diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznjir" "b/data_all_eng_slimpj/shuffled/split2/finalzznjir" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznjir" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn the Standard Model (SM),\n the electroweak symmetry breaking is realized by the negative mass term in the Higgs potential,\n which seems to be artificial because there is nothing to stabilize the electroweak scale.\nIf new physics takes place at a very high energy, e.g. the Planck scale,\n the mass term receives large corrections which are quadratically sensitive to the new physics scale, \n so that the electroweak scale is not stable against the corrections. \nThis is the so-called gauge hierarchy problem.\nIt is well known that supersymmetry (SUSY) can solve this problem. \nSince the mass corrections are completely canceled by the SUSY partners,\n no fine-tuning is necessary to reproduce the electroweak scale correctly, \n unless the SUSY breaking scale is much higher than the electroweak scale.\nOn the other hand, since no indication of SUSY particles has been obtained in the large hadron collider (LHC) experiments, \n one may consider other solutions to the gauge hierarchy problem without SUSY. \n\n\nIn this direction, recently a lot of works have been done in models based on a classically conformal symmetry,\n where an additional $U(1)$ gauge symmetry, e.g., $U(1)_{B-L}$, is added\n \\cite{Hempfling:1996ht}-\\cite{Plascencia:2015xwa}. \nThis direction is based on the argument by Bardeen~\\cite{Bardeen:1995kv} that \n the quadratic divergence in the Higgs mass corrections can be subtracted \n by a boundary condition of some ultraviolet complete theory, \n which is classically conformal, and only logarithmic divergences should be considered\n (see Ref.~\\cite{Iso:2012jn} for more detailed discussions). \nIf this is the case, imposing the classically conformal symmetry to the theory \n is another way to solve the gauge hierarchy problem. \nSince there is no dimensionful parameter in this class of models,\n the gauge symmetry must be broken by quantum corrections. \nThis structure fits the model first proposed by Coleman and Weinberg~\\cite{Coleman:1973jx}, \n where a model is defined as a massless theory and the gauge symmetry is radiatively broken \n by the Coleman-Weinberg (CW) mechanism, generating a mass scale through the dimensional transmutation. \n\n\nIn this paper we propose a classically conformal $U(1)_{B-L}$ extended SM with two Higgs doublets. \nAn SM singlet, $B-L$ Higgs field develops its vacuum expectation value (VEV) by the CW mechanism, \n and the $U(1)_{B-L}$ symmetry is radiatively broken. \nThis gauge symmetry breaking also generates the mass terms for the two Higgs doublets \n through quartic couplings between the two Higgs doublets and the $B-L$ Higgs field. \nWe assume the quartic couplings to be all positive but, nevertheless, the electroweak symmetry breaking \n is triggered through the so-called bosonic seesaw mechanism~\\cite{Calmet:2002rf,Kim:2005qb,Haba:2005jq}, \n which is analogous to the seesaw mechanism for the neutrino mass generation \n and leads to a negative mass squared for the SM-like Higgs doublet. \nA large hierarchy among the quartic Higgs couplings is crucial for the bosonic seesaw mechanism \n to work at the $U(1)_{B-L}$ symmetry breaking scale. \nAlthough it seems unnatural to introduce the large hierarchy by hand, \n we find that the renormalization group evolutions of the quartic Higgs couplings \n dramatically reduce the large hierarchy toward high energies. \nTherefore, once our model is defined at some high energy, say, the Planck scale, \n the large hierarchy at the $U(1)_{B-L}$ symmetry breaking scale \n is naturally realized by a mild hierarchy.\nWe also show that the perturabativity of model couplings and the electroweak \n vacuum stability are maintained up to the Planck scale with a suitable choice \n of the input parameters. \n From the naturalness of the electroweak scale, we find the $U(1)_{B-L}$ gauge symmetry breaking \n scale to be $\\lesssim 100$ TeV, which predicts extra heavy Higgs boson masses to be $\\lesssim 2$ TeV. \nSuch heavy Higgs boson can be tested at the LHC in the near future. \n\n\nIn the next section, we will define our model, and discuss the $U(1)_{B-L}$ symmetry breaking \n as well as the electroweak symmetry breaking by the bosonic seesaw mechanism. \nWe also present the mass spectrum of the model. \nIn Sec.~\\ref{sec:result}, we will analyze the renormalization group evolutions for all couplings of the model \n and present our numerical results. \nWe will see that the hierarchy among the quartic Higgs couplings is dramatically reduced \n toward high energies. \nSec.~\\ref{sec:conclusion} is devoted to conclusion. \n\n\n\n\\section{Model}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|cc|}\\hline\n & $SU(3)_c \\otimes SU(2)_L \\otimes U(1)_Y$ & $U(1)_{B-L}$ \\\\\n\\hline \\hline\n$Q^i$ & (3, 2, 1\/6) & $1\/3$\\\\\n$U^i$ & (3, 1, $2\/3$) & $1\/3$\\\\\n$D^i$ & (3, 1, $-1\/3$) & $1\/3$\\\\\n$L^i$ & (1, 2, $-1\/2$) & $-1$\\\\\n$E^i$ & (1, 1, $-1$) & $-1$\\\\\n$N^i$ & (1, 1, 0) & $-1$\\\\\n$H_1$ & (1, 2, 1\/2) & $0$\\\\\n$H_2$ & (1, 2, 1\/2) & $4$\\\\\n$\\Phi$ & (1, 1, 0) & $2$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Particle contents in our model. \n$i=1,2,3$ is the generation index.}\n\\label{table:1}\n\\end{table}\n\n\nWe consider an extension of the SM with an additional $U(1)_{B-L}$ gauge symmetry. \nThe particle contents of our model are listed in Table \\ref{table:1},\n where two Higgs doublets ($H_1$ and $H_2$) and one SM singlet, $B-L$ Higgs field ($\\Phi$) are introduced. \nAs is well known, the introduction of the three right-handed neutrinos ($N^i$, $i=1$, 2, 3)\n is crucial to make the model free from all the gauge and gravitational anomalies. \nIn addition, we impose a classically conformal symmetry to the model, \n under which the scalar potential is given by\n\\begin{eqnarray}\n\tV &=& \\lambda_1 |H_1|^4 + \\lambda_2 |H_2|^4 + \\lambda_3 |H_1|^2 |H_2|^2 + \\lambda_4 (H_2^\\dagger H_1) (H_1^\\dagger H_2) + \\lambda_\\Phi |\\Phi|^4 \\nonumber \\\\\n\t&& + \\lambda_{H1\\Phi} |H_1|^2 |\\Phi|^2 + \\lambda_{H2\\Phi} |H_2|^2 |\\Phi|^2 + \\left( \\lambda_{\\rm mix} (H_2^\\dagger H_1) \\Phi^2 + h.c. \\right).\n\\end{eqnarray}\nHere, all of the dimensionful parameters are prohibited by the classically conformal symmetry.\nIn this system, the $U(1)_{B-L}$ symmetry must be radiatively broken by quantum effects, i.e., the CW mechanism.\nThe CW potential for $\\Phi$ is described as\n\\begin{eqnarray}\n\tV_\\Phi(\\phi) = \\frac{1}{4} \\lambda_\\Phi(v_\\Phi)\\, \\phi^4\n\t\t\t\t\t\t+ \\frac{1}{8} \\beta_{\\lambda_{\\Phi}}(v_\\Phi)\\, \\phi^4\n\t\t\t\t\t\t\t\\left( \\ln \\frac{\\phi^2}{v_\\Phi^2} -\\frac{25}{6} \\right),\n\\label{CWpotential}\n\\end{eqnarray}\n where $\\Re[\\Phi] = \\phi\/\\sqrt{2}$, and $v_\\Phi=\\langle \\phi \\rangle$ is the VEV of $\\Phi$.\nWhen the beta function $\\beta_{\\lambda_\\Phi}$ is dominated by\n the $U(1)_{B-L}$ gauge coupling ($g_{B-L}$) and the Majorana Yukawa couplings of right-handed neutrinos ($Y_M$) \n as shown in Appendix, the minimization condition of $V_\\Phi$ approximately leads to\n\\begin{eqnarray}\n\t\\lambda_\\Phi \\simeq \\frac{11}{6 \\pi^2}\n\t\t\\left( 6 g_{B-L}^4 - {\\rm tr} Y_M^4 \\right),\n\\label{CW_relation}\n\\end{eqnarray}\n where all parameters are evaluated at $v_\\Phi$. \nThrough the $U(1)_{B-L}$ symmetry breaking, the mass terms of the two Higgs doublets arise \n from the mixing terms between $H_{1,2}$ and $\\Phi$, and the scalar mass squared matrix is read as \n\\begin{eqnarray}\n\t-\\mathcal{L}&=& \\frac{1}{2}\n\t\t\t\t\t(H_1, H_2) \\left( \\begin{array}{cc}\n\t\t\t\t\t\\lambda_{H1\\Phi} v_\\Phi^2 & \\lambda_{\\rm mix} v_\\Phi^2 \\\\\n\t\t\t\t\t\\lambda_{\\rm mix} v_\\Phi^2 & \\lambda_{H2\\Phi} v_\\Phi^2\n\t\t\t\t\\end{array} \\right)\n\t\\left( \\begin{array}{c} H_1 \\\\ H_2 \\end{array} \\right) \\nonumber\\\\\n\t&\\approx& \\frac{1}{2}\n\t\t\t\t\t(H'_1, H'_2) \\left( \\begin{array}{cc}\n\t\t\t\t\t\\lambda_{H1\\Phi} v_\\Phi^2 - \\frac{\\lambda_{\\rm mix}^2}{\\lambda_{H2\\Phi}} v_\\Phi^2 & 0 \\\\\n\t\t\t\t\t0 & \\lambda_{H2\\Phi} v_\\Phi^2\n\t\t\t\t\\end{array} \\right)\n\t\\left( \\begin{array}{c} H'_1 \\\\ H'_2 \\end{array} \\right),\n\\label{matrix}\n\\end{eqnarray}\n where we have assumed a hierarchy among the quartic couplings as\n $0 \\leq \\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix} \\ll \\lambda_{H2\\Phi}$ at the scale $\\mu = v_\\Phi$.