ref{end})\nin the Eqs.~(\\ref{P}) and~(\\ref{In}) and using the Eq.~(\\ref{betaPR})\none can see immediately that $P(\\lambda,v)$ is a solution of the\nproblem (\\ref{P}),~(\\ref{In}) and, according to the Cauchy theorem,\nit is the only classical solution of this problem.\n\n\\section{Concluding remarks}\n\\begin{itemize}\n\\item The exact solution of the Cauchy problem (\\ref{P}),~(\\ref{In})\nis obtained using the algebraic method we have described.\nThe Eq.~(\\ref{P}) is a generalization\nof the equation used by R.Friedrich and J.Peinke\n( see section 1)\nin their description of a turbulent cascade by a\nFokker-Planck equation\nwith coefficients derived by a detailed analysis of\nexperimental data\nof a fluid dynamical experiment.\n\\item If\nthe probability distribution function\n$P(\\lambda,v)$ is known, then\none may derive the properties of\na given stochastic process, in our case -\n the turbulent cascade \\cite{Monin}~$-\\!$~\\cite{Friedrich}.\n\\item For more realistic description of the turbulent cascade\nby a FPE\nit should be desirable to use for\n$D^1(\\lambda, v)$ and $D^2(\\lambda ,v)$ in the Eq.~(\\ref{FPP})\nmore general expressions than these in Eq.~(\\ref{D}),\nfor instance:\\\\\n $ D^1(\\lambda,v) = a_1(\\lambda) - a(\\lambda)v,\\,\\,\\, a(\\lambda)>0$\\quad\nand \\quad\n$D^2(\\lambda,v) = c_1(\\lambda) + c(\\lambda) v^2 $.\n\n\\end{itemize}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{section1}\n\nPathology analysis based on microscopic images is a critical task in medical image computing. In recent years, deep learning of digitalized pathology slide has facilitated the progress of automating many diagnostic tasks, offering the potential to increase accuracy and improve review efficiency. Limited by computation resources, deep learning-based approaches on whole slide pathology images (WSIs) usually train convolutional neural networks (CNNs) on patches extracted from WSIs and aggregate the patch-level predictions to obtain a slide-level representation, which is further used to identify cancer metastases and stage cancer \\cite{Wang2016DeepLF}. Such a patch-based CNN approach has been shown to surpass pathologists in various diagnostic tasks \\cite{Liu2017DetectingCM}\n\nOff-the-shelf CNNs have been shown to be able to accurately classify or segment pathology images into different diagnostic types in recent studies \\cite{huang2018improving, veeling2018rotation}. However, \nmost of these methods are weak in interpretability especially for clinicians, due to a lack of evidence supporting for the decision of interest. During diagnosis, a pathologist often inspects abnormal structures (e.g., large nucleus, hypercellularity) as the evidence for determining whether the glimpsed patch is cancerous. For CAD systems, learning to pinpoint the discriminative evidence can provide precise visual assistance for clinicians. Strong supervision-based feature localization methods require a large number of pathology images annotated in pixel-level or object-level, which are very costly and time-consuming and can be biased by the experiences of the observers. In this paper, we propose a weakly supervised learning (WSL) method that can learn to localize the discriminative evidence for the class-of-interest on pathology images from weakly labeled (i.e. image-level) training data. Our contributions include: \ni) proposing a new CNN architecture with multi-branch attention modules and deep supervision mechanism, to address the difficulty of localizing discrete and small objects in pathology images, \nii) formulating a generalizable approach that leverages gradient-weighted class activation map and saliency map in a complementary way to provide accurate evidence localization, \niii) designing a new attention module which allows capturing spatial attention from various context, \niv) quantitatively and visually evaluating WSL methods on large scale histopathology datasets, and \nv) constructing a new dataset (HPLOC) based on Camelyon16 for effectively evaluating evidence localization performance on histopathology images.\n\n\n\n\\textbf{Related Work.} Recent studies have demonstrated that CNN can learn to localize discriminative features even when it is trained on image-level annotations \\cite{zhou2016learning}. However, these methods are evaluated on natural image datasets (e.g., PASCAL), where the objects of interest are usually large and distinct in color and shape. In contrast, objects in pathology images are usually small and less distinct in morphology between different classes. A few recent studies investigated WSL approaches on medical images, including lung nodule detection and placental ultrasound images \\cite{feng2017discriminative}. These methods employ GAP-based class activation map and require CNNs ending with global average pooling, which degrades the performance of CNNs as a side effect \\cite{zhou2016learning}. \n\n\\section{Methods}\nThe overview of the framework is shown in Fig.{~\\ref{fig:fig1}}. \nThe model is trained to predict the cancer score for a given image, indicating the presence of cancer metastasis.\nIn the test phase, besides giving a binary classification, the model generates a cancerous evidence localization map and performs localization. \n\n \\afterpage{\n \\begin{figure}[t]\n\n \t\\centering\n\n \t\\begin{tabular}{cc}\n \t\t\\includegraphics[scale=0.5]{.\/fw.pdf} &\n \n \t\n\n \t\t\\includegraphics[scale=0.5]{.\/blk.pdf} \\\\%width=\\linewidth\n\n \t\n \t\\end{tabular}\n \t\\caption{ \\textbf{Left:} Framework overview of the proposed WSL method. The under line (e.g., \/1, 16) denotes stride and number of channels.  \\textbf{Right:} A building block for the multi-branch attention based residual module (MA-ResModule). \n \t\n \t} \n \t\\label{fig:fig1}\n \\end{figure}\n}\n \n \\subsection{Cancerous Evidence Localization Networks (CELNet)}\n \\label{subsection_2_3}\n\n\n Given the object of interest is relatively small and discrete, a moderate number of convolutional layers is sufficient for encoding locally discriminative features. As discussed in Section \\ref{section1}, instances on pathology images are similar in morphology and can be densely distributed, the model should avoid over-downsampling in order to pinpoint the cancerous evidence from the densely distributed instances. The proposed CELNet starts with a $3\\times 3$ convolution head followed by 3 Multi-branch Attention-based Residual Modules (MA-ResModule)\n \\footnote{Densely connected module is not employed considering it is comparatively speed-inefficient for WSIs application due to its dense tensor concatenation. } \n \\cite{he2016deep}. Each MA-ResModule is composed of 3 consecutive building blocks integrated with the proposed attention module (MAM) as  shown in Fig.{~\\ref{fig:fig1}} (Right). We use $3\\times 3$ convolution with stride of 2 for downsampling in residual connections instead of $1\\times 1$ convolution to reduce information loss. Batch normalization and ReLU are applied after each convolution layer for regularization and non-linearity\n \n \n \\subsubsection{Multi-branch Attention Module (MAM)}\n To eliminate the effect of background contents and focus on representing the cancerous evidence (which can be sparse), we employ attention mechanism. \n Improved on Convolutional Block Attention Module (CBAM) \n\n , which extracts channel attention and spatial attention of an input feature map in a squeeze and excitation manner, we propose a multi-branch attention module. MAM can better approximate the importance of each location on the feature map by looking at its context at different scales. \n Given a squeezed feature map $F_{sq} $ generated by the channel attention module, we compute and derive a 2D spatial attention map $A_s$ by $ A_s = \\sigma(\\sum_{k'} f^{k' \\times k'} (F_{sq}) ),$\n where $f^{k' \\times k'}$ represents a convolution operation with kernel size of $k' \\times k'$, and $\\sigma$ denotes the sigmoid function. We set $k' \\in \\{3, 5, 7\\}$ in our experiments, corresponding to 3 branches. Hereby, the feature map  $F_{sq} $ is refined by element-wise multiplication with the spatial attention map $A_s$.\n \nMAM is conceptually simple but effective in improving detection and localization performance as demonstrated in our experiments.\n \n\n\n\n \\subsubsection{Deep Supervision}\n Deep supervision \\cite{lee2015deeply} is employed to empower the intermediate layers to learn class-discriminative representations, for building the cancer activation map in a higher resolution.  We achieve this by adding two companion output layers to the last two MA-ResModules, as shown in Fig. \\ref{fig:fig1}. Global max pooling (GMP) is applied to search for the best discriminative features spatially, while global average pooling (GAP) is applied to encourage the network to identify all discriminative parts on the image. Each companion output layer applies GAP and GMP on the input feature map and concatenates the resulting vectors. The cancer score of the input image is derived by concatenating the outputs of the two companion layers followed by a fully convolutional layer (i.e., kernel size $1 \\times 1$) with a sigmoid activation. \n CELNet enjoys efficient inference when applied to test WSIs, as it is fully convolutional and avoids repetitive computation for the overlapping part between neighboring patches. \n \n \n\\subsection{Cancerous Evidence Localization Map (CELM) }\n\\subsubsection{Cancer Activation Map (CAM)}\nGiven an image $I \\in \\mathbb{R}^{H \\times W \\times 3}$, let $y^c = S_c(I)$ represent the cancer score function governed by the trained CELNet (before sigmoid layer). \nA cancer-class activation map $M^c$ shows the importance of each region on the image to the diagnostic value. For a target layer $l$, the CAM $M^c_l$ is derived by taking the weighted sum of feature maps $F_l$  with the weights \\{$\\alpha_{k,l}^c$ \\}, where $\\alpha_{k,l}^c$ represents the importance of $k^{th}$ feature plane. The weights $\\alpha_{k,l}^c$ are computed as $\\alpha_{k,l}^c = Avg_{i,j}( \\frac{\\partial y^c}{\\partial F_l^k(i,j)} )$, i.e., spatially averaging the gradients of  cancer score $y^c$ with respect to the $k^{th}$ feature plane $F_l^k$, which is achieved by back propagation (see Fig.\\ref{fig:fig1}). Thus, the CAM of layer $l$ can be derived by $ M_l^c = ReLU(\\sum_k \\alpha_{k,l}^c F_l^k)$, where ReLU is applied to exclude the features with negative influence on the class of interest \\cite{selvaraju2017grad}. \n\nWe derive two CAMs, $M^c_2$ and $M^c_3$ from the last layer of the second and the third residual module on CELNet respectively (i.e., CAM2 and CAM3 in Fig.\\ref{fig:fig1}). CAM3 can represent discriminative regions for identifying a cancer class  in a relatively low resolution while CAM2 enjoys higher resolution and still class-discriminative under deep supervision. \n\n\\subsubsection{Cancer Saliency Map (CSM)}\nIn contrast with CAM, the cancer-class saliency map shows the contribution of each pixel site to the cancer score $y^c$. This can be approximated by the derivate of a linear function $S^c(I) \\approx w^TI + b$. Thus the pixel contribution is computed as $ w = \\frac{\\partial S^c(I)}{\\partial I} $. Different from \\cite{Simonyan2013DeepIC}, we derive $w$  by the guided back-propagation \\cite{springenberg2014striving} to prevent backward flow of negative gradients. \nFor a RGB image, to obtain its cancer saliency map $M^s \\in \\mathbb{R}^{H \\times W\\times 1} $ from $w  \\in \\mathbb{R}^{H \\times W \\times 3} $, we first normalize $w$ to $[0,1]$ range, followed by greyscale conversion and Gaussian smoothing, instead of simply taking the maximum magnitude of $w$ as proposed in \\cite{Simonyan2013DeepIC}. Thus, the resulting cancer saliency map (see Fig.