\\footnote{\n In our analyses, we will take boundary conditions as $\\lambda_1(v_\\Phi)=\\lambda_2(v_\\Phi)=\\lambda_H(v_\\Phi)$,\n for simplicity, where $\\lambda_H$ is a Higgs quartic coupling in the SM.\n }\nIn the next section, we will show that this hierarchy is dramatically reduced toward high energies \n in their renormalization group evolutions. \nBecause of this hierarchy,\n mass eigenstates $H'_1$ and $H'_2$ are almost composed of $H_1$ and $H_2$, respectively.\nHence, we approximately identify $H'_1$ with the SM-like Higgs doublet. \nNote that even though all quartic couplings are positive, \n the SM-like Higgs doublet obtains a negative mass squared for $\\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix}^2\/\\lambda_{H2\\Phi}$, \n and hence the electroweak symmetry is broken. \nThis is the so-called bosonic seesaw mechanism \\cite{Calmet:2002rf,Kim:2005qb, Haba:2005jq}.\n\n\nIn more precise analysis for the electroweak symmetry breaking, we take into account \n a scalar one-loop diagram through the quartic couplings, $\\lambda_3$ and $\\lambda_4$, \n shown in Fig.~\\ref{one-loop}, and \n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[scale=0.7,clip]{higgs_corr.eps}\n\\end{center}\n\\caption{Scalar one-loop diagram which contributes to the SM-like Higgs doublet mass.}\n\\label{one-loop}\n\\end{figure}\n the SM-like Higgs doublet mass is given by\n\\begin{eqnarray}\n\tm_h^2 &\\simeq& -\\frac{\\lambda_{H1\\Phi}}{2} v_\\Phi^2 + \\frac{\\lambda_{\\rm mix}^2}{2\\lambda_{H2\\Phi}} v_\\Phi^2\n\t\t\t\t\t+ \\frac{\\lambda_{H2\\Phi}}{16\\pi^2} (2\\lambda_3 + \\lambda_4) v_\\Phi^2 \\nonumber\\\\\n\t\t\t&\\simeq& \\lambda_{H2\\Phi} v_\\Phi^2 \\left[ \n\t\t\t\t\\frac{1}{2} \\left(\\frac{\\lambda_{\\rm mix}}{\\lambda_{H2\\Phi}}\\right)^2\n\t\t\t\t+ \\frac{2\\lambda_3 + \\lambda_4}{16\\pi^2} \\right],\n\\label{Higgs}\n\\end{eqnarray}\n where we have omitted the $\\lambda_{H1\\Phi}$ term in the second line, and \n the observed Higgs boson mass $M_h=125$ GeV is given by $M_h=m_h\/\\sqrt{2}$. \n\n\nIn addition to the scalar one-loop diagram, \n one may consider other Higgs mass corrections coming from a neutrino one-loop diagram and \n two-loop diagrams involving the $U(1)_{B-L}$ gauge boson ($Z'$) and the top Yukawa coupling, \n which are, respectively, found to be~\\cite{Iso:2009ss}\n\\begin{eqnarray}\n\t\t\\delta m_h^2 \\sim \\frac{Y_\\nu^2 Y_M^2 v_\\Phi^2}{16 \\pi^2},\\qquad\n\t\t\\delta m_h^2 \\sim \\frac{y_t^2 g_{B-L}^4 v_\\Phi^2}{(16 \\pi^2)^2},\n\\label{deltam}\n\\end{eqnarray}\n where $Y_\\nu$ and $y_t$ are Dirac Yukawa couplings of neutrino and top quark, respectively. \nIt turns out that these contributions are negligibly small\n compared to the scalar one-loop correction in Eq.~(\\ref{Higgs}). \nAs we will discuss in the next section, the quartic couplings $\\lambda_3$ and $\\lambda_4$ \n should be sizable $\\lambda_{3,4} \\gtrsim 0.15$ in order to stabilize the electroweak vacuum. \nThe neutrino one-loop correction is roughly proportional to the active neutrino mass \n by using the seesaw relation, and it is highly suppressed by the lightness of the neutrino mass.\nThe two-loop corrections with the $Z'$ boson is suppressed by a two-loop factor $1\/(16\\pi^2)^2$. \nUnless $g_{B-L}$ is large, the two-loop corrections are smaller than the scalar one-loop correction. \nIn Table~\\ref{table:QC}, we summarize typical orders of magnitude for the three corrections for $v_\\Phi=10$ and $100$ TeV. \nFor the light neutrino mass, we have adopted the seesaw relation, $m_\\nu \\sim (Y_\\nu v_H)^2\/(Y_M v_\\Phi)\\sim 0.1$ eV, \n with the SM-like Higgs field VEV, $v_H=246$ GeV. \nFor both $v_\\Phi=10$ and $100$ TeV, we have fixed \n $\\lambda_3=\\lambda_4=0.15$, $g_{B-L}=0.15$ and $Y_\\nu=2.0 \\times 10^{-6}$, \n while we have used $\\lambda_{H 2 \\Phi}=0.01$ ($10^{-4}$) and $Y_M=0.23$ ($0.023$) \n for $v_\\Phi=10$ ($100$) TeV. \n\n\nThe other scalar masses are approximately given by\n\\begin{eqnarray}\n\tM_\\phi^2 &=& \\frac{6}{11} \\lambda_\\Phi v_\\Phi^2, \\\\\n\tM_H^2 &=& M_A^2 = \\lambda_{H2\\Phi} v_\\Phi^2 + (\\lambda_3 + \\lambda_4) v_H^2, \\\\\n\tM_{H^\\pm}^2 &=& \\lambda_{H2\\Phi} v_\\Phi^2 + \\lambda_3 v_H^2, \n\\end{eqnarray}\n where $M_\\phi$ is the mass of the SM singlet scalar,\n $M_H$ ($M_A$) is the mass of CP-even (CP-odd) neutral Higgs boson,\n and $M_{H\\pm}$ is the mass of charged Higgs boson. \nThe extra heavy Higgs bosons are almost degenerate in mass. \nThe masses of the $Z'$ boson and the right-handed neutrinos are given by\n\\begin{eqnarray}\n\tM_{Z'} &=& 2 g_{B-L} v_\\Phi,\n\\label{MZ} \\\\\n\tM_N &=& \\sqrt{2} y_M v_\\Phi\n\t\t\\simeq \\left[ \\frac{3}{2N_\\nu} \\left( 1 - \\frac{\\pi^2 \\lambda_\\Phi}{11 g_{B-L}^4} \\right) \\right]^{1\/4} M_{Z'},\n\\label{MN}\n\\end{eqnarray}\n where we have used ${\\rm tr}Y_M=N_\\nu y_M$, for simplicity,\n and $N_\\nu$ stands for the number of relevant Majorana couplings.\nIn the following analysis, we will take $N_\\nu = 1$ for simplicity,\n because our final results are almost insensitive to $N_\\nu$. \nIn the last equality in Eq.~(\\ref{MN}), we have used Eq.~(\\ref{CW_relation}).\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\\hline\n$v_\\Phi$ & $10\\,{\\rm TeV}$ & $100\\,{\\rm TeV}$ \\\\ \\hline \\hline\n1-loop with scalar & $\\sim (50\\,{\\rm GeV})^2$ & $\\sim (50\\,{\\rm GeV})^2$\\\\ \\hline\n2-loop with $Z'$ & $(\\mathcal{O}(1)\\,{\\rm GeV})^2$ & $(\\mathcal{O}(10)\\,{\\rm GeV})^2$\\\\ \\hline\n1-loop with neutrino & $(\\mathcal{O}(10^{-3})\\,{\\rm GeV})^2$ & $(\\mathcal{O}(10^{-3})\\,{\\rm GeV})^2$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Typical orders of magnitude of the quantum corrections to the SM-like Higgs doublet mass.}\n\\label{table:QC}\n\\end{table}\n\n\n\\section{Numerical results} \\label{sec:result}\n\nBefore presenting our numerical results,\n we first discuss constraints on the model parameters from the perturbativity \n and the stability of the electroweak vacuum in the renormalization group evolutions. \nIn our analysis, all values of couplings are given at $\\mu=v_\\Phi$.\nFor $v_\\Phi$ at the TeV scale, \n we find the constraint $g_{B-L}\\lesssim 0.3$ to avoid the Landau pole of the gauge coupling \n below the Planck scale, while a more severe constraint $g_{B-L} \\lesssim 0.2$ is obtained \n to avoid a blowup of the quartic coupling $\\lambda_2$ below the Planck scale. \nFrom $g_{B-L} \\lesssim 0.2$ and\n the experimental bound $M_{Z'} > 2.9$ TeV on the $Z'$ boson mass \\cite{Aad:2014cka,Khachatryan:2014fba},\n we find $v_\\Phi > 7.25$\\,TeV.\nThe electroweak vacuum stability, in other words, $\\lambda_H(\\mu) > 0$ for any scales between \n the electroweak scale and the Planck scale, \n can be realized by sufficiently large $\\lambda_3$ and\/or $\\lambda_4$\n as $\\lambda_3 =\\lambda_4 \\gtrsim 0.15$.\nTo keep their perturbativity below the Planck scale, \n $\\lambda_3 =\\lambda_4 \\lesssim 0.48$ must be satisfied,\n while we will find that the naturalness of the electroweak scale leads to a more severe upper bound.\n\n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[clip, height=6cm]{bound.eps} \\hspace{0.5cm}\n\t\\includegraphics[clip, height=6cm]{Heavy_mass.eps}\n\\end{center}\n\\caption{\nThe relation between $v_\\Phi$ and $\\lambda_{H2\\Phi}$ through Eq.~(\\ref{Higgs}) (left panel), \n and the corresponding extra heavy Higgs boson mass spectrum (right panel). \nThe red and blue lines correspond to $\\lambda_{\\rm mix}=0$ and\n $\\lambda_{\\rm mix} = 0.1\\times \\lambda_{H2\\Phi}$, respectively. \nThe shaded region shows the perturbativity bound for $g_{B-L}=0.2$.\nThe vertical lines show the upper bound of $v_\\Phi$,\n at which Higgs mass corrections from the two loop diagrams with the $U(1)_{B-L}$ gauge boson \n become $(10\\,{\\rm GeV})^2$ for $g_{B-L}=0.1$ (left) and $g_{B-L}=0.01$ (right), respectively.\n}\n\\label{bound}\n\\end{figure}\n\nTo realize the hierarchy $\\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix} \\ll \\lambda_{H2\\Phi}$,\n we take $\\lambda_{H1\\Phi}=0$, for simplicity. \nWhen we consider $\\lambda_{\\rm mix}$ in the range of $0< \\lambda_{\\rm mix} < 0.1\\times \\lambda_{H2\\Phi}$,\n the relation between $v_\\Phi$ and $\\lambda_{H2\\Phi}$ obtained by Eq.~(\\ref{Higgs}) \n is shown in the left panel of Fig.~\\ref{bound}. \nHere, we have fixed $\\lambda_3 =\\lambda_4 = 0.15$ as an example.\nThe red and blue lines correspond to the lowest value $\\lambda_{\\rm mix}=0$ and\n the highest value $\\lambda_{\\rm mix} = 0.1\\times \\lambda_{H2\\Phi}$, respectively.\\footnote{\nAlthough the bosonic seesaw mechanism does not work for $\\lambda_{\\rm mix}=0$, \n one may consider the electroweak symmetry breaking through the scalar loop correction shown in Fig.~\\ref{one-loop}.\n}\nThe shaded region shows the perturbativity bound $v_\\Phi > 7.25$\\,TeV.\nThe vertical lines show the upper bound of $v_\\Phi$,\n at which two-loop corrections with the $Z'$ boson to the Higgs mass become $(10\\,{\\rm GeV})^2$\n for $g_{B-L}=0.1$ (left) and $g_{B-L}=0.01$ (right), respectively.\nNote that $\\lambda_{H2\\Phi} v_\\Phi^2$ is almost constant.\nSince all heavy Higgs boson masses are approximately determined by $\\lambda_{H2\\Phi} v_\\Phi^2$,\n they are almost independent of $v_\\Phi$,\n as is shown in the right panel of Fig.~\\ref{bound}.\nThe heavy Higgs boson masses lie in the range between 1\\,TeV and 1.7\\,TeV,\n which can be tested at the LHC in the near future. \n\n\nIn Eq.~(\\ref{Higgs}), it may be natural for the first term from the tree-level couplings \n dominates over the second term from the 1-loop correction. \nThis naturalness leads to the constraint of $\\lambda_3 = \\lambda_4 < 0.26$,\n which is more severe than the perturbativity bound $\\lambda_3 =\\lambda_4 \\lesssim 0.48$ discussed above.\nThis condition is equivalent to the fact that the origin of the negative mass term mainly comes from\n the diagonalization of the scalar mass squared matrix in Eq.~(\\ref{matrix}),\n namely, the bosonic seesaw mechanism.\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[clip, height=6cm]{10TeV.eps} \\hspace{0.5cm}\n \\includegraphics[clip, height=6cm]{100TeV.eps} \n\\end{center}\n\\caption{\nRenormalization group evolutions of the quartic couplings for $v_\\Phi=10$\\,TeV (left) and 100\\,TeV (right).\nThe red, green, and blue lines correspond to\n $\\lambda_{H1\\Phi}$, $\\lambda_{H2\\Phi}$ and $\\lambda_{\\rm mix}$, respectively.\nThe rightmost vertical line shows the reduced Planck scale.\n}\n\\label{running}\n\\end{figure}\n\nNow we present the results of our numerical analysis. \nIn Fig.~\\ref{running}, we show the renormalization group evolutions of the quartic couplings.\nHere, we have taken $\\lambda_{H1\\Phi} = 0$,\n and $\\lambda_{H2\\Phi} = 10^{-2}$ and $10^{-4}$\n for $v_\\Phi=10$\\,TeV (left panel) and 100\\,TeV (right panel), respectively.\nThe red, green, and blue lines correspond to the running of\n $\\lambda_{H1\\Phi}$, $\\lambda_{H2\\Phi}$ and $\\lambda_{\\rm mix}$, respectively.\nThe rightmost vertical line denotes the reduced Planck scale $M_{Pl} = 2.4 \\times 10^{18}$ GeV.\nIn this plot, the other input parameters have been set\n as $g_{B-L} = 0.17$ and $\\lambda_3 = \\lambda_4 = 0.17$\n to realize the electroweak vacuum stability without the Landau pole, and $\\lambda_\\Phi = 10^{-3}$.\nThe value of $\\lambda_1=\\lambda_2=\\lambda_H$ at $\\mu=v_\\Phi$ has been evaluated by \n extrapolating the SM Higgs quartic coupling with $M_h=125$ GeV from the electroweak scale to $v_\\Phi$. \nFor this parameter choice,\n the $Z'$ boson and the right-handed neutrinos have the masses of the same order of magnitude as\n $M_{Z'}=3.4$ $(34)$ TeV and $M_N=2.0$ $(20)$ TeV for $v_\\Phi=10$ $(100)$ TeV,\n while the $B-L$ Higgs boson mass is calculated as $M_\\phi=0.23$ $(2.3)$ TeV. \nAs is well-known, $M_\\phi \\ll M_{Z'}$ is a typical prediction of the CW mechanism.\nThe masses of the heavy Higgs bosons are roughly 1\\,TeV for both $v_\\Phi=10$\\,TeV and 100\\,TeV.\n\n\nIn order for the bosonic seesaw mechanism to work,\n we have assumed the hierarchy among the quartic couplings as \n $\\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix} \\ll \\lambda_{H2\\Phi}$ at the scale $\\mu=v_\\Phi$.\nOne may think it unnatural to introduce this large hierarchy by hand.\nHowever, we find from Fig.~\\ref{running} that the large hierarchy \n between $\\lambda_{H1\\Phi}$ and $\\lambda_{H2\\Phi}$ tends to disappear toward high energies.\nThis is because the beta functions of the small couplings $\\beta_{\\lambda_{H1\\Phi}}$ and $\\beta_{\\lambda_{H2\\Phi}}$ \n are not simply proportional to themselves, but include terms given by other sizable couplings \n (see Appendix for the explicit formulas of their beta functions). \nThis behavior of reducing the large hierarchy in the renormalization group evolutions \n is independent of the choice of the boundary conditions for $g_{B-L}$, $\\lambda_3$, $\\lambda_4$ and $\\lambda_\\Phi$. \nTherefore, Fig.~\\ref{running} indicates that once our model is defined at some high energy, say, the Planck scale, \n the large hierarchy among the quartic couplings, which is crucial for the bosonic seesaw mechanism to work, \n is naturally achieved from a mild hierarchy at the high energy. \n\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|cc|}\\hline\n & $SU(3)_c \\otimes SU(2)_L \\otimes U(1)_Y$ & $U(1)_{B-L}$ \\\\\n\\hline \\hline\n$S_{L,R}$ & (1, 1, 0) & $x$\\\\\n$S'_{L,R}$ & (1, 1, 0) & $x-2$\\\\\n$D_{L,R}$ & (1, 2, 1\/2) & $x$\\\\\n$D'_{L,R}$ & (1, 2, 1\/2) & $x+2$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Additional vector-like fermions. $x$ is a real number.}\n\\label{table:3}\n\\end{table}\n\n\nWe see in Fig.~\\ref{running} that $\\lambda_{\\rm mix}$ is almost unchanged.\nThis is because $\\beta_{\\lambda_{\\rm mix}}$ is proportional to $\\lambda_{\\rm mix}$, \n which is very small (see Appendix).\nHence, the hierarchy between $\\lambda_{\\rm mix}$ and the other couplings gets enlarged at high energies.\nTo avoid this situation and make our model more natural, \n one may introduce additional vector-like fermions listed in Table \\ref{table:3}, for example.\\footnote{\nAs another possibility, one may think that some symmetry forbids the $\\lambda_{\\rm mix}$ term\n and it is generated via a small breaking.}\nAlthough $x$ is an arbitrary real number, we assume $x\\neq 1$ to distinguish the new fermions from the SM leptons.\nThese fermions have Yukawa couplings as\n\\begin{eqnarray}\n\t-{\\mathcal L}_V &=& Y_{SS} \\overline{S_L} \\Phi S'_R + Y_{SD} \\overline{S'_R} H_2^\\dagger D'_L\n\t\t\t\t\t+ Y_{DD} \\overline{D'_L} \\Phi D_R + Y_{DS} \\overline{D_R} H_1 S_L \\nonumber \\\\\n\t\t\t\t\t&& + Y'_{SS} \\overline{S_R} \\Phi S'_L + Y'_{SD} \\overline{S'_L} H_2^\\dagger D'_R\n\t\t\t\t\t+ Y'_{DD} \\overline{D'_R} \\Phi D_L + Y'_{DS} \\overline{D_L} H_1 S_R\n\t\t\t\t\t+ h.c.,\n\\end{eqnarray}\n so that $\\beta_{\\lambda_{\\rm mix}}$ includes\n terms of $Y_{SS} Y_{SD} Y_{DD} Y_{DS}$ and $Y'_{SS} Y'_{SD} Y'_{DD} Y'_{DS}$,\n which are not proportional to $\\lambda_{\\rm mix}$.\nThese terms originate the diagram shown in Fig.\\,\\ref{mix}.\n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[scale=0.7,clip]{mix_corr.eps}\n\\end{center}\n\\caption{One-loop diagram due to the additional fermions, which is relevant to $\\beta_{\\lambda_{\\rm mix}}$.}\n\\label{mix}\n\\end{figure}\nAccordingly, the minimization condition of $V_\\Phi$ is modified to\n\\begin{eqnarray}\n\t\\lambda_\\Phi \\simeq \\frac{11}{6 \\pi^2}\n\t\t\\left[ 6 g_{B-L}^4 - {\\rm tr} Y_M^4\n\t\t- \\frac{1}{8}\\left( Y_{SS}^4 + Y_{SS}^{\\prime 4} + 2Y_{DD}^4 + 2Y_{DD}^{\\prime 4} \\right) \\right].