{~\\ref{fig:vis}} (b)) is far less noisy and more focus on class-related objects than the original one proposed in \\cite{Simonyan2013DeepIC}.\n\n\n\n\\subsubsection{Complementary Fusion}\nThe generated CAMs coarsely display discriminative regions for identifying a cancer class (see Fig.\\ref{fig:vis} (c)), while the CSM is fine-grained, sensitive and represents pixelated contributions for the identification (see Fig.\\ref{fig:vis} (b)). To combine the merits of them for precise cancerous evidence localization, we propose a complementary fusion method. First, CAM3 and CAM2 are combined to obtain a unified cancer activation map $M^c \\in \\mathbb{R}^{H \\times W\\times 1}$   as $M^c = \\alpha f_u(M^c_3) + (1- \\alpha) f_u(M^c_2)$, where $f_u$ denotes a upsampling function by bilinear interpolation, and  the coefficient $\\alpha$ in range [0,1] is confirmed by validation.\nThe CELM is derived by complementarily fusing CSM and CAM as $M = \\beta (M^c \\odot M^s) + (1 - \\beta) M^c$,\nwhere $\\odot$ denotes element-wise product, and  the coefficient $\\beta$ captures the reliability of the point-wise multiplication of CAM and CSM, and the value of $\\beta$ is estimated by cross-validation in experiments. \n\n\n\n\n\\section{Experiments \\& Results}\nWe first evaluate the detection performance of the proposed model as for clinical requirements, followed by evidence localization evaluations.  \n\n\\subsection{Datasets and Experimental Setup}\nThe detection performance of the proposed method is validated on two benchmark datasets, PCam\\cite{veeling2018rotation} and Camelyon16 \\footnote{https:\/\/camelyon16.grand-challenge.org}. \n\n\\textbf{PCam:} The PCam dataset contains 327,680 lymph node histopathology images of size $96 \\times 96$ with binary class labels indicating the presence of cancer metastasis, split into 75\\% for training, 12.5\\% for validation, and 12.5\\% for testing as originally proposed. The class distribution in each split is balanced (1:1). For a fair comparison, following \\cite{veeling2018rotation}, we perform image augmentation by random 90-degree rotations and horizontal flipping during training.\n\n\\textbf{Camelyon16:} The Camelyon16 dataset includes 270 H\\&E stained WSIs (160 normal and 110 cancerous cases) for training and 129 WSIs held out for testing (80 normal and 49 cancerous cases) with average image size about $65000 \\times 45000$, where regions with cancer metastasis are delineated in cancerous slides. To apply our CELNet on WSIs, we follow the pipeline proposed in \\cite{Liu2017DetectingCM}, including WSI pre-processing, patch sampling and augmentation, heatmap generation, and slide-level detection tasks. For slide-level classification, we take the maximum tumor score among all patches as the final slide-level prediction. For tumor region localization, we apply non-suppression maximum algorithm on the tumor probability map aggregated from patch predictions to iteratively extract tumor region coordinates. We work on the WSI data at 10$\\times$ resolution instead of  40$\\times$ with the available computation resources.\n\n\tIn our experiments, all models are trained using binary cross-entropy loss with L2 regularization of $10^{-5}$ to improve model generalizability, and optimized by SGD with Nesterov momentum of 0.9 with a batch size of 64 for 100 epochs. The learning rate is initialized with $10^{-4}$ and is halved at 50 and 75 epochs. We select model weights with minimum validation loss for test evaluation.  \n\n\\subsection{Classification Results}\t\n\nAs Tbl.{~\\ref{table1}} shows, CELNet consistently outperforms ResNet, DenseNet, and P4M-DenseNet \\cite{veeling2018rotation} in histopathologic cancer detection on the PCam dataset. \n\n P4M-DenseNet uses less parameters due to parameter sharing in the p4m-equivariance. \n\n For auxiliary experiments, we perform ablation studies and visual analysis. From Tbl.{~\\ref{table1}} , we observe that our attention module brings 1.77\\% accuracy gain, which is larger than the gain brought by CBAM \\cite{woo2018cbam}. Both the CAM and CELM on CELNet are mainly activated for the cancerous regions (see Fig.\\ref{fig:vis} (c) and (d)). These subfigures indicate that CELNet is effective in extracting discriminative evidence for histopathologic classification. \n \n\\begin{table}[h]\n\t\\centering\n\t\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={left,top} }}]{table}[\\FBwidth]\n\t\t{\n\t\t\\caption{Quantitative comparisons on the PCam test set. P4M-DenseNet  \\cite{veeling2018rotation}:  current SoTA method for  the PCam benchmark, CELNet: our method, $^{-}$: removal of the proposed multi-branch attention module, +CBAM: integration with convolutional block attention module \\cite{woo2018cbam}.  \n\t\t}}\n\t\t{\n\t\t\t\\begin{tabular}{l  c c c }\n\t\t\t\t\\toprule\n\t\t\t\tMethods\t\t\t\t\t\t\t&   Acc \t\t& AUC \t& \\#Params \\\\\n\t\t\t\t\\midrule    \n\t\t\t\tResNet18 \\cite{he2016deep}\t\t\t& \t  88.73\t\t\t&\t95.36\t\t& 11.2M\t\\\\\n\t\t\t\tDenseNet \\cite{veeling2018rotation}\t\t\t\t&\t 87.20  & 94.60\t& 902K   \\\\\n\t\t\t\tP4M-DenseNet    & 89.80 \t&\t96.30\t& 119K \\\\\n\t\t\t\t\\midrule  \n\t\t\t\tCELNet\t\t  \t\t\t    &\t \\textbf{91.87}\t& \\textbf{97.72} & 297K\t \\\\\n\t\t\t\tCELNet$^{-}$ \t \t\t\t&\t90.10 \t\t& 96.45\t& 292K\t\t\\\\\n\t\t\t\tCELNet$^{-}$ +CBAM &\t  90.86 \t& 97.17 & \t  296K\t\\\\\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t\\label{table1}\n\t\t}\n\\end{table}\n\n\nOn slide-level detection tasks, as shown in Tbl.\\ref{table2}, our CELNet based approach achieves higher classification performance (1.7\\%) in terms of AUC than the baseline method \\cite{Liu2017DetectingCM}, and  outperforms previous state-of-the-art methods in slide-level tumor localization performance in terms of FROC score. The results illustrate that instead of using off-the-shelve CNNs as the core patch-level model for histopathologic slide detection, adopting CELNet can potentially bring larger performance gain. CELNet is more parameter-efficient as shown in Tbl.\\ref{table1} and testing a slide on Camelyon16 takes about 2 minutes on a Nvidia 1080Ti GPU. \n\\begin{table}[h]\n\t\\centering\n\t\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={left,top} }}]{table}[\\FBwidth]\n\n\t{\\caption{Quantitative comparisons of slide-level classification performance (AUC) and slide-level tumor localization performance (FROC) on the Camelyon16 test set. *: The Challenge Winner uses $40\\times$ resolution while results of other methods are based on $10\\times$. \n\t}}\n\t{\n\t\t\\begin{tabular}{l  c c  }\n\t\t\t\\toprule\n\t\t\tMethods\t\t\t\t\t\t\t&   AUC \t\t& FROC \t \\\\\n\t\t\t\\midrule    \n\t\t\tP4M-DenseNet\t\t\t& \t  -\t\t\t\t\t\t\t\t\t\t\t\t&\t84.0 \t\t\t\\\\\n\t\t\tLiu \\cite{Liu2017DetectingCM}    & 96.5 \t\t\t\t\t\t&\t79.3  \t \\\\\n\t\t\tChallenge Winner$^*$ \\cite{Wang2016DeepLF}    & \\textbf{99.4} \t&\t80.7\t \\\\\n\t\t\tPathologist \t\t\t\t&\t  \t\t\t96.6 \t\t\t\t\t\t\t& 73.3\t\\\\\n\t\t\tCELNet\t\t  \t\t\t    &\t  97.2 & \\textbf{84.8} \t \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{table2}\n\t}\n\\end{table}\n\n\n\\subsection{Weakly Supervised Localization and Results}\nGiven that the trained CELNet can precisely classify a pathology image, here we aim to investigate its performance in localizing the supporting evidence based on the proposed CELM. To achieve this, based on Camelyon16, we first construct a dataset with region-level annotations for cancer metastasis, namely HPLOC, \nand develop the metrics for measuring localization performance on HPLOC. \n\n\\textbf{HPLOC:} The HPLOC dataset contains 20,000 images of size $96 \\times 96$ with segmentation masks for cancerous region. Each image is sampled from the test set of Camelyon16 and contains both cancerous regions and normal tissue in the glimpse, which harbors the high quality of the Camelyon16 dataset.\n\n\n\\textbf{Metrics:} To perform localization, we generate segmentation masks from CELM\/CAM\/CSM by thresholding and smoothing (see Fig.\\ref{fig:vis} (e)). If a segmentation mask intersects with the cancerous region by at least 75\\%\n\\footnote{The annotated contour in Camelyon16 is usually enlarged to surround all tumors. }\n, it is defined as a true positive. Otherwise, if a segmentation mask intersects with the normal region by at least 75\\%, it is considered as a false positive. Thus, we can use precision and recall score to quantitatively assess the localization performance of different WSL methods, where the results are summarized in Tbl.\\ref{table3}. \n\n\\begin{table}[h]\n\t\\centering\n\t\\floatbox[{\\capbeside\\thisfloatsetup{capbesideposition={left,top} }}]{table}[\\FBwidth]\n\t{\\caption{Quantitative comparisons for different weakly supervised localization methods on the HPLOC dataset. Ours: CELNet + CELM. MAM and DS are short for multi-branch attention module and deep supervision respectively. \n\t}}\n\t{\n\t\t\\begin{tabular}{l  c c }\n\t\t\t\\toprule\n\t\t\tMethods\t\t\t\t& Precision & Recall \\\\\n\t\t\t\\midrule   \n\t\t\tResNet18 + Backprop \\cite{Simonyan2013DeepIC}     \t\t& 79.8\t& 85.5 \\\\ \n\t\t\tResNet18 + GradCAM \\cite{selvaraju2017grad}\t\t  \t& \t 85.6\t& 82.4   \\\\\n\t\t\t\\midrule  \n\t\t\tOurs\t\t  \t\t&\t \\textbf{91.6}\t& 87.3  \t \\\\\n\t\t\tOurs w\/o MAM\t \t\t&\t88.1 \t\t\t\t& 85.6\t\\\\\n\t\t\tOurs w\/o DS\t\t&\t  90.5 \t\t& \t\\textbf{87.7}\t\\\\\n\t\t\tCELNet + GradCAM  &\t  91.0\t\t& \t85.4\t\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{table3}\n\t}\n\\end{table}\n\n\n\\begin{figure}[t\n\t\\centering\n\n\t\\begin{tabular}{cccccc}\n\n\n\n\n\n\n\n\n\n\t\t\\includegraphics[scale=0.5]{2a.png}  &\n\t\t\\includegraphics[scale=0.5]{2e_sm.png} &\n\t\t\\includegraphics[scale=0.5]{2f_cam.png} &\n\t\t\\includegraphics[scale=0.5]{2b_celm.png} &\n\n\t\t\\includegraphics[scale=0.5]{2c_loc_on_img.png} &\n\t\t\\includegraphics[scale=0.5]{2d_mask.png}\n \\\\\n\t\t(a) Input  & (b) CSM  & (c) CAM & (d) CELM  & (e) Localization  & (f) GT   \\\\\n\t\\end{tabular}\n\n\t\\caption{ Evidence localization results of our WSL method on the HPLOC dataset.\n\t\t (a) Input glimpse, (b) Cancer Saliency Map, (c) Cancer Activation Map, (d) CELM: Cancerous Evidence Localization Map, (e) Localization results based on CELM,  where the localized evidence is highlighted for providing visual assistance, (f) GT: ground truth, white masks represent tumor regions and the black represents normal tissue} \n\n\t\\label{fig:vis}\n\n\\end{figure}\n\n\nWe observe that our WSL method based on CELNet and CELM consistently performs better than the back propagation-based approach \\cite{Simonyan2013DeepIC} and the class activation map-based approach \\cite{selvaraju2017grad}. Note that we used ResNet18 \\cite{he2016deep} as the backbone for the compared methods because it achieves better classification performance and provides higher resolution for GradCAM (12$\\times$12) as compared to DenseNet (3 $\\times$ 3) \\cite{veeling2018rotation}. \nWe perform ablation studies to further evaluate the key components of our method in Tbl.