\n\\label{CW_relation2}\n\\end{eqnarray}\nFrom the conditions $\\lambda_\\Phi>0$ and $g_{B-L}<0.2$,\n the additional Yukawa contribution should satisfy\n $Y_{SS}^4 + Y_{SS}^{\\prime 4} + 2Y_{DD}^4 + 2Y_{DD}^{\\prime 4} \\lesssim 3\\times(0.4)^4$.\nNote that vector-like fermions masses are dominantly generated by $v_\\Phi$,\n and they are sufficiently heavy to avoid the current experimental bounds.\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[clip, height=7cm]{100TeV_f.eps}\n\\end{center}\n\\caption{Runnings of quartic couplings for $v_\\Phi=100$\\,TeV\n with additional vector-like fermions.\nThe input parameters are the same as before.}\n\\label{running2}\n\\end{figure}\n\n\nFig.~\\ref{running2} shows the runnings of the quartic couplings for $v_\\Phi=100$ TeV\n with the additional vector-like fermions. \nThe input parameters are the same as before,\n while we have taken the Yukawa couplings as\n $Y_{SS}=Y_{SD}=Y_{DD}=Y_{DS}=0.2$ and $Y'_{SS}=Y'_{SD}=Y'_{DD}=Y'_{DS}=0.1$ at $\\mu=v_\\Phi$, \n for simplicity. \nToward high energies, $|\\lambda_{\\rm mix}|$ becomes larger,\n and the hierarchy with the other couplings becomes mild. \nWe can see that $\\lambda_{H1\\Phi}$ is negative below $\\mu\\simeq10^8$ GeV,\n because the contributions of additional Yukawa couplings to $\\beta_{\\lambda_{H1\\Phi}}$ are effective\n below $\\mu\\simeq10^8$ GeV.\nAbove the scale, the contribution of $U(1)_{B-L}$ couplings becomes effective,\n and then $\\lambda_{H1\\Phi}$ becomes positive.\nAs a result, the large hierarchy at the $U(1)_{B-L}$ symmetry breaking scale can be realized \n with a mild hierarchy at some high energy. \nWe expect that a ultraviolet complete theory, which provides the origin of \n the classical conformal invariance, takes place at the high energy. \n \n\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nWe have investigated a classically conformal $U(1)_{B-L}$ extension of the Standard Model \n with two electroweak Higgs doublet fields. \nThrough the Coleman-Weinberg mechanism, the $U(1)_{B-L}$ symmetry is radiatively broken, \n and a mass scale is generated via the dimensional transmutation. \nThis symmetry breaking is the sole origin of all dimensionful parameters in the model, \n and the mass terms of the two Higgs doublet fields are generated \n through their quartic couplings with the $B-L$ Higgs field. \nAll generated masses are set to be positive but, nevertheless, \n the electroweak symmetry breaking is realized by the bosonic seesaw mechanism. \nIn order for the bosonic seesaw mechanism to work,\n we need a large hierarchy among the two Higgs doublet masses, \n which originates from a large hierarchy among the quartic couplings. \nAlthough it seems unnatural to introduce the large hierarchy by hand at the $U(1)_{B-L}$ symmetry breaking scale, \n we have found through analysis of the renormalization group evolutions of the quartic couplings \n that this hierarchy is dramatically reduced towards high energies. \nTherefore, once our model is defined at some high energy, for example, the Planck scale, \n in other words, the origin of the classically conformal invariance is provided \n by some ultraviolet complete theory at the Planck scale, \n the bosonic seesaw mechanism is naturally realized with a mild hierarchy among the quartic couplings. \nThe requirements for the perturbativity of the running couplings and the electroweak vacuum stability \n in the renormalization group analysis as well as for the naturalness of the electroweak scale, \n we have identified the regions of model parameters such as \n $g_{B-L}(v_\\Phi)\\lesssim0.2$, $0.15 \\lesssim \\lambda_3(v_\\Phi) = \\lambda_4(v_\\Phi) \\lesssim 0.23$, \n and $v_\\Phi \\lesssim 100$ TeV. \nWe have also found that all heavy Higgs boson masses are almost independent of $v_\\Phi$,\n and lie in the range between $1$ TeV and $1.7$ TeV, which can be tested at the LHC in the near future. \n \n \n\n\n\\subsection*{\\centering Acknowledgment} \\label{Acknowledgement}\nN.O. would like to thank the Particle Physics Theory Group of Shimane University\n for hospitality during his visit.\nThis work is partially supported by Scientific Grants\n by the Ministry of Education, Culture, Sports, Science and Technology (Nos. 24540272, 26247038, and 15H01037)\n and the United States Department of Energy (DE-SC 0013680).\nThe work of Y.Y. is supported\n by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists\n (Grant No. 26$\\cdot$2428).\n\n\n\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec1}\nLet $\\Omega$ be a bounded domain in $\\rn$, with $C^{1,\\alpha}$ boundary. We consider the following quasilinear elliptic problem involving the $A$-laplacian operator\n\\begin{equation}\\label{problem}\n\\begin{cases}\n- \\Delta_A u =f(x,u, \\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\nu =0 & {\\rm on}\\,\\,\\,\n\\partial \\Omega \\,,\n\\end{cases}\n\\end{equation}\nwhere $A:[0,\\infty)\\to [0,\\infty)$ is a convex function, vanishing at $0$, $A\\in C^2((0,+\\infty))$, \nand $f:\\Omega\\times\\r\\times\\rn\\to\\r$ is a Carath\\'{e}odory function. The $A$-laplacian operator is defined by $\\Delta_A u={\\rm div}\\left(A'(|\\nabla u|)\\frac{\\nabla u}{|\\nabla u|}\\right)$. The properties of the function $A$ guarantee that $\\Delta_A u$ makes sense also when $|\\nabla u|=0$. \\\\\nA wide class of operators can be incorporated in \\eqref{problem}. The $p$-Laplacian and the $(p,q)$-Laplacian, for which $A(t)=t^p$ and $A(t)=t^p+t^q$, $t\\geq 0$, respectively, are the most known, but we can also consider functions like $A(t)=(\\sqrt{1+t^2}-1)^\\gamma$, for $t\\geq 0$ and $\\gamma> 1$ or $A(t)=t^p\\lg(1+t)$, for $t\\geq 0$ and $p>1$. All the $\\Delta_A$ corresponding to the functions $A$ considered above appear in many physical contests, like nonlinear elasticity and plasticity theory.\\\\ \nThe presence of the gradient in the nonlinear term, called convection term, makes variational methods not applicable. Among the techniques used to study problems with a convection term, we cite: topological degree method (\\cite{BalFil, Ruiz}), theory of pseudomonotone operators (\\cite{GW}), fixed point theorems (\\cite{BNV, Zou}), sub and super solution methods (\\cite{FMMT, FM, G, NgSch}), approximation methods (\\cite{Tanaka}), or a combination of the techniques above (\\cite{BaTo1, FMP, MotWin}).\\\\\nWe deal with existence, regularity and sign of the solutions to \\eqref{problem}. Results in this direction can be found in \\cite{BalFil, FMMT, FMP, MotWin, NgSch, Ruiz, Tanaka, Zou}. In the papers above there are various growth conditions on $f$, with respect to each variable, which make it necessary to use different methods to approach the problem, depending on the behavior of the convective term.\\\\\nIn all the papers cited above, the abstract framework is the classical Sobolev space $W_0^{1,p}(\\Omega)$ and the growth conditions with respect $(s,\\xi)\\in \\r\\times \\rn$ are of polynomial type. By contrast, we work in Orlicz spaces and take into account a class of operators which, although they depend only on the gradient, cannot be treated in the Sobolev spaces. Furthermore, this allows for $f$ a wider choice than that seen above. Roughly speaking, for a problem with the $p$-Laplacian, a function $f(x,s,\\xi)=-c+\\frac{|s|^{p^*-1}}{\\lg(1+|s|)}+a(|s|)|\\xi|^p$ (see Theorem \\ref{main2}) is allowed. This does not happen if we consider standard growths.\\\\\nIn \\cite{BalFil, FMP, Ruiz, Zou} the authors establish the existence and the regularity of positive solutions for a problem involving the $p$-Laplacian. In \\cite{BalFil, Ruiz} \nthe convection term is a continuous, nonnegative function with subcritical growth with respect to $u$ and growth less then $p$ with respect to $\\nabla u$. In \\cite{FMP} the growth of the convection term is at most $p-1$ with respect to $u$ and $\\nabla u$, while in\n \\cite{Zou} the convection term is superlinear for $(s,\\xi)\\to (0,0)$ and its growth is at most $p$ with respect to $u$ and strictly less than $p$ with respect to $\\nabla u$. \nAn existence and regularity result for the $p$-Laplacian with a convection term that can be singular at $0$ can be found in \\cite{NgSch}. The existence of a suitable pair of sub and supersolutions plays a crucial role in their proof. In general, sub and supersolution methods allow to study also the case of a singular convection term, provided the interval of sub and supersolutions does not contain the singular point.\\\\\nIn \\cite{MotWin} the authors\ngive also sign information on the solutions. They use sub and supersolution methods, in combination with variational techniques, for an operator that can be treated as the $(p,q)$-laplacian. \\\\ \nExistence and regularity results, for a general operator $A(x,\\nabla u)$ can be found in \\cite{NgSch, Tanaka}, and in \\cite{FMMT} for $A(x,u,\\nabla u)$. In \\cite{Tanaka} the convection term is a continuous function with growth less than $p-1$ with respect to $u$ and $|\\nabla u|$, while in \\cite{FMMT, NgSch} the growth is at most $p$ with respect to $|\\nabla u|$.\\\\\nLet's make some more detailed comments on our new existence and regularity results to \\eqref{problem} (Theorems \\ref{main}, \\ref{main1} and \\ref{main2}). \nIn Theorem \\ref{main} we assume the existence of an ordered pair of sub and supersolutions $\\underline u,\\,\\overline u \\in W^{1,\\infty}(\\Omega)$ and use Theorem 3.6 in \\cite{BaTo1} to prove the existence of a regular solution to \\eqref{problem}. The growth condition on $f$, in Theorem \\ref{main}, is weaker than that used the Theorem 3.6 in \\cite{BaTo1}.\nA limit in the use of the method of sub and supersolutions is due to the fact that establishing their existence may not be easy. So we give two existence results, Theorems \\ref{main1} and \\ref{main2}, where a unified hypothesis on $f$ guarantees the existence of a suitable pair of sub and supersolutions and enable us to apply Theorem \\ref{main} to obtain the existence of a regular constant sign solution.\\\\\nThe paper is arranged a follows: in Section \\ref{sec2} we give the basic definitions and collect some auxiliary results. Our main theorems are proved in Section \\ref{sec3}. Finally, in Section \\ref{sec4}, we present some examples in which it is easy to verify the existence of constant sub and supersolutions. \n\n\\section{Preliminaries}\\label{sec2}\n\nIn this Section we give the main definitions on Young functions and define the Orlicz Sobolev spaces that we use in the sequel. For a comprehensive treatment of Young functions and Orlicz spaces we refer the reader to \\cite{Chl, KrRu, RR1, RR2}. We also collect some auxiliary results for the proof of the main theorems.\n\\begin{definition} A function $A: [0, \\infty ) \\to [0, \\infty]$ is called a Young function if it is convex, vanishes at $0$, and is neither identically equal to $0$, nor to infinity (in $(0,+\\infty)$).\n\\end{definition}\nFor Young functions\n\\begin{equation}\\label{convexity}\nA(\\lambda t) \\leq \\lambda A(t)\\quad \\hbox{for $\\lambda \\leq 1$ and $t \\geq 0$.}\n\\end{equation}\n\\begin{definition}\\label{conjygate} The Young conjugate of a Young function $A$ is the Young function $\\widetilde A$ defined as\n$$\\widetilde A (s) = \\sup\\{ st - A(t):\\ t \\geq 0\\} \\quad \\hbox{for $s \\geq 0$.}$$\n\\end{definition}\n\n\\begin{definition} A Young function $A$ is said to satisfy the $\\Delta_2$-condition near infinity (briefly $A\\in \\Delta_2$ near infinity) if it is finite valued and there exist two constants $K\\geq 2$ and $M\\geq 0$ such that\n\\begin{equation}\\label{delta2young}\nA(2t)\\leq KA(t)\\quad \\hbox{for }\\ t\\geq M\\,.\n\\end{equation}\n\\end{definition}\n\\begin{definition} The function $A$ is said to satisfy the $\\nabla_2$-condition near infinity (briefly $A\\in\\nabla_2$ near infinity) if there exist two constants $K>2$ and $M\\geq 0$ such that\n\\begin{equation}\\label{nabla2young}\nA(2t)\\geq KA(t)\\quad \\hbox{for }\\ t\\geq M\\,.\n\\end{equation}\n\\end{definition}\nIf \\eqref{delta2young} or \\eqref{nabla2young} holds with $M=0$, then $A$ is said to satisfy the $\\Delta_2$-condition (globally), or the $\\nabla_2$-condition (globally), respectively. \nGiven a Young function $A \\in C^1([0,+\\infty))$, define the quantities \n\\begin{equation}\\label{i}\n\tp_A=inf_{t>0}\\frac {t\\cdot A'(t)}{A(t)},\\ \\hbox{and}\\quad q_A=sup_{t>0}\\frac {t\\cdot A'(t)}{A(t)}\\,.\n\\end{equation}\nThe conditions\n\\begin{equation*}\\label{ndg}\n\tp_A>1\\ \\hbox{and}\\quad q_A<+\\infty \n\\end{equation*}\nare equivalent to the fact that $A\\in\\nabla_2\\cap\\Delta_2$ (globally). \n\n\\par\nWe give basic definitions and the main properties on the Orlicz spaces.\nLet $\\Omega$ be a measurable set in $\\rn$, with $n\\geq 1$. Given a Young function $A$, the Orlicz space $L^A(\\Omega)$ is the set of all measurable functions $u:\\Omega\\to\\r$ such that the Luxemburg norm\n$$\\|u\\|_{L^A(\\Omega)}=\\inf\\bigg\\{\\lambda>0:\\int_\\Omega A\\big(\\tfrac 1\\lambda|u|\\big)\\,dx\\leq 1 \\bigg\\}$$\nis finite. The functional $\\|\\cdot\\|_{L^A(\\Omega)}$ is a norm on $L^A(\\Omega)$, and the latter is a Banach space (see \\cite{Adams}).\\\\\nIf $A$ is a Young function, then a generalized H\\\"older inequality\n\\begin{equation}\\label{holderyoung}\n\\int _\\Omega |u v|\\,dx \\leq 2\\|u\\|_{L^A (\\Omega)} \\|v\\|_{L^{\\widetilde A}(\\Omega)}\n\\end{equation}\nholds for every $u\\in L^A (\\Omega)$ and $v\\in L^{\\widetilde A}(\\Omega)$.\\\\\n\nIf $A\\in \\Delta_2$ globally (or $A\\in \\Delta_2$ near infinity and $\\Omega$ has finite measure) then \n\\begin{equation}\\label{intfinito}\n \\int_{\\Omega} A(k|u|)dx <+\\infty \\ \\hbox{for all}\\ u\\in L^A(\\Omega),\\ \\hbox{all}\\ k\\geq 0\\,. \n\\end{equation}\nLet $\\Omega$ be an open set in $\\rn$ with $|\\Omega|<\\infty$. The isotropic Orlicz-Sobolev spaces \n$W^{1,A}_0(\\Omega)$ is defined as\n\\begin{align*}\nW^{1,A}_0(\\Omega)=\\{u:\\Omega \\to \\r: &\\, \\hbox{the continuation of $u$ by $0$ outside $\\Omega$} \\\\ & \\hbox{is weakly differentiable in $\\rn$, \\, |u|,\\, $|\\nabla u| \\in L^A (\\Omega)$}\\}.\n\\end{align*}\nThe space $W_0^{1,A}(\\Omega)$ equipped with the norm \n$$\\|u\\|_{W^{1,A}_0(\\Omega)} = \\| |\\nabla u| \\|_{L^A (\\Omega)}.$$\nis a Banach space. This norm is equivalent to the standard one\n$$\\|u\\|_{W_0^{1,A}(\\Omega)} = \\|u\\|_{L^A(\\Omega)}+\\||\\nabla u|\\|_{L^A (\\Omega)}\\,.$$\n\nFor the Young function $A$ in \\eqref{problem}, we assume:\n\\begin{itemize}\n\\item [[ A1]] \\ $A\\in C^2(]0,+\\infty[)$ (this implies $A'\\in C^1([0,+\\infty[)$);\n \\item [[ A2]] there exist two positive constants $\\delta$, $g_0>0$ such that\n\\begin{equation}\\label{dg}\n \\delta\\leq \\frac{tA''(t)}{A'(t)}\\leq g_0\\quad \\hbox{for}\\ t>0\\,.\n\\end{equation}\n\\end{itemize}\n\nWe point out that, \\eqref{dg} guarantees that $A'(0)=0$ and $A\\in \\nabla_2\\cap\\Delta_2$ globally. In fact integrating \\eqref{dg},\n\\begin{equation*}\n\\left(\\frac{t}{t_0}\\right)^{\\delta}\\leq \\frac{A'(t)}{A'(t_0)}\\leq \\left(\\frac{t}{t_0}\\right)^{g_0}\\quad \\hbox{for}\\ t>t_0>0\\,.\n\\end{equation*}\nChoosing $t=2t_0$\n\\begin{equation*}\n\t2^\\delta A'(t_0)\\leq A'(2t_0) \\leq 2^{g_0}A'(t_0)\\quad \\hbox{for}\\ t>t_0>0\\,.\n\\end{equation*}\nThus\n\\begin{equation*}\nA(2t)=\\int_0^{2t}A'(\\tau)d\\tau=2\\int_0^{t}A'(2s)ds\\leq 2^{g_0+1}\\int_0^{t}A'(s)ds=2^{g_0+1}A(t)\\quad \\hbox{for all}\\; t>0,\n\\end{equation*}\nand\n\\begin{equation*}\nA(2t)=\\int_0^{2t}A'(\\tau)d\\tau=2\\int_0^{t}A'(2s)ds\\geq 2^{\\delta+1} \\int_0^{t}A'(s)ds=2^{\\delta+1} A(t)\\quad \\hbox{for all}\\; t>0.\n\\end{equation*}\n\\par\nWe investigate the existence and the regularity of the solutions to problem \\eqref{problem}. The proof of the existence is based on sub and supersolution methods, while the main tool for the regularity is Theorem 1.7 of \\cite{Li1} and the remark immediately after the statement (see also \\cite[Theorem 1]{Li}), that we recall below.