\\ref{table3}. We observe the effectiveness of the proposed multi-branch attention module in increasing the localization accuracy. \nThe deep supervision mechanism effectively improves the precision in localization despite slightly lower recall score, which can be caused by the regularization effect on the intermediate layers, that is, encouraging the learning of discriminative features for classification but also potentially discouraging the learning of some low-level histological patterns. \nWe observe that using CELM can improve the recall score and precision, which indicates that CELM allows better discovery of cancerous evidence than using GradCAM. \nWe present the visualization results in Fig. {\\ref{fig:vis}}, the cancerous evidence \nis represented as large nucleus and hypercellularity in the images, which are precisely captured by the CELM. Fig.\\ref{fig:vis}(e) visualizes the localization results by overlaying the segmentation mask generated from CELM onto the input image, which demonstrates the effectiveness of our WSL method in localizing cancerous evidence.\n\n\n\\section{Discussion \\& Conclusions}\nIn this paper, we have proposed a generalizable method for localizing cancerous evidence on histopathology images. \nUnlike the conventional feature-based approaches, the proposed method does not rely on specific feature descriptors but learn discriminative features for localization from the data. \nTo the best of our knowledge, investigating weakly supervised CNNs for cancerous evidence localization and quantitatively evaluating them on large datasets have not been performed on histopathology images. \nExperimental results show that our proposed method can achieve competitive classification performance on histopathologic cancer detection, and more importantly, provide reliable and accurate cancerous evidence localization using weakly training data, which reduces the burden of annotations. We believe that such an extendable method can have a great impact in detection-based studies in microscopy images and help improve the accuracy and interpretability for current deep learning-based pathology analysis systems. \n\n\\bibliographystyle{splncs04}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn continuum thermodynamics, the constitutive theories are based, besides some general invariance principles, on the second law of thermodynamics, which states that in every admissible process the entropy production has to be non-negative \\cite{Truesdell}.\n\nA rigorous procedure for the exploitation of the entropy principle has been developed for the first time in 1963 by Coleman and Noll \\cite{Coleman-Noll}, and later by Coleman and Mizel \\cite{Coleman-Mizel}. In both papers, the authors assumed the unbalance of entropy \nin the classical form, say the Clausius-Duhem inequality; in this inequality, the entropy flux  is taken as the ratio between the heat flux and the absolute temperature. Later on, M\\\"uller \\cite{Muller} proposed an extension of the entropy inequality, allowing a more general expression for the entropy flux, thus obtaining the thermodynamic compatibility for wider classes of materials. A slightly different approach has been applied by other authors who accepted the classical Clausius-Duhem inequality, but proposed a more general form of the local balance of energy \\cite{Gurtin,Dunn-Serrin,Dunn}. Furthermore,  in 1972, Liu  \\cite{Liu} developed a different procedure for the analysis of the entropy principle, based on the method of Lagrange multipliers.\n\nIn all these papers, the basic assumption is that the second law of thermodynamics restricts the constitutive equations and not the thermodynamic processes. Hence, the constitutive relations are required to be such that the entropy inequality be satisfied for all solutions of the thermodynamic field equations. This assumption is purely mathematical and from a physical point of view may have two\ndifferent interpretations \\cite{Muschik-Ehrentraut}:\n\\begin{itemize}\n\\item all solutions of the balance equations have to satisfy the second law;\n\\item there are solutions of the balance equations which satisfy the second law, and other ones which do not. \n\\end{itemize}\n\nThe first interpretation requires that the constitutive equations must be assigned in such a way the entropy inequality is satisfied along arbitrary processes, whereas the second one means that we have to exclude from the set of solutions of the balance equations those which are not physically achievable, since they do not satisfy the second law of thermodynamics. In \\cite{Muschik-Ehrentraut},  the authors\nproposed a way to choose between the two statements through an \\emph{amendment to the second law}, by expliciting the nearly self-evident, but never precisely formulated, postulate that there are no reversible process directed towards non-equilibrium. \nBy means of this amendment, they were able to prove that, necessarily, the second law of thermodynamics restricts the constitutive equations and not the processes. Such a result justifies, from the physical point of view, the approach to the exploitation of second law \nthrough  Coleman-Noll and Liu procedures (see also \\cite{Muschik,Muschik-Papenfuss-Ehrentraut}).\n\nIn a series of papers \\cite{Cimmelli-2007,CST-JMP-2009,CST-PRSA-2010,COT-JMP-2011},\nthe two classical Coleman-Noll and Liu procedures have been extended in order to use as constraints in\nthe entropy inequality both the balance equations and their gradient extensions up to a suitable finite order. \nThis approach, successfully used in many applications of physical interest (see, for instance, \n\\cite{CST-JNET-2010,COP-Elasticity-2011,COP-IJNLM-2013,COP-CMT-2015,OPR-2016,CGOP-miscele-2020}), \nrevealed essential in order to ensure the compatibility of non-local constitutive relations with second law of thermodynamics \nwithout modifying \\emph{a priori} the entropy inequality or the energy balance through the introduction of extra-terms.\nIn particular, the extended Liu technique requires to add to the entropy inequality a linear combination of the field equations and of the spatial gradients of the latter (in the following they are called extended equations), up to the order of the gradients entering the state space. The coefficients of this linear combination are the Lagrange multipliers and depend upon the state variables only. Thus,  the number of independent constraints to be taken into account is never less than that of the unknown elements entering the constitutive equations as independent state variables.  \n\nIn this paper,  we start discussing the Liu extended procedure from an abstract mathematical point of view, \nand then apply the results to some\nphysical instances of continua with non-local constitutive equations. In fact, we consider a system of first order balance laws in one space dimension sufficiently general to contain the equations governing the thermodynamical processes occurring in a continuous medium. We assume that this system involves some functions whose constitutive equations are allowed to be non-local, \\emph{i.e.}, we admit the possibility that the \\emph{state space} includes the gradients of (not necessarily all) the field variables up to the order $r\\ge 1$. The compatibility of the constitutive equations with an entropy-like inequality  is then discussed. Once we expand the derivatives in the\nentropy-like inequality and impose as constraints the balance equations and some of their gradient extensions,\na set of sufficient conditions, such that the entropy-like inequality is not violated, are  derived. \nFurthermore, the results are applied to two physical instances of continua: \nthe first example is concerned with a fluid with a scalar internal variable and constitutive equations with first order \nnon-localities, the second one to a fluid whose  state space includes also the second order spatial derivative of the mass density (Korteweg fluid \\cite{Korteweg}).\n\nThe plan of the paper is the following. In Section~\\ref{sec:balance}, we define  mathematically the problem, and introduce the notation that\nwill be used throughout the paper. In Section~\\ref{sec:liu}, we discuss the extended Liu procedure and state the theorem providing the sufficient conditions in order the entropy-like inequality be satisfied for all thermodynamical processes. In Section~\\ref{sec:applications},\nwe give two non-trivial applications of the procedure in meaningful physical situations, and solve the thermodynamical conditions providing explicitly a solution for the constitutive equations. Finally, Section~\\ref{sec:conclusions} contains some concluding remarks.\n\n\\section{Balance equations for a continuous medium}\n\\label{sec:balance}\nPhysical laws describing the mechanical as well as the thermodynamical \nproperties of continuous media are usually expressed in terms of balances of some physical quantities (mass, linear and angular momentum, energy, etc.). Here, for simplicity, we consider the case of one-dimensional continuous media and postpone to a forthcoming paper the multi-dimensional case. \n\nLet \n\\[\n\\mathbf{u}\\equiv(u_1(t,x),\\ldots,u_n(t,x))\n\\] \nbe the vector of $n$ field variables, depending on time $t$ and space $x$, describing a one-dimensional continuum. \n\nIt is well known that the general governing equations of a continuum are underdetermined since they involve some constitutive functions that specify the particular continuum we are dealing with. \nThe various constitutive quantities (for instance, the Cauchy stress tensor or the heat flux) depend on the so called \\emph{state variables}: the state variables include (some of) the field variables when a local constitutive theory is adopted, or,  when non-local constitutive theories are considered, also the spatial derivatives up to a finite order $r$ of some field variables. In the following, we shall be concerned with\na non-local constitutive theory. \nIn fact, in our framework, the state variables will be the elements of the set \n\\[\n\\displaystyle \\mathcal{Z}=\\bigcup_{k=0}^r \\mathcal{Z}^{(k)},\n\\]\nwhere\n\\[\n\\begin{aligned}\n&\\mathcal{Z}^{(0)}\\subseteq \\{u_1,\\ldots,u_n\\}\\equiv \\mathcal{U}^{(0)},\\\\\n&\\mathcal{Z}^{(k)}\\subseteq \\left\\{\\frac{\\partial^k u_1}{\\partial x^k},\\ldots,\\frac{\\partial^k u_n}{\\partial x^k}\\right\\}\\equiv \\mathcal{U}^{(k)}, \\qquad k=1,\\ldots,r,\n\\end{aligned}\n\\]\nwhere $r\\ge 1$.\nFinally, let us denote with $\\mathbf{z}$ the vector whose $N$ components belong to the set $\\mathcal{Z}$ of state variables, and $\\mathbf{z}^\\star$ the vector whose $N^\\star$ components belong to the set $\\mathcal{Z}^\\star=\\mathcal{U}^{(0)}\\bigcup \\mathcal{Z}$.\n\nIn local form, the thermomechanical description of a continuum leads to consider a system of partial differential equations having the form\n\\begin{equation}\n\\label{generalbalance}\n\\mathcal{E}_i\\equiv\n\\frac{D\\Phi_i(\\mathbf{u})}{D t}+\\frac{D \\left(\\Psi_i(\\mathbf{u})+\\chi_i(\\mathbf{z}^\\star)\\right)}{D x}-\\Gamma_i(\\mathbf{z}^\\star)=0, \\qquad i=1,\\ldots, n,\n\\end{equation}\nwhere the differential operators $D\/Dt$ and $D\/Dx$, acting on a composite function $F$ depending on $t$ and $x$ through some quantities $w_1(t,x),\\ldots, w_n(t,x)$\n(in the paper, these variables are the field variables or the elements of the state space), by chain rule, are defined as follows\n\\[\n\\frac{D F}{Dt}=\\sum_{j=1}^n\\frac{\\partial F}{\\partial w_j}\\frac{\\partial w_j}{\\partial t},\\qquad\n\\frac{D F}{Dx}=\\sum_{j=1}^n\\frac{\\partial F}{\\partial w_j}\\frac{\\partial w_j}{\\partial x}.\n\\]\nThese operators do not correspond to total derivatives, and we introduce them in order to avoid confusion when stating Theorem 1, see below.