\n\\begin{proposition}(see \\cite[Theorem 1.7]{Li1})\\label{Lib}\nLet $\\Omega$ be a bounded domain in $\\r^n$ with $C^{1,\\alpha}$ boundary, for some $0<\\alpha\\leq 1$. Let $g:[0,+\\infty[\\to [0,+\\infty[$ be a $C^1$, increasing function, satisfying $0<\\delta\\leq \\frac{tg'(t)}{g(t)}\\leq g_0$, for $t>0$, and let $G(t)=\\int_0^t g(\\tau)d\\tau$. Consider the problem\n$$div({\\cal A}(x,u,\\nabla u))+B(x,u,\\nabla u)=0\\ \\hbox{in}\\ \\Omega\\,.$$\nSuppose ${\\cal A}$ and $B$ satisfy the structure conditions (here $a_{ij}(x,z,\\eta)=\\frac{\\partial {\\cal A}^i}{\\partial\\eta_j}$) \n\\begin{itemize}\n\t\\item[$(a)$] $\\sum_{i,j=1}^na_{ij}(x,z,\\eta)\\xi_i\\xi_j\\geq \\frac{g(|\\eta|)}{|\\eta|}|\\xi|^2$\n\\item[$(b)$] $\\sum_{i,j=1}^n|a_{ij}(x,z,\\xi)|\\leq \\Lambda\\frac{g(|\\xi|)}{|\\xi|}$\n\\item[$(c)$] $|{\\cal A}(x,z,\\xi)-{\\cal A}(y,w,\\xi)|\\leq \\Lambda_1(1+g(|\\xi|)(|x-y|^\\alpha+|z-w|^\\alpha)$\n\\item[$(d)$] $|B(x,z,\\xi)|\\leq \\Lambda_1(1+g(|\\xi|)|\\xi|),$\n\\end{itemize}\nfor some positive constants $\\Lambda$, $\\Lambda_1$, $M_0$, for all $x$ and $y\\in \\Omega$, for all $z,w\\in [-M_0,M_0]$ and for all $\\xi \\in \\mathbb{R}^n$.\t\nThen, any solution $u\\in W^{1,G}(\\Omega)$, with $|u|\\leq M_0$ in $\\Omega$, is $C^{1,\\beta}(\\overline{\\Omega})$ for some positive $\\beta$. Moreover \n\\begin{align}\\label{boundc1}\n\t\\|u\\|_{C^{1,\\beta}(\\overline{\\Omega})}\\leq C(\\alpha, \\Lambda, \\delta, g_0, n, \\Lambda_1, g(1), \\Omega, M_0)\n\\end{align}\n\\end{proposition}\n\n\\begin{lemma}\\label{LemLib} Let $A$ be a Young function satisfying $[A1]$ and $[A2]$. Put $\\Phi(\\xi)=A(|\\xi|)$. Then \n\\begin{eqnarray}\\label{1.10a''}\n\t\t\\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j\n\t\t\\geq\\min\\{\\delta, 1\\}\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{1.10b}\n\\sum_{i,j=1}^n\\left|\\partial_{ij}\\Phi(\\eta)\\right|\n\t\\leq [2\\max\\{|\\delta-1|,|g_0-1|\\}+n]\\frac{A'(|\\eta|)}{|\\eta|}\\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,,\n\\end{eqnarray} \t\n\\end{lemma}\nand $\\nabla \\Phi=\\cal A$ satisfies conditions $a--c$ in Proposition \\ref{Lib}, with $g(t)=\\min\\{\\delta,\\,1\\}A'(t)$, $\\Lambda= \\frac{\\lambda}{\\min\\{\\delta,\\,1\\}}$, where $\\lambda=2\\max\\{|\\delta-1|,|g_0-1|\\}+n$.\\\\\n{\\bf Proof}\nFrom \\eqref{dg} \n\\begin{equation*}\n(\\delta -1) \\frac{A'(|\\eta|)}{|\\eta|}\\leq A''(|\\eta|)-\\frac{A'(|\\eta|)}{|\\eta|} \\leq (g_0-1)\\frac{A'(|\\eta|)}{|\\eta|}\n\\ \\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation*}\nAlso, $\\partial_i\\Phi(\\eta)=A'(|\\eta|)\\frac{\\eta_i}{|\\eta|}$, and\n\\begin{equation*}\n\\partial_{ij}\\Phi(\\eta)= A''(|\\eta|)\\frac{\\eta_i\\eta_j}{|\\eta|^2} +A'(|\\eta|)\\left(\\frac{\\delta_{ij}}{|\\eta|} -\\frac{\\eta_i\\eta_j}{|\\eta|^3}\\right)\\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation*}\nThus\n\\begin{align*}\n\\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j= &\\sum_{i,j=1}^n \\left(A''(|\\eta|)\\frac{\\xi_i\\eta_i\\xi_j\\eta_j}{|\\eta|^2} +A'(|\\eta|)\\frac{\\delta_{ij}\\xi_i\\xi_j}{|\\eta|}- A'(|\\eta|)\\frac{\\xi_i\\eta_i\\xi_i\\eta_j}{|\\eta|^3}\\right)\\\\\n&=\\left(\\frac{A''(|\\eta|)}{|\\eta|^2}- \\frac{A'(|\\eta|)}{|\\eta|^3}\\right)(\\xi,\\eta)^2+\n\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\\\\n &\\geq (\\delta -1)\\frac{A'(|\\eta|)}{|\\eta|^3}(\\xi,\\eta)^2+\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,.\\qquad\\qquad\\qquad\\qquad\n\\end{align*}\nIf $\\delta\\geq 1$ \n\\begin{equation}\\label{1.10a}\n \\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j\\geq \\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation}\nIf $\\delta<1$ \n\\begin{eqnarray}\\label{1.10a'}\n \\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j\\geq (\\delta -1)\\frac{A'(|\\eta|)}{|\\eta|^3}|\\xi|^2|\\eta|^2+\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\\\\n =\\delta\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,.\\nonumber\n\\end{eqnarray}\nPutting together \\eqref{1.10a} and \\eqref{1.10a'}, we get \\eqref{1.10a''}.\\\\\nConsider\n\\begin{equation*}\n|\\partial_{ij}\\Phi(\\eta)|\\leq\\frac{|\\eta_i||\\eta_j|}{{|\\eta|^2}} \\left|A''(|\\eta|) -\\frac{A'(|\\eta|)}{|\\eta|}\\right|+\n\t\\delta_{ij}\\frac{A'(|\\eta|)}{|\\eta|} \\quad \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation*}\t\n\nThus\n\\begin{eqnarray*}\n\\sum_{i,j=1}^n\\left|\\partial_{ij}\\Phi(\\eta)\\right|\\leq\\frac{\\left(\\sum_{i=1}^n|\\eta_i|\\right)^2}{{|\\eta|^2}} \\left|A''(|\\eta|) -\\frac{A'(|\\eta|)}{|\\eta|}\\right|+\nn\\frac{A'(|\\eta|)}{|\\eta|}\\nonumber \\\\\n\\leq\\left[2\\max\\{|\\delta-1|,|g_0-1|\\}+n\\right]\\frac{A'(|\\eta|)}{|\\eta|}\\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{eqnarray*} \nSo \\eqref{1.10b} holds with $\\lambda=2\\max\\{|\\delta-1|,|g_0-1|\\}+n$.\n\\qed\n\\section{Main results}\\label{sec3}\nIn this section first we give two existence and regularity results (Theorems \\ref{main1} and \\ref{main2}), in which we assume a global growth condition on $f$, unilateral with respect to $s\\in \\r$. In Theorem \\ref{main1} we require that $f$ satisfies some conditions for $(x,s,\\xi)\\in \\Omega \\times [0,+\\infty)\\times \\rn$ and obtain the existence of a nonnegative solution. Similarly, in Theorem \\ref{main2}, $f$ satisfies some conditions for $(x,s,\\xi)\\in \\Omega \\times (-\\infty, 0]\\times \\rn$ that guarantee the existence of a non-positive solution.\\\\\n\\noindent Here is the definition of weak solution to \\eqref{problem}. \n\\begin{definition}\nA function $u\\in \\w0$ is a weak solution to problem \\eqref{problem} if\n\\begin{equation*}\n\\int_{\\Omega}A'(|\\nabla u|)\\cdot\\frac{\\nabla u}{|\\nabla u|}\\cdot\\nabla v dx= \\int_{\\Omega}f(x,u,\\nabla u)vdx\n\\end{equation*}\nfor all $v\\in \\w0$. \n\\end{definition}\nFor the first two Theorems, we assume\n$$({\\cal{H}}):\n\\left\\{\n\\begin{array}{l}\n\ta:[0,+\\infty[\\to [0,+\\infty[ \\ \\hbox{is a locally essentially bounded function};\\\\\n\t\\rho_1,\\rho_2:\\Omega\\to [0,+\\infty[\\ \\hbox{are two measurable functions},\\ \\rho_1,\\,\\rho_2\\in L^\\infty(\\Omega)\\ \\hbox{and}\\\\ \\rho_2 (x)>0\\ \\hbox{on a set of positive measure};\\\\\n\tg_1,g_2:[0,+\\infty[\\to [0,+\\infty[\\ \\hbox{are two non-decreasing functions such that}\\ g_1(0)=g_2(0)=0\\\\\n\t\\hbox{and there exist}\\ s_0>0,\\,k_1\\in \\left ]0,\\omega_n^{\\frac{1}{n}}|\\Omega|^{-\\frac{1}{n}}\\right[,\\ \\hbox{such that}\\ \n\tg_1(|s|)|s|\\leq A(k_1|s|)\\ \\hbox{for all}\\ |s|\\geq s_0\\,.\n\\end{array}\n\\right.\n$$\nHere $\\omega_n$ is the measure of the unit ball in $\\rn$.\n\\begin{theorem}\\label{main1}\nLet $\\Omega$ be a bounded domain in $\\rn$ with $C^{1,\\alpha}$ boundary. \nLet $A: [0,+\\infty[\\to [0,+\\infty[$ be a Young function, satisfying $[A1]$ and $[A2]$. Let $f:\\Omega\\times \\mathbb{R}\\times \\mathbb{R}^n\\to \\mathbb{R}$ be a Carath\\'{e}odory function fulfilling\n\t\\begin{equation}\\label{growth f2+}\n\t\t\\rho_2(x)-g_2(s)-a(s)A'(|\\xi|)|\\xi|\\leq f(x,s,\\xi)\\leq \\rho_1(x)+g_1(s)\\ \\hbox{ for a.e.}\\ x\\in \\Omega,\\ \\hbox{all}\\ s\\geq 0,\\ \\hbox{all}\\ \\xi\\in\\rn\\,.\n\t\n\t\\end{equation}\nThe functions $a,\\,\\rho_1,\\,\\rho_2,\\,g_1,\\,g_2$ are as in $(\\cal{H})$.\n\tThen problem \\eqref{problem} has a nontrivial, nonnegative solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$.\\\\\n\n\tIf, in addition, there exist $\\overline{\\delta} >0$ and $k_3>0$ such that\n\t$g_2(s)s\\leq A(k_3s)$ for every $s\\in (0,\\overline{\\delta})$, then $u>0$ in $\\Omega$.\n\\end{theorem}\n\\begin{theorem}\\label{main2}\nLet $\\Omega$ be a bounded domain in $\\rn$ with $C^{1,\\alpha}$ boundary.\nLet $A: [0,+\\infty[\\to [0,+\\infty[$ be a Young function, satisfying $[A1]$ and $[A2]$. Let $f:\\Omega\\times \\mathbb{R}\\times \\mathbb{R}^n\\to \\mathbb{R}$ be a Carath\\'{e}odory function fulfilling\n\\begin{equation}\\label{growth f2}\n-\\rho_1(x)-g_1(|s|)\t\\leq f(x,s,\\xi)\\leq -\\rho_2(x)+g_2(|s|)+a(s)A'(|\\xi|)|\\xi|\\ \\hbox{for a.e.