\nIn the equations (\\ref{generalbalance}),\n\\begin{itemize}\n\\item $\\Phi_i(\\mathbf{u})$ are some densities, depending at most on the field variables;\n\\item the fluxes are split as sums of functions $\\Psi_i(\\mathbf{u})$ depending at most on the field variables, and \n$\\chi_i(\\mathbf{z}^\\star)$  depending at most on the field variables and the state variables;\n\\item $\\Gamma_i(\\mathbf{z}^\\star)$ are the production terms depending at most on the field variables and the state variables. \n\\end{itemize}\n\nIt is well known that in general the constitutive functions have to satisfy some universal principles  \n(invariance with respect to rigid motions, time translation, \nscale changes of fundamental quantities, Galilei or Lorentz transformations, etc.). In such a framework, general representation theorems for isotropic scalar, vectorial or tensorial constitutive equations have to be taken into account \\cite{Wang1,Wang2,Smith1,Smith2,Smith3}. \nAdditional constraints are imposed by the second law of thermodynamics, which requires that the admissible processes must be such that the entropy production is non-negative.\n\nTherefore, in our general framework, we have to exploit the compatibility of constitutive relations with an entropy-like inequality, assumed in the form\n\\begin{equation}\\label{entropyineq1}\n\\frac{D s(\\mathbf{z})}{D t}+\\frac{D(v s(\\mathbf{z}) +J_s(\\mathbf{z}))}{D x}\\ge 0,\n\\end{equation}\nwhere  $s$ and $J_s$, which are functions depending on the state variables, represent the entropy and the entropy flux, respectively, and $v$ the velocity. In the case where the velocity does not appear among the field variables (for instance, in the case of a model of rigid heat conductors), \nthe entropy-like inequality is assumed to be\n\\begin{equation}\\label{entropyineq2}\n\\frac{D s(\\mathbf{z})}{D t}+\\frac{D J_s(\\mathbf{z})}{D x}\\ge 0.\n\\end{equation}\n\nIn what follows, we will use the entropy-like inequality in the form \\eqref{entropyineq1};   \nin the cases where the velocity does not belong to the field variables, the corresponding results are obtained simply by setting $v=0$.\n\nFrom the analysis of the entropy inequality, some restrictions on the form of constitutive \nequations can be obtained.\nClassically, the  restrictions placed by the entropy principle on the constitutive functions are found by using the Coleman-Noll  procedure \\cite{Coleman-Noll,Coleman-Mizel}, or the Liu  one \\cite{Liu}.\nBoth procedures have to be extended in order to manage non-local constitutive equations. \nThe entropy principle imposes that the inequality \\eqref{entropyineq1} must be satisfied for arbitrary \nthermodynamic processes \\cite{CJRV-2014,JCL-2010}. To find a set of conditions which are at least sufficient for the \nfulfilment of such a constraint, we apply an extended Liu procedure recently developed in a series of papers \n\\cite{Cimmelli-2007,CST-JMP-2009,CST-PRSA-2010,COT-JMP-2011}, \nincorporating new restrictions consistent with higher order non-local constitutive \ntheories. In fact, in order to exploit the second law, we use as constraints both the balance equations for the unknown fields and their extended equations up to the order of the \nderivatives entering the state space.\n\nSimple mathematical considerations may clarify the necessity of imposing as additional constraints in the \nentropy inequality the gradients of the balance equations when dealing with non-local constitutive equations. \n\nThe thermodynamic processes are solutions of the balance equations, and, if these solutions are smooth enough, \nare trivially solutions of their differential consequences  (see also \\cite{Rogolino-Cimmelli-2019}). \nSince the entropy inequality \\eqref{entropyineq1} has to be satisfied in arbitrary smooth processes,  then it is \nnatural, from a mathematical point of view, to use the differential consequences of the equations governing those \nprocesses as constraints for such an inequality. \nNext Section will be devoted to the description of the extended Liu procedure in this general framework.\n \n\\section{Extended Liu procedure}\n\\label{sec:liu}\nIn this Section, we introduce a general scheme in order to apply the extended Liu technique in the \ncase of $r$-th order ($r\\ge 1$) non-local constitutive equations.\n\nWe consider this rather general case essentially for two reasons. The first reason is to have a unified framework good enough to be applied to different models with non-local constitutive equations of arbitrary order. The second one is of computational nature. In fact, in dealing with applied problems, it is relevant both the derivation of the thermodynamic restrictions arising from the entropy inequality, and, whenever this is possible, a more or less explicit \ncharacterization of the constitutive equations. Since the thermodynamic restrictions may have  a lengthy expression, it is convenient to use a computer algebra system for their possible solution. In order to be able to have a flexible computer algebra package that can be used in many different cases, a general approach reveals useful if not necessary. In fact, we developed some general routines in the CAS Reduce \\cite{Reduce} that implement the algorithm at the core of extended Liu approach. \n\nFirst of all, we need to compute the spatial derivatives of the fundamental balance equations. \nFrom  the system \\eqref{generalbalance}, developing the first order time and space derivatives, one has:\n\\begin{equation}\n\\mathcal{E}_i\\equiv \\frac{\\partial \\Phi_i}{\\partial u_j}\\frac{\\partial u_j}{\\partial t}+\\frac{\\partial \\Psi_i}{\\partial u_j}\\frac{\\partial u_j}{\\partial x}+\n\\frac{\\partial \\chi_i}{\\partial z^\\star_{\\alpha}}\\frac{\\partial z^\\star_\\alpha}{\\partial x}-\\Gamma_i=0,\n\\end{equation}\nwhere the Einstein summation convention over repeated indices is used. \nIn order to write in general the $m$-th order spatial derivative of the balance laws, let us recall \n\\cite{Mishkov} a formula giving an expression for the $m$-th derivative of a composite function when the argument is a vector with an arbitrary number of components. This formula is a generalization of the well known Fa\\`a di Bruno's formula \\cite{FaadiBruno,Roman}.\n\nThe following theorem is no more than a simple rewriting of the main result contained in \\cite{Mishkov}.\n\\begin{theorem}\nLet $\\mathbf{w}=(w_1(t,x),\\ldots,w_s(t,x))$ be a vector, and $F(\\mathbf{w}(t,x))$ a composite function for which all the needed derivatives are defined, then\n\\begin{equation}\n\\label{faabruno}\n\\begin{aligned}\n&\\frac{D^m F(\\mathbf{w}(t,x))}{D x^m}=\n\\sum_{J_0}\\sum_{J_1}\\cdots\\sum_{J_m}\n\\frac{m!}{\\prod_{i=1}^m(i!)^{k_i}\\prod_{i=1}^m\\prod_{j=1}^s q_{ij}!}\\times\\\\\n&\\qquad\\times \\frac{\\partial^k F}{\\partial w_1^{p_1}\\partial w_2^{p_2}\\cdots\\partial w_s^{p_s}}\n\\prod_{i=1}^m \\left(\\frac{\\partial^i w_1}{\\partial x^i}\\right)^{q_{i1}}\n\\left(\\frac{\\partial^i w_2}{\\partial x^i}\\right)^{q_{i2}}\\cdots \n\\left(\\frac{\\partial^i w_s}{\\partial x^i}\\right)^{q_{is}},\n\\end{aligned}\n\\end{equation}\nwhere the various sums are over all nonnegative integer solutions of the Diophantine equations\n\\begin{equation*}\n\\begin{aligned}\n&\\sum_{J_0} \\rightarrow k_1+2k_2+\\ldots +mk_m = m,\\\\\n&\\sum_{J_1} \\rightarrow q_{11}+q_{12}+\\ldots +q_{1s} = k_1,\\\\\n&\\sum_{J_2} \\rightarrow q_{21}+q_{22}+\\ldots +q_{2s} = k_2,\\\\\n&\\ldots\\\\\n&\\sum_{J_m} \\rightarrow q_{m1}+q_{m2}+\\ldots +q_{ms} = k_m,\n\\end{aligned}\n\\end{equation*}\nand it is\n\\begin{equation*}\n\\begin{aligned}\n&p_j=q_{1j}+q_{2j}+\\ldots+q_{mj}, \\qquad j=1,\\ldots,s,\\\\\n&k=p_1+p_2+\\ldots+p_s=k_1+k_2+\\ldots+k_m. \\qquad \\square\n\\end{aligned}\n\\end{equation*}\n\\end{theorem}\n\nBy introducing the $m$-th order $(m=1,\\ldots,r)$ spatial derivatives of the balance laws, we get\n\\begin{equation}\n\\label{estese}\n\\begin{aligned}\n&\\frac{D^m \\mathcal{E}_i}{D x^m}\\equiv\\sum_{h=0}^k\\binom{m}{h}\\left[\\frac{D^{h}}{D x^{h}}\\left(\\frac{\\partial \\Phi_i(\\mathbf{u})}{\\partial u_j}\\right)\\frac{\\partial^{m-h+1}u_j}{\\partial t \\partial x^{m-h}} \\right.\\\\\n&\\left.+\\frac{D^{h}}{D x^{h}}\\left(\\frac{\\partial \\Psi_i(\\mathbf{u})}{\\partial u_j}\\right)\\frac{\\partial^{m-h+1}u_j}{\\partial x^{m-h+1}}+\\frac{D^h}{D x^h}\n\\left(\\frac{\\partial \\chi_i(\\mathbf{z}^\\star)}{\\partial z^\\star_\\alpha}\\right)\\frac{\\partial^{m-h+1}z^\\star_\\alpha}{\\partial x^{m-h+1}}\\right]\\\\\n&-\\frac{D^m \\Gamma_i(\\mathbf{z}^\\star)}{D x^{m}}=0.\n\\end{aligned}\n\\end{equation}\nIn order to take into account in the entropy inequality the restrictions determined by the field equations and their spatial derivatives, let us introduce the Lagrange multipliers $\\Lambda^{(k)}_i$\n$(i=1,\\ldots,n,\\; k=0,\\ldots,r)$ associated to \n$\\displaystyle\\frac{D^k\\mathcal{E}_i}{D x^k}$. Therefore, the entropy inequality \\eqref{entropyineq1} becomes\n\\begin{equation}\\label{dis}\n\\frac{\\partial s(\\mathbf{z})}{\\partial z_\\alpha}\\frac{\\partial z_\\alpha}{\\partial t}+v\\frac{\\partial s(\\mathbf{z})}{\\partial z_\\alpha}\\frac{\\partial z_\\alpha}{\\partial x}+s(\\mathbf{z})\\frac{\\partial v}{\\partial x}+\\frac{\\partial J_s(\\mathbf{z})}{\\partial z_\\alpha}\\frac{\\partial z_\\alpha}{\\partial x}-\\sum_{i=1}^n\\sum_{k=0}^r\n\\Lambda_i^{(k)}\\frac{D^k \\mathcal{E}_i}{D x^k}\\geq 0.\n\\end{equation}\n\nBy expanding the derivatives in \\eqref{dis}, and using formulae \\eqref{faabruno} and \\eqref{estese}, a straightforward though tedious computation provides a very long expression that turns out to be a polynomial in some derivatives of field variables not belonging to the state space with coefficients depending at most on the field and state variables. This polynomial must be non-negative! Once \\eqref{dis} has been expanded, we may distinguish the derivatives of field variables therein appearing,\nand not entering the state space, in two different classes:\n\\begin{itemize}\n\\item \\emph{highest derivatives}: time derivatives of the field variables, time derivatives of the spatial derivatives (up to the order $r$) of the field variables, and spatial derivatives of highest order: it is easily ascertained that these highest derivatives appear linearly with coefficients depending on the field and state variables;\n\\item \\emph{higher derivatives}: spatial derivatives whose order is not maximal but higher than that of the derivatives entering the state space: it is easily recognized that these higher derivatives appear in powers of degree up to $r+1$, with coefficients depending on the field and state variables.\n\\end{itemize}\n\nLet us define the following sets:\n\\[\n\\begin{aligned}\n&\\widehat{\\mathcal{Z}}^{(k)}=\\left\\{ w \\in \\mathcal{Z}^{(k)} \n\\;:\\; \\frac{\\partial^h w}{\\partial x^h}\\notin \\mathcal{Z},\\; h=1,\\ldots,r-k\\right\\},\\quad k=0,\\ldots,r-1,\\\\\n&\\widehat{\\mathcal{Z}}^{(r)}\\equiv \\mathcal{Z}^{(r)},\n\\end{aligned}\n\\]\nand\n\\[\n\\begin{aligned}\n&\\widehat{\\mathcal{Z}}=\\bigcup_{k=0}^r \\widehat{\\mathcal{Z}}^{(k)}.\n\\end{aligned}\n\\]\nThe highest derivatives are the time derivatives of the field variables and of their spatial derivatives up to the order $r$, together with the $(r+1)$th order spatial derivatives of the elements belonging to the set $\\widehat{\\mathcal{Z}}$.\n\nTherefore, denoting with  $\\boldsymbol\\zeta$ the vector whose components $\\zeta_i$ are the highest derivatives, and with \n$\\boldsymbol\\eta$ the vector whose components $\\eta_j$ are the higher derivatives, \nthe entropy inequality \\eqref{dis} can be cast in the following compact form:\n\\begin{equation}\n\\begin{aligned}\nA_i(\\mathbf{z}^\\star) \\zeta_i&+B^{(r+1)}_{j_1\\ldots j_{r+1}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r+1}}+\nB^{(r)}_{j_1\\ldots j_{r}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r}}\\\\\n&+\\ldots+B^{(2)}_{j_1j_{2}}(\\mathbf{z}^\\star)\\eta_{j_1}\\eta_{j_{2}}+B^{(1)}_{j_1}(\\mathbf{z}^\\star)\\eta_{j_1}+B^{(0)}(\\mathbf{z}^\\star)\\ge 0,\n\\end{aligned}\n\\end{equation}\nwhere the coefficients $A_i$, $B^{(r+1)}_{j_1\\ldots j_{r+1}}$, $B^{(r)}_{j_1\\ldots j_{r}}$,\\ldots,\n$B^{(2)}_{j_1j_{2}}$,  $B^{(1)}_{j_1}$ and $B^{(0)}$ may depend upon the field variables and the elements entering the state space.\nThis inequality must be satisfied for every thermodynamical process. \n\nFirst, let us observe that nothing prevents to have a thermodynamic process where $B^{(0)}=0$.\nMoreover, since we used in the entropy inequality all the constraints imposed by\nthe field equations together with their spatial derivatives, the highest and higher derivatives may assume arbitrary values. \nConsequently, we may give a set of conditions that are sufficient in order the inequality \\eqref{dis} be fulfilled for every thermodynamical process. These sufficient conditions provide constraints on the constitutive equations.\n\n\\begin{theorem}\\label{theorem2}\nLet $\\boldsymbol\\zeta=(\\zeta_1,\\ldots,\\zeta_p)$ be the vector of highest derivatives, and $\\boldsymbol\\eta=(\\eta_1,\\ldots,\\eta_q)$ the vector of higher derivatives.  \nLet \n\\begin{itemize}\n\\item $A_i(\\mathbf{\\mathbf{z}^\\star})$ are $p$ functions of $\\mathbf{z}^\\star$;\n\\item $B^{(k)}_{j_1\\ldots j_{k}}(\\mathbf{z}^\\star)$, with $k=1,\\ldots,r+1$, are $\\binom{q-k+1}{k}$ functions of $\\mathbf{z}^\\star$; \n\\item $B^{(0)}(\\mathbf{z}^\\star)$ is a function of $\\mathbf{z}^\\star$.\n\\end{itemize}\nThe inequality\n\\begin{equation}\n\\begin{aligned}\nA_i(\\mathbf{z}^\\star) \\zeta_i&+B^{(r+1)}_{j_1\\ldots j_{r+1}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r+1}}+\nB^{(r)}_{j_1\\ldots j_{r}}(\\mathbf{z}^\\star)\\eta_{j_1}\\cdots \\eta_{j_{r}}\\\\\n&+\\ldots+B^{(2)}_{j_1j_{2}}(\\mathbf{z}^\\star)\\eta_{j_1}\\eta_{j_{2}}+B^{(1)}_{j_1}(\\mathbf{z}^\\star)\\eta_{j_1}+B^{(0)}(\\mathbf{z}^\\star)\\ge 0,\n\\end{aligned}\n\\end{equation}\nholds for arbitrary vectors $\\boldsymbol\\zeta$ and $\\boldsymbol\\eta$ if \n\\begin{enumerate}\n\\item $A_i=0$;\n\\item $B^{(2k-1)}_{j_1\\ldots j_{2k-1}}=0$, $k=1,\\ldots,\\lfloor{\\frac{r+2}{2}}\\rfloor$\\footnote{$\\lfloor{x}\\rfloor$ denotes the greatest integer less than $x$.};\n\\item $B^{(2k)}_{j_1\\ldots j_{2k}}\\eta_{j_1}\\cdots \\eta_{j_{2k}}$, $k=1,\\ldots,\\lfloor{\\frac{r+1}{2}}\\rfloor$,\n nonnegative for all $\\boldsymbol\\eta$;\n\\item $B^{(0)}\\ge 0$. $\\qquad \\square$\n\\end{enumerate}\n\n\\end{theorem}\n\nDue to Theorem~\\ref{theorem2}, by imposing that the coefficients of $u_{j,t}$, $u_{j,t x}$, \\ldots, $u_{j,t,\\underbrace{x\\ldots x}_{r}}$, where the indices ${(\\cdot)}_{,t}$ and ${(\\cdot)}_{,x}$ denote the\npartial derivatives with respect to time and space, respectively,\nare vanishing, we obtain:\n\\begin{equation}\n\\begin{aligned}\n&\\frac{\\partial s}{\\partial u_j}-\\sum_{i=1}^n\\sum_{k=0}^r\\Lambda_i^{(k)}\\frac{D^k}{Dx^k}\\left(\\frac{\\partial\\Phi_i}{\\partial u_j}\\right)=0,\\\\\n&\\frac{\\partial s}{\\partial u_{j,x}}-\\sum_{i=1}^n\\sum_{k=1}^r \\binom{k}{k-1}\\Lambda_i^{(k)}\\frac{D^{k-1}}{Dx^{k-1}}\\left(\\frac{\\partial\\Phi_i}{\\partial u_j}\\right)=0,\\\\\n&\\frac{\\partial s}{\\partial u_{j,xx}}-\\sum_{i=1}^n\\sum_{k=2}^r \\binom{k}{k-2}\\Lambda_i^{(k)}\\frac{D^{k-2}}{Dx^{k-2}}\\left(\\frac{\\partial\\Phi_i}{\\partial u_j}\\right)=0,\\\\\n&\\ldots\\\\\n&\\frac{\\partial s}{\\partial u_{j,\\underbrace{x\\ldots x}_{r}}}-\\sum_{i=1}^n\\Lambda_i^{(r)}\\frac{\\partial\\Phi_i}{\\partial u_j}=0,\n\\end{aligned}\n\\end{equation}\nthat allow us the determination of the Lagrange multipliers. The coefficients of the remaining highest derivatives, \n\\begin{equation}\n\\sum_{i=1}^n\\Lambda_i^{(r)}\\left(\\frac{\\partial\\Psi_i}{\\partial \\widehat{z}_\\alpha}+\\frac{\\partial\\chi_i}{\\partial \\widehat{z}_\\alpha}\\right)=0,\\qquad \\widehat{z}_\\alpha\\in\\widehat{\\mathcal{Z}},\n\\end{equation}\nprovide conditions to be used together with the constraints coming from \nthe arbitrariness of the higher derivatives to restrict the constitutive equations. After all these restrictions have been derived, the residual entropy inequality, say\n\\begin{equation} \nB^{(0)}\\ge 0,\n\\end{equation}\nremains providing further constraints.\n\nIt is evident that the general restrictions here derived can not be discussed\nfrom a physical point of view, but they are essential in writing the computer algebra program that almost automatically computes the restrictions placed by second law.\n\nIn the next Section, we provide some examples of physical interest where the procedure here described can be applied, and we discuss the physical meaning of the results.\n\n\\section{Applications}\n\\label{sec:applications}\nHere, we consider two physical examples of fluids whose constitutive equations involve first or second order non-localities, \\emph{i.e.}, special instances of higher grade fluids. \nIn modern terminology, a fluid is said to be of grade $(r+1)$ \n\\cite{Dunn-Serrin,Truesdell_Noll,Dunn-Rajagopal,Gouin2019} if the constitutive quantities are allowed to depend on gradients of order  $r$.  In recent years, these higher grade \nfluids have been employed, for instance, to model capillarity effects \n\\cite{Gouin1985a,Gouin1985b}, or to analyze the structure of liquid-vapor phase transitions \nunder both static\n\\cite{Aifantis-Serrin-1,Aifantis-Serrin-2} and dynamic \\cite{Slemrod-1,Slemrod-2} conditions. \n\nAs observed in \\cite{Dunn-Serrin,Dunn}, these fluids are, in general, incompatible with the restrictions placed by the second law of thermodynamics. In order to find a remedy to such an incompatibility, these authors proposed a generalization of the classical local balance of energy by postulating the existence of a rate of supply of mechanical energy, the so called interstitial working; in such a framework, the entropy flux has the classical form as the ratio between the heat flux and the absolute temperature. Nevertheless, the same authors \\cite{Dunn-Serrin} remarked that the interstitial working can be removed but at the cost of introducing an entropy extra-flux \\cite{Muller} in order to satisfy the second law of thermodynamics. \n\nIn the applications we consider below, the assumptions we make consist in taking the local energy balance in the classical form without including any extra-term; moreover, we write the entropy inequality without specializing the form of the entropy flux: as will be seen, the expression of entropy flux will arise as a consequence of the extended procedure when solving the constraints placed by the second law.\n\n\\subsection{Fluid of grade 2 with a scalar internal variable}\nLet us consider a fluid of grade 2 whose description involves, in addition to the basic fields of mass density, velocity and internal energy, an internal variable. The latter may describe an additional internal degree of freedom of the material, for instance representative of a suitable scalar microstructure \\cite{Capriz,OS-2008} or another extensive property.\n\nThe governing equations we consider read\n\\begin{equation}\n\\label{fluid-grade-2}\n\\begin{aligned}\n&\\mathcal{E}_1\\equiv\\frac{D\\rho}{D t} + \\frac{D (\\rho v)}{D x}=0, \\\\\n&\\mathcal{E}_2\\equiv\\frac{D(\\rho v)}{D t} + \\frac{D (\\rho v^2  - T)}{D x}=0,\\\\\n&\\mathcal{E}_3\\equiv\\frac{D}{D t}\\left(\\rho\\varepsilon+\\rho \\frac{v^2}{2}\\right) +\\frac{D }{D x}\\left(\\rho v\\varepsilon+\\rho \\frac{v^3}{2}-Tv+ q\\right)=0,\\\\\n&\\mathcal{E}_4\\equiv\\frac{D(\\rho\\gamma)}{Dt}+\\frac{D(\\rho v\\gamma+\\phi)}{Dx}=0,\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ is the mass density, $v$ the velocity, $\\varepsilon$ the internal energy per unit mass, and $\\gamma$ an internal state variable;\nmoreover, the Cauchy stress $T$, the heat flux $q$, and the flux $\\phi$  of internal variable must be assigned by \nmeans of suitable constitutive equations such that for every admissible process the entropy inequality \n\\begin{equation}\n\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\ge 0\n\\end{equation}\nbe satisfied, being $s$ and $J_s$ (to be assigned as constitutive quantities) the specific entropy and  the entropy flux, respectively.\n\nWe assume the state space spanned by\n\\begin{equation}\n\\mathcal{Z}=\\{\\rho,\\varepsilon,\\gamma,\\rho_{,x},v_{,x},\\varepsilon_{,x},\\gamma_{,x}\\}.\n\\end{equation}\n\nAs shown in the previous Section, the exploitation of second law of thermodynamics is here performed by taking into \naccount the constraints \nimposed on the thermodynamic processes by the balance equations and their first order extensions; these \nconstraints are imposed by introducing some  Lagrange multipliers.\nTherefore, the entropy inequality becomes\n\\begin{equation}\n\\label{entropyconstrained}\n\\begin{aligned}\n&\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\\\\n&\\quad- \\Lambda^{(0)}_1 \\mathcal{E}_1- \\Lambda^{(0)}_2 \\mathcal{E}_2- \\Lambda^{(0)}_3 \\mathcal{E}_3\n- \\Lambda^{(0)}_4 \\mathcal{E}_4\\\\\n&\\quad-\\Lambda^{(1)}_1\\frac{D\\mathcal{E}_1}{Dx}-\\Lambda^{(1)}_2\\frac{D\\mathcal{E}_2}{Dx}\n-\\Lambda^{(1)}_3\\frac{D\\mathcal{E}_3}{Dx}-\\Lambda^{(1)}_4\\frac{D\\mathcal{E}_4}{Dx} \\geq 0.\n\\end{aligned}\n\\end{equation}\nFor the sake of clarity, we present the details of the computation we are required to do in applying the extended Liu procedure. \n\nExpanding the derivatives in the entropy inequality \\eqref{entropyconstrained}, we obtain a very long expression, say\n\\begin{align*}\n&\\left(\\rho\\frac{\\partial s}{\\partial\\rho}-\\Lambda^{(0)}_1\\right)\\rho_{,t}-\\left(\\rho\\Lambda^{(0)}_2+\\rho_{,x}\\Lambda^{(1)}_2\\right)v_{,t}+\\left(\\rho\\frac{\\partial s}{\\partial\\varepsilon}-\\rho\\Lambda^{(0)}_3-\\rho_{,x}\\Lambda^{(1)}_3\\right)\\varepsilon_{,t}\\allowdisplaybreaks\\\\\n&\\quad +\\left(\\rho\\frac{\\partial s}{\\partial\\gamma}-\\rho\\Lambda^{(0)}_4-\\rho_{,x}\\Lambda^{(1)}_4\\right)\\gamma_{,t}\n+\\left(\\rho\\frac{\\partial s}{\\partial\\rho_{,x}}-\\Lambda^{(1)}_1\\right)\\rho_{,tx}\\allowdisplaybreaks\\\\\n&\\quad+\\rho\\left(\\frac{\\partial s}{\\partial v_{,x}}-\\Lambda^{(1)}_2\\right)v_{,tx}\n+\\rho\\left(\\frac{\\partial s}{\\partial \\varepsilon_{,x}}-\\Lambda^{(1)}_3\\right)\\varepsilon_{,tx}\n+\\rho\\left(\\frac{\\partial s}{\\partial \\gamma_{,x}}-\\Lambda^{(1)}_4\\right)\\gamma_{,tx}\\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial T}{\\partial\\rho_{,x}}\\Lambda^{(1)}_2- \\frac{\\partial q}{\\partial\\rho_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial\\rho_{,x}}\\Lambda^{(1)}_4 \\right)\\rho_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial T}{\\partial v_{,x}}\\Lambda^{(1)}_2- \\frac{\\partial q}{\\partial v_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial v_{,x}}\\Lambda^{(1)}_4 \\right)v_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad +\\left(\\frac{\\partial T}{\\partial \\varepsilon_{,x}}\\Lambda^{(1)}_2 - \\frac{\\partial q}{\\partial \\varepsilon_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial \\varepsilon_{,x}}\\Lambda^{(1)}_4\\right)\\varepsilon_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad +\\left(\\frac{\\partial T}{\\partial \\gamma_{,x}}\\Lambda^{(1)}_2- \\frac{\\partial q}{\\partial \\gamma_{,x}}\\Lambda^{(1)}_3 \n- \\frac{\\partial\\phi}{\\partial \\gamma_{,x}}\\Lambda^{(1)}_4 \\right)\\gamma_{,xxx}\\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial ^{2}T}{\\partial \\rho_{,x}^{2}}  \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial \\rho_{,x}^{2}}  \\Lambda^{(1)}_3\n-\\frac{\\partial ^{2}\\phi }{\\partial \\rho_{,x}^{2}}  \\Lambda^{(1)}_4 \n\\right)  \\rho_{,xx}^{2}\\allowdisplaybreaks\\\\\n&\\quad+2 \\left(\\frac{\\partial ^{2}T}{\\partial \\rho_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_2   \n- \\frac{\\partial ^{2}q}{\\partial \\rho_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_3\n-\\frac{\\partial ^{2}\\phi }{\\partial \\rho_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_4\\right)  \\rho_{,xx}  v_{,xx}\\allowdisplaybreaks\\\\  \n&\\quad+2\\left( \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\rho_{,x}}    \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_3\n- \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\rho_{,x}}    \\Lambda^{(1)}_4  \\right) \\rho_{,xx} \\varepsilon _{,xx}    \\allowdisplaybreaks\\\\\n&\\quad+2 \\left( \\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_2 \n-\\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_3 \n- \\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}\\partial \\rho_{,x}} \\Lambda^{(1)}_4\\right)  \\rho_{,xx}\\gamma_{,xx}  \\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial ^{2}T}{\\partial v_{,x}^{2}}  \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial v_{,x}^{2}}  \\Lambda^{(1)}_3 \n-\\frac{\\partial ^{2}\\phi }{\\partial v_{,x}^{2}}  \\Lambda^{(1)}_4\\right)  v_{,xx}^{2}\\allowdisplaybreaks\\\\\n&\\quad+2\\left( \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial v_{,x}}  \\Lambda^{(1)}_2 \n- \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial v_{,x}} \\Lambda^{(1)}_3 \n- \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial v_{,x}} \\Lambda^{(1)}_4\\right) v_{,xx} \\varepsilon _{,xx}   \\allowdisplaybreaks\\\\\n&\\quad+2\\left(  \\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}\\partial v_{,x}}  \\Lambda^{(1)}_2  \n- \\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}\\partial v_{,x}}   \\Lambda^{(1)}_3  \n- \\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}\\partial v_{,x}}\\Lambda^{(1)}_4\\right)  v_{,xx}\\gamma_{,xx} \\allowdisplaybreaks\\\\\n&\\quad+ \\left(\\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}^{2}} \\Lambda^{(1)}_2\n- \\frac{\\partial ^{2}q}{ \\partial \\varepsilon _{,x}^{2}} \\Lambda^{(1)}_3 \n-\\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}^{2}} \\Lambda^{(1)}_4\\right) \\varepsilon _{,xx}^{2}  \\allowdisplaybreaks\\\\\n&\\quad+2  \\left(\\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\gamma_{,x}}\\Lambda^{(1)}_2  \n- \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\gamma_{,x}}  \\Lambda^{(1)}_3 \n- \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\gamma_{,x}}\\Lambda^{(1)}_4 \\right) \\varepsilon _{,xx}  \\gamma_{,xx}\n  \\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}^{2}}   \\Lambda^{(1)}_2\n- \\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}^{2}}   \\Lambda^{(1)}_3\n-\\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}^{2}}  \\Lambda^{(1)}_4\\right) \\gamma_{,xx}^{2} \\allowdisplaybreaks\\\\\n&\\quad+\\left(\\frac{\\partial s}{\\partial \\rho_{,x}}  \\rho  v+\\frac{\\partial J_s}{\\partial \\rho_{,x}} +\\frac{\\partial T}{\\partial \\rho_{,x}}  \\Lambda^{(0)}_2- \\frac{\\partial q}{\\partial \\rho_{,x}}  \\Lambda^{(0)}_3-\\frac{\\partial \\phi }{\\partial \\rho_{,x}}  \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad-\\Lambda^{(1)}_1 v+\\Lambda^{(1)}_2\\left(\\frac{\\partial T}{\\partial \\rho }  +2 \\frac{\\partial ^{2}T}{\\partial \\rho \\partial \\rho_{,x}}  \\rho_{,x}  +2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\rho_{,x}}  \\varepsilon _{,x}    \n+2 \\frac{\\partial ^{2}T}{\\partial \\gamma\\partial \\rho_{,x}}  \\gamma_{,x}\\right) \\allowdisplaybreaks \\\\\n&\\qquad +\\Lambda^{(1)}_3\\left(\\frac{\\partial T}{\\partial \\rho_{,x}}  v_{,x}-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\rho_{,x}}  \\varepsilon _{,x}  -2  \\frac{\\partial ^{2}q}{\\partial \\gamma\\partial \\rho_{,x}}  \\gamma_{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\rho \\partial \\rho_{,x}}   \\rho_{,x}  \n-\\frac{\\partial q}{\\partial \\rho }  \\right)    \\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4\\left(  -2  \\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon \\partial \\rho_{,x}}  \\varepsilon _{,x}    \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\gamma\\partial \\rho_{,x}}  \\gamma_{,x}   \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial \\rho_{,x}}  \\rho_{,x} -\\frac{\\partial \\phi }{\\partial \\rho } \\right)\n\\right) \\rho_{,xx} \\allowdisplaybreaks\\\\\n&\\quad+ \\left(\\frac{\\partial s}{\\partial v_{,x}}  \\rho   v  \n+\\frac{\\partial J_s}{\\partial v_{,x}}  \n+\\frac{\\partial T}{\\partial v_{,x}}  \\Lambda^{(0)}_2  \n- \\frac{\\partial q}{\\partial v_{,x}}  \\Lambda^{(0)}_3  \n-\\frac{\\partial \\phi }{\\partial v_{,x}}  \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\left(-\\Lambda^{(1)}_1  \\rho   \n-\\Lambda^{(1)}_2  \\left(\\rho   v  \n+2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial v_{,x}}  \\varepsilon _{,x}  \n+2 \\frac{\\partial ^{2}T}{\\partial \\gamma\\partial v_{,x}}  \\gamma_{,x}  \n+2 \\frac{\\partial ^{2}T}{\\partial \\rho \\partial v_{,x}}    \\rho_{,x}\\right)\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_3  \\left(T +\\frac{\\partial T}{\\partial v_{,x}}   v_{,x}\n-2 \\frac{\\partial ^{2}q}{\\partial \\rho \\partial v_{,x}}   \\rho_{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial v_{,x}}  \\varepsilon _{,x} \n-2 \\frac{\\partial ^{2}q}{\\partial \\gamma\\partial v_{,x}}  \\gamma_{,x} \\right) \\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4\\left(-2\\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial v_{,x}}\\rho_{,x}\n-2  \\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon \\partial v_{,x}}  \\varepsilon _{,x}  \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\gamma\\partial v_{,x}}  \\gamma_{,x}  \\right) \\right) v_{,xx}\\allowdisplaybreaks\\\\\n&\\quad+ \\left(\\frac{\\partial s}{\\partial \\varepsilon _{,x}}   \\rho   v\n+\\frac{\\partial J_s}{\\partial \\varepsilon _{,x}} \n+\\frac{\\partial T}{\\partial \\varepsilon _{,x}}  \\Lambda^{(0)}_2\n -\\frac{\\partial q}{\\partial \\varepsilon _{,x}}  \\Lambda^{(0)}_3\n-\\frac{\\partial \\phi }{\\partial \\varepsilon _{,x}} \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_2\\left(2  \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\varepsilon _{,x}}  \\varepsilon _{,x} \n+ \\frac{\\partial T}{\\partial \\varepsilon }  \n+2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\gamma}  \\gamma_{,x} \n+2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon _{,x}\\partial \\rho } \\rho_{,x}\\right)\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_3\\left(-2\\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\varepsilon _{,x}}  \\varepsilon _{,x}  \n+\\frac{\\partial T}{\\partial \\varepsilon _{,x}}  v_{,x}\n-  \\rho   v\n-\\frac{\\partial q}{\\partial \\varepsilon } \n-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\gamma} \\gamma_{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon _{,x}\\partial \\rho } \\rho_{,x}\\right) \\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4\\left(-2  \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon \\partial \\varepsilon _{,x}}  \\varepsilon _{,x} \n-\\frac{\\partial \\phi }{\\partial \\varepsilon } \n-2  \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\gamma}   \\gamma_{,x} \n-2 \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon _{,x}\\partial \\rho } \\rho_{,x}\\right)\\right) \\varepsilon _{,xx}   \\allowdisplaybreaks\\\\  \n&\\quad+ \\left(\\frac{\\partial s}{\\partial \\gamma_{,x}}  \\rho   v\n+\\frac{\\partial J_s}{\\partial \\gamma_{,x}}  \n+\\frac{\\partial T}{\\partial \\gamma_{,x}}  \\Lambda^{(0)}_2\n- \\frac{\\partial q}{\\partial \\gamma_{,x}}  \\Lambda^{(0)}_3\n-\\frac{\\partial \\phi }{\\partial \\gamma_{,x}}  \\Lambda^{(0)}_4\\right.\\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_2\\left(2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\gamma_{,x}}  \\varepsilon _{,x} \n+2  \\frac{\\partial ^{2}T}{\\partial \\gamma\\partial \\gamma_{,x}}  \\gamma_{,x} \n+\\frac{\\partial T}{ \\partial \\gamma}  \n+2  \\frac{\\partial ^{2}T}{\\partial \\gamma_{,x}\\partial \\rho }  \\rho_{,x}\\right)  \\allowdisplaybreaks\\\\\n&\\qquad+\\Lambda^{(1)}_3\\left(-2 \\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\gamma_{,x}}  \\varepsilon _{,x}  \n-2 \\frac{\\partial ^{2}q}{\\partial \\gamma\\partial \\gamma_{,x}}  \\gamma_{,x}  \n-\\frac{\\partial q}{\\partial \\gamma}  \n-2  \\frac{\\partial ^{2}q}{\\partial \\gamma_{,x}\\partial \\rho }  \\rho_{,x}\n+\\frac{\\partial T}{\\partial \\gamma_{,x}} v_{,x}\\right)\\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\Lambda^{(1)}_4 \\left(-2  \\frac{\\partial ^{2}\\phi }{\\partial \\varepsilon \\partial \\gamma_{,x}}  \\varepsilon _{,x} \n-2\\frac{\\partial ^{2}\\phi }{\\partial \\gamma\\partial \\gamma_{,x}}  \\gamma_{,x}  \n-\\frac{\\partial \\phi }{\\partial \\gamma} \n-2 \\frac{\\partial ^{2}\\phi }{\\partial \\gamma_{,x}\\partial \\rho }  \\rho_{,x}\n-\\rho  v\\right)\\right)\\gamma_{,xx}\\allowdisplaybreaks\\\\\n& \\rho v\\left(\\frac{\\partial s}{\\partial \\rho }\\rho_{,x} \n+ \\frac{\\partial s}{\\partial \\varepsilon }  \\varepsilon _{,x} \n+ \\frac{\\partial s}{\\partial \\gamma}  \\gamma_{,x}\\right)\n+\\frac{\\partial J_s}{\\partial \\rho }  \\rho_{,x}\n+\\frac{\\partial J_s}{ \\partial \\varepsilon }  \\varepsilon _{,x}\n+\\frac{\\partial J_s}{ \\partial \\gamma}  \\gamma_{,x}\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(0)}_1  \\left(v\\rho_{,x}+\\rho   v_{,x}\\right)\n-\\Lambda^{(0)}_2 \\left(\\rho   v  v_{,x}-\\frac{\\partial T}{\\partial \\rho } \\rho_{,x}\n- \\frac{\\partial T}{\\partial \\varepsilon }  \\varepsilon _{,x} \n-\\frac{\\partial T}{ \\partial \\gamma}  \\gamma_{,x} \\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(0)}_3\\left(\\rho   v\\varepsilon _{,x}  -T  v_{,x}+ \\frac{\\partial q}{\\partial \\rho }  \\rho_{,x}\n+\\frac{\\partial q}{ \\partial \\varepsilon }  \\varepsilon _{,x}  \n+\\frac{\\partial q}{\\partial \\gamma}  \\gamma_{,x}\\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(0)}_4 \\left(\\rho v\\gamma_{,x} +\\frac{\\partial \\phi }{\\partial \\rho }   \\rho_{,x} \n+\\frac{\\partial \\phi }{\\partial \\varepsilon }  \\varepsilon _{,x}  \n+\\frac{\\partial \\phi }{\\partial \\gamma}  \\gamma_{,x}  \\right)\n-2  \\Lambda^{(1)}_1  \\rho_{,x}  v_{,x}\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(1)}_2\\left(v\\rho_{,x} v_{,x}+\\rho   v_{,x}^{2}\n-\\frac{\\partial ^{2}T}{\\partial \\rho ^{2}}   \\rho_{,x}^{2}\n-2 \\frac{\\partial ^{2}T}{ \\partial \\rho \\partial \\varepsilon} \\rho_{,x}  \\varepsilon _{,x}   \n-2  \\frac{\\partial ^{2}T}{\\partial \\rho \\partial \\gamma} \\rho_{,x}  \\gamma_{,x} \\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.- \\frac{\\partial ^{2}T}{\\partial \\varepsilon ^{2}}  \\varepsilon _{,x}^{2}  \n-2 \\frac{\\partial ^{2}T}{\\partial \\varepsilon \\partial \\gamma}  \\varepsilon _{,x}  \\gamma_{,x}  \n-\\frac{\\partial ^{2}T}{ \\partial \\gamma^{2}}  \\gamma_{,x}^{2} \\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(1)}_3\\left(\n( v\\rho_{,x}  +\\rho   v_{,x})\\varepsilon _{,x}   \n-\\left(\\frac{\\partial T}{\\partial \\rho }   \\rho_{,x}  \n+ \\frac{\\partial T}{\\partial \\varepsilon }  \\varepsilon _{,x}    \n+\\frac{\\partial T}{\\partial \\gamma}  \\gamma_{,x} \\right)   v_{,x}\\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\frac{\\partial ^{2}q}{\\partial \\rho ^{2}}  \\rho_{,x}^{2}\n+2  \\frac{\\partial ^{2}q}{\\partial \\rho\\partial \\varepsilon} \\rho_{,x} \\varepsilon _{,x}  \n+2  \\frac{\\partial ^{2}q}{\\partial \\rho\\partial\\gamma}  \\rho_{,x}\\gamma_{,x} \\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.