}\\, x\\in \\Omega,\\ \\hbox{all}\\ s\\leq 0,\\ \\hbox{all}\\ \\xi\\in\\rn\\,,\n\n\t\\end{equation}\nwhere the functions $a,\\,\\rho_1,\\,\\rho_2,\\,g_1,\\,g_2$ are as in $(\\cal{H})$.\n\tThen problem \\eqref{problem} has a nontrivial, non-positive solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$.\\\\\n\n\tIf, in addition, there exist $\\overline{\\delta} >0$ and $k_3>0$ such that\n\t$g_2(s)s\\leq A(k_3s)$ for every $s\\in (0,\\overline{\\delta})$, then $u<0$ in $\\Omega$.\n\\end{theorem}\n\\begin{remark}\nIn \\cite{BalFil}, Theorem 1, the authors prove the existence of a positive solution for a problem with the $p$-Laplacian, and a convection term $f$ satisfying the hypotheses of Theorem \\ref{main2}.\n\\end{remark}\nFor the proof of the Theorems above, we need an abstract existence result, where sub and supersolutions come into play.\\\\\nThe definition of sub and supersolution in general domains, for which a trace theory may not hold, can be found in \\cite{BaTo1}. Our hypotheses on $\\Omega$ allow to adopt the classical definition (see \\cite[Theorem 3.1]{Ctrace}).\\\\\nWe say that $\\overline u\\in W^{1,A}(\\Omega)$ is a supersolution to \\eqref{problem} if $\\overline u_{|\\partial \\Omega}\\geq 0$ (in the sense of traces) and\n\\begin{equation*}\\label{supersolution}\n\t\\int_{\\Omega}A'(|\\nabla\\overline u|)\\cdot\\frac{\\nabla\\overline u}{|\\nabla\\overline u|}\\cdot\\nabla v dx\\geq \\int_{\\Omega}f(x,\\overline u,\\nabla \\overline u)vdx\n\\end{equation*}\nfor all $v\\in W_0^{1,A}(\\Omega)$, $v\\geq 0$ a.e. in $\\Omega$.\\\\\nWe say that $\\underline u\\in W^{1,A}(\\Omega)$ is a subsolution to \\eqref{problem} if $\\underline u_{|\\partial \\Omega}\\leq 0$ (in the sense of traces) and\n\\begin{equation*}\\label{subsolution}\n\t\\int_{\\Omega}A'(|\\nabla\\underline u|)\\cdot\\frac{\\nabla \\underline u}{|\\nabla \\underline u|}\\cdot \\nabla v dx\\leq \\int_{\\Omega}f(x,\\underline u,\\nabla \\underline u)vdx\n\\end{equation*}\nfor all $v\\in W_0^{1,A}(\\Omega)$, $v\\geq 0$ a.e. in $\\Omega$.\\\\\nFor the next Theorem we assume that problem \\eqref{problem} has a subsolution and supersolution, $\\underline{u}$, $\\overline{u}\\in W^{1,\\infty}(\\Omega)$, with $\\underline{u}(x)<\\overline{u}(x)$ for all $x\\in \\Omega$.\nAlso, $f:\\Omega\\times \\r\\times \\rn\\to \\r$ is a Carath\\'{e}odory function satisfying the following growth condition:\n\\begin{itemize}\n\\item [(H)] there exists a function $\\sigma\\in L^{\\infty}(\\Omega)$ and a constant $a>0$, such that\n$$\n|f(x,s,\\xi)|\\leq \\sigma (x)+a A'(|\\xi|)|\\xi|\\quad \\hbox{for a.e.}\\ x\\in \\Omega,\\ \\hbox{all}\\ s\\in [\\underline u(x),\\overline u(x)],\\ \\hbox{all}\\ \\xi\\in \\rn\\, .\n$$\n\\end{itemize}\n\\noindent The local condition on $f$, with respect to $s$, is sufficient for our purposes. The use of the method of sub and supersolutions requires an a priori analysis of the problem. Only once the existence of sub and supersolutions has been established does one proceed to search for the existence of a solution.\n\n\n\n\\begin{theorem}\\label{main}\nLet $\\Omega$ be a bounded domain in $\\rn$ with $C^{1,\\alpha}$ boundary.\nLet $A: [0,+\\infty[\\to [0,+\\infty[$ be a Young function satisfying $[A1]$ and $[A2]$. Let $\\underline{u}$, $\\overline{u}\\in W^{1,\\infty}(\\Omega)$ be as above and assume that $f$ satisfies hypothesis (H). Then problem \\eqref{problem} admits at least a solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$. Moreover $\\underline u(x)\\leq u(x)\\leq \\overline u(x)$ a.e in $\\Omega$.\n\\end{theorem}\n{\\bf Proof}. Let $M=\\max\\{\\| \\overline u\\|_\\infty,\\| \\underline u\\|_\\infty\\}$ and $R>\\max\\{\\|\\nabla \\overline u\\|_\\infty,\\|\\nabla \\underline u\\|_\\infty\\}$. Consider the truncated function $f_R$ defined by\n\\begin{equation*}\\label{fR}\n f_R(x,s,\\xi)=\\left\\{\\begin{array}{cc}\n f(x,s,\\xi) & \\hbox{if}\\ |\\xi|\\leq R,\\\\\n f(x,s,\\xi)\\cdot\\frac{A'(R)R}{A'(|\\xi|)|\\xi|} & \\hbox{if}\\ |\\xi|> R \\,,\n \\end{array}\n \\right.\n\\end{equation*}\nand the problem\n\\begin{equation}\\label{problem1}\n\\begin{cases}\n-\\Delta_A(u)=f_R(x,u,\\nabla u) & {\\rm in}\\,\\,\\,\\Omega \\\\\nu =0 & {\\rm on}\\,\\,\\,\\partial \\Omega \\,.\n\\end{cases}\n\\end{equation}\nIn view of the choice of $R$, $\\underline u$ and $\\overline u$ are a subsolution and a supersolution to \\eqref{problem1} respectively. Using the monotonicity of $A'$ we deduce that $|f_R(x,s,\\xi)|\\leq \\sigma (x)+a A'(R)R$, for a.e. $x\\in \\Omega$, all $s\\in [\\underline u(x),\\overline u(x)]$, all $\\xi\\in \\rn$. From Theorem 3.6 in \\cite{BaTo1} problem \\eqref{problem1} admits a solution $u\\in W^{1,A}_0(\\Omega)$ with $\\underline u(x)\\leq u(x)\\leq \\overline u(x)$ a.e in $\\Omega$. Thus $u\\in L^\\infty(\\Omega)$.\\\\ \n\\noindent The functions $A$ and $f$ satisfy the hypotheses of Proposition \\ref{Lib}, with $\\Lambda_1=\\max\\{\\|\\sigma\\|_\\infty, \\min\\{\\delta,1\\}^{-1}a\\}$ (see also Lemma \\ref{LemLib}). Since $|f_R|\\leq |f|$ the same holds for $f_R$ whatever $R$ is.\nDue to Proposition \\ref{Lib} there exist two positive constants $0<\\beta\\leq 1$ and $C$, independent from $R$, such that any solution to \\eqref{problem1} belongs in $C_0^{1,\\beta}(\\overline \\Omega)$ and $\\|u\\|_{C_0^{1,\\beta}(\\overline \\Omega)}\\leq C$. Choosing $R>C$ we deduce that $u$ is a solution to \\eqref{problem}.\n\\qed\n\\begin{remark}\nWhen the solution $u$ has constant sign, it should be interest to verify if it is positive (or negative) in $\\Omega$. The maximum principle by Pucci-Serrin (see \\cite [Theorem 3.5] {PS}) is a powerful tool, as it ensures that (under some proper conditions on $A$ and $f$) \nany nonnegative solution to \\eqref{problem} is positive in $\\Omega$. \nA quite standard situation occurs when $f$ is bounded below by suitable monotone functions and $A\\in\\Delta_2$ near $0$, as the following corollary shows. \\end{remark}\n\\begin{corollary}\\label{cor1}\nUnder the hypotheses of Theorem \\ref{main}, assume that $f$ satisfies\n\\begin{equation}\\label{growthPS}\n\tf(x,s,\\xi)\\geq -a A'(|\\xi|)-b(s)\\ \\hbox{for all} \\; x\\in \\Omega,\\ s>0,\\ \\xi\\in \\mathbb{R}^n,\\; |\\xi|\\leq 1,\n\\end{equation}\nwhere $a>0$, $b:[0,+\\infty[ \\to [0,+\\infty[$ is a function increasing in $(0,\\overline{\\delta})$ (for some $\\overline{\\delta}>0$), $b(0)=0$, and $b(s)=\\frac{A(ks)}{s}$ for $s\\in (0,\\overline\\delta)$ and some $k>0$.\nThen any nonnegative, nontrivial solution to \\eqref{problem} is positive.\n\\end{corollary}\n{\\bf Proof.}\nLet $u\\in W^{1,A}_0(\\Omega)$ be a nonnegative, nontrivial solution to \\eqref{problem}. Theorem \\ref{main} ensures that $u\\in C_0^{1,\\beta}(\\overline{\\Omega})$. In order to prove that $u>0$ in $\\Omega$ we use Theorem 5.3.1 of \\cite{PS}.\\\\\nConditions $(A1)'$ and $(A2)$ of the Theorem cited above hold, because $A\\in C^2((0,+\\infty))$, $A'(0)=0$, and $s\\mapsto A'(s)$ is strictly increasing. Conditions $(F2)$ and $(B1)$ are satisfied too.\\\\\nIt remains to verify condition $(1.1.5)$ of Theorem 5.3.1 of \\cite{PS}. \nPut $B(s)=\\int_0^s b(t)dt$. Due to the monotonicity of $\\frac{A(t)}{t}$, for $s\\in(0,\\overline\\delta)$, it holds \n$$B(s)=\\int_0^s \\frac{A(kt)}{t}dt\\leq \\int_0^s \\frac{A(ks)}{s}dt=A(ks)\\,.$$\nIf $h\\in \\mathbb{N}$ is such that $k<2^h$ then, in view of \\eqref{delta2young}\n$$A(ks)\\leq K^h A(s)\\quad \\hbox{for all}\\; s\\geq 0\\,.$$ \nLet $b_1=\\max\\{p_A-1,K^h\\}$. \nThen, for $s\\in (0,\\overline{\\delta})$, using \\eqref{convexity} and the inequality above\n\\begin{align*}\n\tH(s)=sA'(s)-A(s)&\\geq (p_A-1)A(s)=\\frac{p_A-1}{b_1}b_1A(s)\\nonumber\\\\\n\t\\geq b_1 A\\left(\\frac{(p_A-1)s}{b_1}\\right)\\geq K^hA\\left(\\frac{(p_A-1)s}{b_1}\\right)&\\geq A\\left(\\frac{k(p_A-1)s}{b_1}\\right) \\geq B\\left(\\frac{(p_A-1)s}{b_1}\\right)\\,, \n\\end{align*}\nor equivalently\n\\begin{equation*}\nH\\left(\\frac{b_1s}{p_A-1}\\right)\\geq B(s)\\quad \\hbox{for}\\ 00$ in $\\Omega$.