+\\frac{\\partial ^{2}q}{\\partial \\varepsilon ^{2}}  \\varepsilon _{,x}^{2}\n+2\\frac{\\partial ^{2}q}{\\partial \\varepsilon \\partial \\gamma}  \\varepsilon _{,x}  \\gamma_{,x} \n+\\frac{\\partial ^{2}q}{\\partial \\gamma^{2}}  \\gamma_{,x}^{2}  \\right)\\allowdisplaybreaks\\\\\n&\\quad-\\Lambda^{(1)}_4\\left((v \\rho_{,x}+\\rho   v_{,x})\\gamma_{,x}    \n+\\frac{\\partial ^{2}\\phi }{\\partial \\rho ^{2}}   \\rho_{,x}^{2}\n+2  \\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial \\varepsilon}   \\rho_{,x} \\varepsilon _{,x} \\right.\\allowdisplaybreaks\\\\\n&\\qquad\\left.+2\\frac{\\partial ^{2}\\phi }{\\partial \\rho \\partial \\gamma}  \\rho_{,x} \\gamma_{,x}    \n+\\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon ^{2}}  \\varepsilon _{,x}^{2} \n+2  \\frac{\\partial ^{2}\\phi }{ \\partial \\varepsilon \\partial \\gamma}  \\varepsilon _{,x}  \\gamma_{,x}  \n+\\frac{\\partial ^{2}\\phi }{\\partial \\gamma^{2}}  \\gamma_{,x}^{2} \\right)\\ge 0,\n\\end{align*}\nwhere we can distinguish the \\emph{highest derivatives}, say\n\\[\n\\{\\rho_{,t},v_{,t},\\varepsilon_{,t},\\gamma_{,t},\\rho_{,tx},v_{,tx},\\varepsilon_{,tx},\\gamma_{,tx},\\rho_{,xxx},v_{,xxx},\\varepsilon_{,xxx},\\gamma_{,xxx}\\},\n\\]\nand the \\emph{higher derivatives}, say\n\\[\n\\{\\rho_{,xx},v_{,xx},\\varepsilon_{,xx},\\gamma_{,xx}\\}.\n\\]\nAs expected, the entropy inequality is linear in the highest derivatives and quadratic in the higher ones; \nthe coefficients are at most functions of the field and the state variables.\nBy vanishing the coefficients of the highest derivatives, we determine the Lagrange multipliers, say\n\\begin{equation}\n\\begin{aligned}\n&\\Lambda_1^{(0)}=\\rho\\frac{\\partial s}{\\partial\\rho}, \\qquad \n&&\\Lambda_2^{(0)}=-\\frac{\\rho_{,x}}{\\rho}\\frac{\\partial s}{\\partial v_x},\\\\\n&\\Lambda_3^{(0)}=\\frac{\\partial s}{\\partial\\varepsilon}-\\frac{\\rho_{,x}}{\\rho}\\frac{\\partial s}{\\partial\\varepsilon_{,x}}, \\qquad \n&&\\Lambda_4^{(0)}=\\frac{\\partial s}{\\partial\\gamma}-\\frac{\\rho_{,x}}{\\rho}\\frac{\\partial s}{\\partial\\gamma_{,x}},\\\\\n&\\Lambda_1^{(1)}=\\rho\\frac{\\partial s}{\\partial \\rho_{,x}},\\qquad\n&&\\Lambda_2^{(1)}=\\frac{\\partial s}{\\partial v_{,x}},\\\\\n&\\Lambda_3^{(1)}=\\frac{\\partial s}{\\partial \\varepsilon_{,x}},\\qquad\n&&\\Lambda_4^{(1)}=\\frac{\\partial s}{\\partial \\gamma_{,x}},\n\\end{aligned}\n\\end{equation}\nas well as the following restrictions involving the entropy, the Cauchy stress tensor, the heat flux and the flux of internal variable:\n\\begin{equation}\n\\begin{aligned}\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial \\rho_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial \\rho_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial \\rho_{,x}}=0,\\\\\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial v_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial v_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial v_{,x}}=0,\\\\\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial \\varepsilon_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial \\varepsilon_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial \\varepsilon_{,x}}=0,\\\\\n&\\frac{\\partial s}{\\partial v_{,x}}\\frac{\\partial T}{\\partial \\gamma_{,x}}-\n\\frac{\\partial s}{\\partial \\varepsilon_{,x}}\\frac{\\partial q}{\\partial \\gamma_{,x}}-\n\\frac{\\partial s}{\\partial \\gamma_{,x}}\\frac{\\partial \\phi}{\\partial \\gamma_{,x}}=0.\n\\end{aligned}\n\\end{equation}\nThe latter, joined with the conditions obtained by annihilating the coefficients of the linear terms  in the higher derivatives, provide the restrictions on the constitutive functions. These conditions can be solved so that we are able to provide an explicit solution that is proved to satisfy the remaining restrictions expressed as inequalities.\nTo proceed further, we start by taking the specific entropy as the sum of an equilibrium term and a non-equilibrium part expressed as a  \nquadratic form in the gradients entering the state space; this quadratic part must be semidefinite negative in order to verify the principle of maximum entropy at equilibrium. By using some routines written in the Computer Algebra System Reduce \\cite{Reduce}, we obtain the following solution to all the  thermodynamic restrictions:\n\\begin{equation}\\label{entr_1}\n\\begin{aligned}\ns&=s_0(\\rho,\\varepsilon)+s_1(\\rho,\\gamma)\\rho_{,x}^2,\\\\\nT&=\\rho^2\\frac{\\partial s_0}{\\partial \\rho}\\left(\\frac{\\partial s_0}{\\partial\\varepsilon}\\right)^{-1}\n+\\tau_1(\\rho,\\varepsilon,\\gamma)v_{,x}\\\\\n&+\\rho^2\\left(\\frac{\\partial s_0}{\\partial\\varepsilon}\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{-1}\n\\left(s_1\\frac{\\partial^2 s_1}{\\partial\\rho\\partial\\gamma}-\\frac{\\partial s_1}{\\partial\\rho}\\frac{\\partial s_1}{\\partial\\gamma}\\right)\\rho_{,x}^2\\\\\n&+\\rho^2\\left(\\frac{\\partial s_0}{\\partial\\varepsilon}\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{-1}\n\\left(s_1\\frac{\\partial^2 s_1}{\\partial\\gamma^2}-2\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^2\\right)\\rho_{,x}\\gamma_{,x},\\\\\nq&= q_1(\\rho,\\varepsilon,\\gamma)\\varepsilon_{,x}+q_2(\\rho,\\varepsilon,\\gamma)\\rho_{,x}+q_3(\\rho,\\varepsilon,\\gamma)v_{,x},\\\\\n\\phi&=\\left(2\\frac{\\partial s_1}{\\partial\\gamma}\\rho_{,x}\\right)^{-1}\n\\left(q_2\\frac{\\partial s_0}{\\partial \\varepsilon}+f(\\rho,\\varepsilon,\\gamma)+2k\\frac{\\partial s_1}{\\partial\\gamma}\\rho_{,x} -2\\rho^2s_1v_{,x}\\right),\\\\\nJ_s&=q\\frac{\\partial s_0}{\\partial\\varepsilon}-\\frac{1}{2}\\left(q_2\\frac{\\partial s_0}{\\partial \\varepsilon}+f(\\rho,\\varepsilon,\\gamma)-2\\rho^2s_1 v_{,x}\\right)\\rho_{,x},\n\\end{aligned}\n\\end{equation}\nwhere the function $s_1(\\rho,\\gamma)$ must be a negative function in order the principle of maximum entropy at the equilibrium be satisfied.\nMoreover, the reduced entropy inequality becomes \n\\begin{equation}\n\\label{reduced1}\n\\begin{aligned}\n&(q_1\\varepsilon_{,x}+q_2\\rho_{,x}+q_3v_{,x})\\left(\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\rho_{,x}+\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\varepsilon_{,x}\\right)+\\tau_1\\frac{\\partial s_0}{\\partial\\varepsilon} v_{,x}^2\\\\\n&-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(\\frac{\\partial g(\\rho,\\varepsilon)}{\\partial \\rho}\\rho_{,x}+\\frac{\\partial g(\\rho,\\varepsilon)}{\\partial \\varepsilon}\\varepsilon_{,x}\\right)\\rho_{,x}\\geq 0,\n\\end{aligned}\n\\end{equation}\nalong with the constraint\n\\begin{equation}\nq_2\\frac{\\partial s_0}{\\partial\\varepsilon}+f(\\rho,\\varepsilon,\\gamma)-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}g(\\rho,\\varepsilon)=0,\n\\end{equation}\nwhere $f(\\rho,\\varepsilon,\\gamma)$ and $g(\\rho,\\varepsilon)$ are arbitrary functions of their arguments.\n\nThe inequality \\eqref{reduced1} is satisfied if and only if the following conditions hold true:\n\\begin{equation}\n\\label{diffconstrgen}\n\\begin{aligned}\n&q_1\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\geq 0,\\qquad\\tau_1\\frac{\\partial s_0}{\\partial\\varepsilon} \\geq 0,\\\\\n&q_2\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\frac{\\partial g}{\\partial \\rho}\\geq 0,\\qquad q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1\\geq 0,\\\\\n&\\left(q_3^2\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}-4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_2\\right)\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}+4\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\frac{\\partial s_0}{\\partial\\varepsilon}\\frac{\\partial g}{\\partial \\rho}\\tau_1\\leq 0,\\\\\n&\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}\\right)^2+4\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\frac{\\partial g}{\\partial\\rho} q_1+\\frac{\\partial s_1}{\\partial\\gamma}\\left(\\frac{\\partial g}{\\partial\\varepsilon}\\right)^2\\\\\n&-2\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}+q_1\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}\\right)\n\\frac{\\partial g}{\\partial\\varepsilon}\\leq 0,\\\\\n&\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial\\rho \\partial\\varepsilon}\\right)^2\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1\\right)\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\frac{\\partial g}{\\partial\\rho}\\\\\n&-\\left(\\frac{\\partial s_1}{\\partial\\gamma}\\right)^{\\frac{1}{2}}\\left(q_3^2\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-2\\left(q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}+q_1\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\right)\\frac{\\partial s_0}{\\partial \\varepsilon}\\tau_1\\right)\\frac{\\partial g}{\\partial \\varepsilon}\\\\\n&+\\frac{\\partial s_1}{\\partial \\gamma}\\frac{\\partial s_0}{\\partial\\varepsilon}\\left(\\frac{\\partial g}{\\partial\\varepsilon}\\right)^2 \\tau_1\\leq 0.\n\\end{aligned}\n\\end{equation}\n\nThe solution so recovered contains some degrees of freedom that can be fixed in order to model specific physical situations. The constitutive equation \\eqref{entr_1}$_1$ can be interpreted as an extension of the equilibrium constitutive equation to non-equilibrium  situations \\cite{Muschik-Ehrentraut}.\nIn what follows, we limit ourselves to discuss in detail the case where $g(\\rho,\\varepsilon)=0$, whereupon the constraints \n\\eqref{diffconstrgen} simplify as\n\\begin{equation}\n\\label{lastrestr1}\n\\begin{aligned}\n&q_2\\frac{\\partial^2 s_0}{\\partial \\rho \\partial\\varepsilon}\\geq 0,\\qquad q_1\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\geq 0,\\qquad \\tau_1\\frac{\\partial s_0}{\\partial\\varepsilon}\\geq 0,\\\\\n&\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\left(4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_2-q_3^2\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}\\right)\\geq 0,\\\\\n&\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\left(4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1-q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\right)\\geq 0,\n\\end{aligned}\n\\end{equation}\ntogether with \n\\begin{equation}\n\\label{relationq1q2}\nq_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial \\rho\\partial\\varepsilon}=0.\n\\end{equation}\n\nSome comments about the constitutive relations so characterized are in order. \nIn equilibrium situations, in which the gradients of the field variables (except at most the density $\\rho$) vanish, let us define the absolute temperature $\\theta$ by the classical thermodynamical relation $\\displaystyle \\frac{1}{\\theta}=\\frac{\\partial s_0}{\\partial\\varepsilon}$. \nUnder the hypothesis of invertibility of \n$\\theta$ with respect to $\\varepsilon$, which is allowed by the positivity of the specific heat \n$\\displaystyle c = \\frac{\\partial \\varepsilon}{\\partial\\theta}$, the internal energy $\\varepsilon$ can be\nexpressed as function of the arguments $\\rho$ and $\\theta$. Thus,  differentiating with respect to $\\rho$ the\ncondition\n\\[\n\\frac{\\partial s_0(\\rho,\\varepsilon(\\rho,\\theta))}{\\partial \\varepsilon}-\\frac{1}{\\theta}=0,\n\\]\nwe get\n\\begin{equation}\n\\label{sign}\n\\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}+\\frac{\\partial^2 s_0}{\\partial \\varepsilon^2}\\frac{\\partial\\varepsilon}{\\partial\\rho}=0,\n\\end{equation}\nthat, used in \\eqref{relationq1q2}, provides\n\\begin{equation}\nq_2=-q_1\\frac{\\partial\\varepsilon}{\\partial\\rho}.\n\\end{equation}\n\nThus, the heat flux reduces to\n\\begin{equation}\nq = q_1\\frac{\\partial\\varepsilon}{\\partial\\theta}\\theta_{,x}+q_3v_x,\n\\end{equation}\n\\emph{i.e.}, when $q_3=0$, we have the classical Fourier law of heat conduction. \n\nSince \n\\[\n\\frac{\\partial s_0}{\\partial\\varepsilon}=\\frac{1}{\\theta}>0, \\qquad\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}=-\\frac{1}{\\theta^2}\\frac{\\partial\\theta}{\\partial\\varepsilon}< 0,\n\\]\nit is $\\tau_1\\geq 0$, and, as physics prescribes,  $q_1\\leq 0$.\nAs far as $q_2$ is concerned, its sign is the same as that of $\\displaystyle\\frac{\\partial\\varepsilon}{\\partial\\rho}$, which in turn, from \\eqref{sign}, has the same sign as  \n$\\displaystyle \\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}$; thus, due to $\\displaystyle\\frac{\\partial\\varepsilon}{\\partial\\rho}\\geq 0$, it is $q_2\\geq 0$. \nFinally, the last two inequalities in \\eqref{lastrestr1} provide\n\\begin{equation}\n\\label{onlyone}\nq_3^2\\leq 4 \\left(\\frac{\\partial^2 s_0}{\\partial \\varepsilon^2}\\right)^{-1}\\frac{\\partial s_0}{\\partial\\varepsilon}q_1\\tau_1.\n\\end{equation}\nLast, but not the least, it is worth of being observed that the entropy flux contains the classical term $\\displaystyle \\frac{q}{\\theta}$  and an additional term (extra-flux) coming from the application of the procedure  without the need of postulating it at the beginning. Finally, at equilibrium the flux $\\phi$ of the internal variable $\\gamma$ reduces to a constant, and\nthe stress tensor $T$ assumes the classical local form if the mass density is constant. \n\n\\subsection{Korteweg fluids}\nHere, we consider the case of a fluid of grade 3 \\cite{Truesdell_Noll}; in fact,  we include in the state space the second spatial derivative of the mass density \\cite{Dunn-Serrin}. For this class of fluids, Korteweg \n\\cite{Korteweg} proposed the Cauchy stress tensor to be given by a constitutive equation like\n\\begin{equation} \n\\label{kort}\nT_{ij}=\\left(-p+\\sum_{k=1}^3\\left(\\alpha_1\\frac{\\partial^2\\rho}{\\partial x_k^2}+\\alpha_2\\frac{\\partial\\rho}{\\partial x_k}\n\\frac{\\partial\\rho}{\\partial x_k}\\right)\\right)\\delta_{ij}+\\alpha_3\\frac{\\partial\\rho}{\\partial x_i}\n\\frac{\\partial\\rho}{\\partial x_j}+\\alpha_4\\frac{\\partial^2\\rho}{\\partial x_i\\partial x_j},\n\\end{equation}\nwhere $\\rho$ denotes the mass density, $p$  the pressure of the fluid, and $\\alpha_i$, $i=1,\\ldots,4$,  \nsuitable material functions depending on density and temperature. These fluids received a moderate attention in the \nliterature after the pioneering paper by Dunn and Serrin \\cite{Dunn-Serrin}, where the compatibility with the basic tenets \nof rational continuum thermodynamics \\cite{Truesdell} has been extensively studied. They have been studied also in \n\\cite{CST-JMP-2009,COT-JMP-2011,CST-JNET-2010,COP-Elasticity-2011} \nby means of an extended Liu procedure, and by \nHeida and M\\'alek \\cite{Heida-Malek} following a different methodology. \n\nBy limiting to the one-dimensional case, the balance equations read \n\\begin{equation}\n\\label{korteweg}\n\\begin{aligned}\n&\\mathcal{E}_1\\equiv\\frac{D\\rho}{D t} + \\frac{D (\\rho v)}{D x}=0, \\\\\n&\\mathcal{E}_2\\equiv\\frac{D(\\rho v)}{D t} + \\frac{D (\\rho v^2  - T)}{D x}=0,\\\\\n&\\mathcal{E}_3\\equiv\\frac{D}{D t}\\left(\\rho\\varepsilon+\\rho \\frac{v^2}{2}\\right) +\\frac{D }{D x}\\left(\\rho v\\varepsilon+\\rho \\frac{v^3}{2}-Tv+ q\\right)=0,\n\\end{aligned}\n\\end{equation}\nwhere $\\rho$ is the mass density, $v$ the velocity, and $\\varepsilon$ the internal energy per unit mass;\nmoreover, the stress $T$ and the heat flux $q$ must be assigned by means of constitutive equations.\nThe constitutive equations must be such that for every admissible process the entropy inequality \n\\begin{equation}\n\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\ge 0,\n\\end{equation}\nbeing $s$ the specific entropy and $J_s$ the entropy flux, is satisfied.\n\nLet us assume the state space spanned by\n\\begin{equation}\n\\mathcal{Z}\\equiv\\{\\rho, \\varepsilon, \\rho_{,x}, \\varepsilon_{,x}, v_{,x}, \\rho_{,xx}\\}.\n\\end{equation}\n\nIn this case, the exploitation of second law of thermodynamics requires that we take into account the constraints \nimposed on the thermodynamic processes by the balance equations together with their first and second order spatial derivatives; \nnevertheless, we observe that, since the unique second order spatial derivative belonging to the state space is $\\rho_{,xx}$, the unique second order extension we need to use as constraint is $\\displaystyle\\frac{D^2\\mathcal{E}_1}{Dx^2}$.\nThus, the entropy inequality writes:\n\\begin{equation}\n\\label{entropyconstrainedkorteweg}\n\\begin{aligned}\n&\\rho\\left(\\frac{Ds}{D t}+v\\frac{Ds}{Dx}\\right)+ \\frac{DJ_s}{Dx} \\\\\n&\\quad- \\Lambda^{(0)}_1 \\mathcal{E}_1- \\Lambda^{(0)}_2 \\mathcal{E}_2- \\Lambda^{(0)}_3 \\mathcal{E}_3\\\\\n&\\quad-\\Lambda^{(1)}_1\\frac{D\\mathcal{E}_1}{Dx}-\\Lambda^{(1)}_2\\frac{D\\mathcal{E}_2}{Dx}\n-\\Lambda^{(1)}_3\\frac{D\\mathcal{E}_3}{Dx}-\\Lambda^{(2)}_1\\frac{D^2\\mathcal{E}_1}{Dx^2} \\geq 0.\n\\end{aligned}\n\\end{equation}\n\nBy expanding the derivatives in \\eqref{entropyconstrainedkorteweg}, we obtain a huge expression that we omit to report here;  \nit results linear in the  highest derivatives, say\n\\[\n\\{\\rho_{,t},v_{,t},\\varepsilon_{,t},\\rho_{,tx},v_{,tx},\\varepsilon_{,tx},\\rho_{,txx},v_{,txx},\\varepsilon_{,txx},v_{,xxxx},\\varepsilon_{,xxxx},\\rho_{,xxxxx}\\},\n\\]\nand cubic in  the higher derivatives, say\n\\[\n\\{v_{,xx},\\varepsilon_{,xx},\\rho_{,xxx},v_{,xxx},\\varepsilon_{,xxx},\\rho_{,xxxx}\\}.\n\\]\n\nTo proceed further, let us write the specific entropy as the sum of the equilibrium part  defined for homogeneous states and\na semidefinite negative quadratic form (in order to satisfy the principle of maximum entropy at the equilibrium) in the gradients appearing in the state space.  At this stage we do not specify the form of the entropy flux.\n\nThe restrictions imposed by entropy inequality can be solved and provide the following solution:\n\\begin{equation}\n\\begin{aligned}\\label{entr_gen}\n&s = s_0(\\rho,\\varepsilon)+s_1(\\rho)\\rho_{,x}^2,\\\\\n&q = q_1(\\rho,\\varepsilon)\\varepsilon_{,x}+q_2(\\rho,\\varepsilon)\\rho_{,x}+q_3(\\rho,\\varepsilon)v_{,x},\\\\\n&T = \\rho^2\\left(\\frac{\\partial s_0}{\\partial \\varepsilon}\\right)^{-1}\\left(\\frac{\\partial s_0}{\\partial \\rho}-\\frac{\\partial s_1}{\\partial \\rho}\\rho_{,x}^2-2 s_1\\rho_{,xx}\\right)+\\tau_1(\\rho,\\varepsilon)v_{,x},\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{constrkdv}\n\\begin{aligned}\n&q_1\\le 0, \\qquad q_2\\ge 0, \\qquad \\tau_1\\ge 0,\\\\\n&q_2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}-q_1\\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}=0,\\\\\n&4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_2-q_3^2\\frac{\\partial^2 s_0}{\\partial\\rho\\partial\\varepsilon}\\geq 0,\\\\\n&4\\frac{\\partial s_0}{\\partial\\varepsilon}\\tau_1 q_1-q_3^2\\frac{\\partial^2 s_0}{\\partial\\varepsilon^2}\\leq 0,\n\\end{aligned} \n\\end{equation}\nand $s_1\\leq 0$ in order the principle of maximal entropy at the equilibrium be satisfied.\nFinally, the entropy flux turns out to be\n\\begin{equation}\n\\begin{aligned}\\label{flux_entr}\nJ_s&=\\frac{\\partial s_0}{\\partial\\varepsilon}\\left(q_1\\varepsilon_{,x}+q_2\\rho_{,x}+q_3v_{,x}\\right)+2\\rho^2 s_1\\rho_{,x}v_{,x}=\\\\\n&=\\frac{q}{\\theta}+2\\rho^2 s_1\\rho_{,x}v_{,x},\n\\end{aligned}\n\\end{equation}\nwhere $\\theta$, defined as usual as\n\\begin{equation}\n\\frac{1}{\\theta}=\\frac{\\partial s_0}{\\partial\\varepsilon},\n\\end{equation}\nis the absolute temperature.\n\nLet us observe that, in the expression \\eqref{flux_entr}  of entropy flux, the second contribution  represents  the  entropy extra-flux \\cite{Muller}, related to the gradients of density and velocity.\nMoreover, a similar reasoning as in previous subsection shows that it is $\\displaystyle q_2=-q_1\\frac{\\partial\\varepsilon}{\\partial\\rho}$ so that, choosing $q_3=0$, we recover the classical Fourier law for heat flux. Also, the last two inequalities \nin \\eqref{constrkdv} provide relation \\eqref{onlyone}.\nFinally, if we consider the classical case, \\emph{i.e.}, $s = s_0(\\rho,\\varepsilon)$, the stress tensor $T$ is expressed in local form, and the classical constitutive equation for the entropy flux $\\displaystyle J_s=\\frac{q}{\\theta}$ is recovered.\n\nTherefore,  in order to obtain a constitutive equation for the stress tensor containing first and second order derivatives of mass density we need to assume that $s$ depends on the gradient of $\\rho$, and the entropy flux involves an extra-flux.\nAs a last remark, a stress tensor depending on the gradients of the density can be obtained only if we use the extended\nprocedure for the exploitation of entropy inequality \\cite{CGOP-miscele-2020}.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper, we discussed the extended Liu procedure in order to investigate the restrictions placed by an entropy inequality\non the constitutive equations of a continuum whose state space contains spatial derivatives of the unknown fields. The analysis is\nperformed first from a purely mathematical viewpoint by considering a system of balance laws sufficiently general to contain\nthe governing equations of continua, and sufficient conditions for the fulfilment of an entropy-like inequality have been derived. Then, we specialized the results to two physical cases with first or second order  non-local \nconstitutive equations. Remarkably, in the application of the procedure we do not modify the energy balance equation with the  inclusion of extra-terms (like the interstitial  working), and we do not prescribe \\emph{a priori} the form of entropy flux, whose expression arises as a result of\nthe method; in the applications we considered, the procedure provides an entropy flux decomposed in the classical term and an extra-flux.\n\nWe limited ourselves to one-dimensional models, and in the considered physical applications we were able to solve the\nconstraints imposed by the exploitation of the entropy inequality, so determining an explicit expression of the constitutive functions\nby assuming  an expansion of the specific entropy at first order in the squared gradients of the field variables entering the state space.\n\nThe procedure in principle allows us to fix the constitutive equations (for stress tensor, heat flux, \\ldots), according to experiments, and determine the form of the entropy flux algorithmically. \nIt is trivial to observe that the procedure requires a huge amount of computation, increasing with the order of non-localities;\nhowever, such computations can be almost automatically carried out by using a Computer Algebra System.\n\nWork is in progress about the investigation of the extended Liu procedure for multi-dimensional continuous media, where\nother general principles of representation theory \\cite{Smith3} of vectorial and tensorial quantities need to be considered. \n\n\\section*{Acknowledgments}\nThe authors acknowledge partial supported by G.N.F.M. of ``Istituto Nazionale di Alta Matematica'' and University of Messina. The authors gratefully thank dr. Luca Amata for drawing their attention to the paper \\cite{Mishkov}.\n \\medskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}