\\qed\n\nNow we accomplish with the proof of Theorems \\ref{main1} and \\ref{main2}.\\\\\n{\\bf Proof of Theorem \\ref{main1}}. \nFrom the proof of Theorem 3.3 of \\cite{BaTo1} we know that there exists a nontrivial solution $\\overline{u}\\geq 0$, to the problem\n\t$$\n\t- \\Delta_A(u) =\\rho_1(x)+g_1(|u|)\\,.\n\t$$\nTheorem 3 of \\cite{Cbound} guarantees that $\\overline{u}$ is bounded. Finally, from Proposition \\ref{Lib}, we have that $\\overline{u}\\in C_0^{1,\\beta}(\\overline{\\Omega})$.\nThe inequalities in \\eqref{growth f2+} show that $\\overline{u}$ is a supersolution to problem \\eqref{problem} and $\\underline{u}=0$ is a subsolution to problem \\eqref{problem}. The assumptions on $\\rho_2$ guarantee that $\\underline{u}=0$ is not a solution. If we put $\\sigma(x)=\\max\\{\\rho_1(x)+g_1(\\overline{u}(x)) ,\\; \\rho_2(x)+g_2(\\overline{u}(x))\\}$ for a.e. $x\\in \\Omega$, then \\eqref{growth f2+} leads to\n$$\n|f(x,s,\\xi)|\\leq \\sigma(x)+a(s)A'(|\\xi|)|\\xi|\\quad \\hbox{for a.e.} \\; x\\in \\Omega, \\; \\hbox{all}\\ s\\in [0,\\overline{u}(x)],\\; \\hbox{all}\\ \\xi\\in \\mathbb{R}^n\\,.$$\nLet $I=[0,\\sup_\\Omega \\overline u]$ and $a=\\|a\\|_{L^\\infty(I)}$. Then $a<+\\infty$ and $a(s)\\leq a$ for a.e. $s\\in I$. Let $I_0\\subset I$ be a set of null measure, such that $a(s)>a$ for all $s\\in I_0$. For $s\\in I_0$ it holds\n\\begin{align}\n|f(x,s,\\xi)|=\\lim_{t\\to s}|f(x,t,\\xi)|=\\liminf_{t\\to s}|f(x,t,\\xi)|& \\leq \\sigma(x)+\\liminf_{t\\to s}a(t)A'(|\\xi|)|\\xi|\\nonumber\\\\\n\t\\leq \\sigma(x)+aA'(|\\xi|)|\\xi| &\\quad \\hbox{for a.e.} \\; x\\in \\Omega, \\; \\xi\\in \\mathbb{R}^n\\,.\n\\end{align}\nThus \n\\begin{align}\n|f(x,s,\\xi)|\\leq \\sigma(x)+aA'(|\\xi|)|\\xi|&\\quad \\hbox{for a.e.} \\; x\\in \\Omega, \\; s\\in [0,\\overline u(x)],\\; \\xi\\in \\mathbb{R}^n\\,.\n\\end{align}\nSo, from Theorem \\ref{main}, problem \\eqref{problem} admits at least a nontrivial solution $u\\in C_0^{1,\\beta}(\\overline{\\Omega})$ such that $0\\leq u\\leq \\overline{u}$.\\\\\nNow we prove that, under the additional condition on $g_2$, $u>0$ in $\\Omega$. We set $b(s)=\\max \\{g_2(s),\\frac{A(k_3s)}{s}\\}$, for $s\\geq 0$ and observe that the left inequality in \\eqref{growth f2+} guarantees that we can apply Corollary \\ref{cor1}.\n\\qed\n\\noindent {\\bf Proof of Theorem \\ref{main2}}. It is enough to put $f_1(x,s,\\xi)=-f(x,-s,-\\xi)$ and to use Theorem \\ref{main1} for $f_1$.\\qed\n\n\\section{Examples}\\label{sec4}\nThis section is devoted to some examples with different Young functions and various nonlinearities.\\\\\n\\noindent In the first two examples we consider the problem\n\\begin{equation}\\label{pqf}\n\\ \\begin{cases}\n- {\\rm div}\\left((|\\nabla u|^{p-2}+|\\nabla u|^{q-2})\\nabla u\\right) =f(x,u,\\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\nu =0 & {\\rm on}\\,\\,\\,\\partial \\Omega \\, ,\n\\end{cases}\n\\end{equation}\nwhere $10\\}|>0$, $|\\{x\\in\\Omega:h<0\\}|>0$.\\\\\nSet\n\\begin{equation*}\nf(x,s,\\xi)= g(s)h(x)+a(s)A'(|\\xi|)(|\\xi|)\\ \\hbox{for}\\ (x,s,\\xi)\\in \\Omega \\times \\r\\times \\rn\\,.\n\\end{equation*}\nThen $u_1=s_1$ and $u_2=s_2$ are a subsolution and a supersolution to \\eqref{pqf}, respectively. Also $u\\equiv 0$ is not a solution nor a sub or a supersolution and $f$ satisfies condition (H) \n with $\\sigma (x)=|h(x)|\\max_{[s_1,s_2]}|g(s)|$ and $a =\\max_{[s_1,s_2]}|a(s)|$. By Theorem \\ref{main}, problem \\eqref{pqf} has a nontrivial solution $u\\in C^{1,\\beta}_0(\\overline\\Omega)$ with $s_1\\leq u\\leq s_2$ a.e. in $\\Omega$.\n\n\\end{example}\n\n\\begin{example}\nLet $a:\\r\\to\\r$ and $g:\\r\\to\\r$ be two continuous functions and let $h:\\Omega\\to \\r$ be an essentially bounded function. Assume that $h(x)\\geq 0$ (or $h(x)\\leq 0$) in $\\Omega$, $h(x)>0$ on a set of positive measure, $g(s_1)=0$ for some $s_1>0$, $g(s)\\neq 0$ for all $s\\in [0,s_1[$, and $g(0)h(x)\\geq 0$ in $\\Omega$.\\\\\nSet\n\\begin{equation*}\n\tf(x,s,\\xi)= g(s)h(x)+a(s)A'(|\\xi|)(|\\xi|)\\ \\hbox{for}\\ (x,s,\\xi)\\in \\Omega \\times \\r\\times \\rn\\,.\n\\end{equation*}\nThen $u_1\\equiv0$ and $u_2\\equiv s_1$ are a subsolution and a supersolution to \\eqref{pqf}, respectively. Also $u\\equiv 0$ is not a solution and $f$ satisfies condition (H) with $\\sigma (x)=|h(x)|\\max_{[0,s_1]}|g(s)|$ and $a =\\max_{[0,s_1]}|a(s)|$. By Theorem \\ref{main}, problem \\eqref{pqf} has a nontrivial solution $u\\in [0,s_1]$, $u\\in C^{1,\\beta}_0(\\overline\\Omega)$. From Theorem 5.3.1 in \\cite{PS}, $u>0$ in $\\Omega$ (note that $g(s)h(x)\\geq 0$ in $\\Omega\\times[0,s_1]$).\n\\end{example}\n\\begin{example} \nConsider the problem \n\\begin{equation}\\label{problempq}\n\\begin{cases}\n- {\\rm div}(\\lg(1+|\\nabla u|^{q})|\\nabla u|^{p-2}\\nabla u) =f(x,u,\\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\nu =0 & {\\rm on}\\,\\,\\,\n\\partial \\Omega \\,,\n\\end{cases}\n\\end{equation}\nwith $q>0$, $p>1$.\\\\ \nLet $r\\geq q$, $\\overline \\delta >0$ and define $b:[0,+\\infty[\\to [0,+\\infty[$ as\n\\begin{equation*}\n b(s)=\\left\\{\\begin{array}{cc}\n s^{p+q-1}&\\hbox{if}\\ s\\in [0,\\overline \\delta], \\\\\n s^{p+r-1}& \\hbox{if}\\ s> \\overline \\delta\\,.\n \\end{array}\\right.\n\\end{equation*}\nAssume that $f:\\Omega\\times\\r\\times\\rn\\to\\r$ is a Carath\\'{e}odory function satisfying\n\\begin{itemize}\n\\item [$(f_0)$] there exists $\\sigma >0$ such that $f(x,\\sigma,0)\\leq 0$ a.e. in $\\Omega$;\n\\item [$(f_1)$] $f(x,0,0)\\geq 0$ a.e. in $\\Omega$, with strict inequality on a set of positive measure;\n\\item [$(f_2)$] there exist $k>0$ such that\n \\begin{equation*}\\label{strong1}\nf(x,s,\\xi)\\geq-k(|\\xi|^{p-1}\\lg(1+|\\xi|^q)+b(s))\\ \\hbox{for}\\ x\\in\\Omega,\\ \\hbox{all}\\ s\\geq 0\\ \\hbox{and all}\\ \\xi\\in \\rn,\\ \\hbox{with}\\ |\\xi|\\leq 1\\,.\n \\end{equation*}\n\\item [$(f_3)$] there exists $c >0$ such that \n$|f(x,s,\\xi)|\\leq c(1+|\\xi|^p\\lg(1+|\\xi|^q))\\ \\hbox{for all}\\ x\\in\\Omega,\\ s\\in \\r, \\ \\xi\\in\\rn$.\n\\end{itemize}\nFor the function $A'$ in problem \\eqref{problempq} condition \\eqref{dg} holds with $\\delta=p-1$, $g_0=p-1+q$ and \n$A(t)\\approx t^{p+q}$ for $t$ small.\nWe can apply Theorem \\ref{main} and Corollary \\ref{cor1} to obtain the existence of a positive solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$, and $u\\leq \\sigma$ in $\\Omega$.\n\\end{example}\nThe example above extends in two directions Theorem 6 of \\cite{FMMT}: it allows an higher growth for $f$ and permit also the choice $r=q$ in the lower bound for $f$.\n\n\\begin{example} \nConsider the problem \\begin{equation}\\label{Apq}\n\t\\begin{cases}\n\t\t- {\\rm div}\\left(\\frac{|\\nabla u|^{p-2}\\nabla u}{\\lg^q(1+|\\nabla u|)}\\right) =f(x,u,\\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\n\t\tu =0 & {\\rm on}\\,\\,\\,\n\t\t\\partial \\Omega \\,,\n\t\\end{cases}\n\\end{equation}\nwith $p>1$, $p-q-1>0$.\nLet $\\rho\\in L^{\\infty}(\\Omega)$ and $g_1,g_2:[0,+\\infty[\\to [0,+\\infty[$ be two unbounded, nondecreasing functions, such that $g_1(0)=g_2(0)=0$. Also, let $a_1,a_2: \\r \\to [0,+\\infty[$ be two locally essentially bounded functions and let $c_1,c_2>0$.\\\\\nAssume that $f:\\Omega\\times\\r\\times\\rn\\to\\r$ is a Carath\\'{e}odory function satisfying\n\\begin{eqnarray*}\n-c_1+g_1(|s|)-a_1(s)\\frac{|\\xi|^{p}}{\\lg^{q}(1+|\\xi|)}\\leq f(x,s,\\xi)\\leq -c_2+g_2(|s|)\\rho(x)+a_2(s)\\frac{|\\xi|^{p}}{\\lg^{q}(1+|\\xi|)}\\\n\\end{eqnarray*}\n$\\hbox{for}\\ (x,s,\\xi)\\in \\Omega \\times \\r\\times \\rn \\,.$ \\\\\nWe show that problem \\eqref{Apq} has a nontrivial solution $u\\leq 0$ in $\\Omega$.\\\\\nFor the function $A'$ in problem \\eqref{Apq} condition \\eqref{dg} holds with $\\delta =p-1-q$, $g_0=p-1$.\nIf $k:=\\inf\\{s>0\\,:\\, g_1(s)\\geq c_1\\}$, then $\\underline u\\equiv -k$ is a subsolution to \\eqref{Apq}, and $\\overline u\\equiv 0$ is a supersolution but not a solution to \\eqref{Apq}. Let $a=max \\{\\|a_1\\|_{L^\\infty ([-k,0])},\\ \\|a_2\\|_{L^\\infty ([-k,0])}\\}$, $\\sigma(x)=max\\{c_1, -c_2+g_2(k)\\rho(x)\\}$. Then \n\\begin{equation*}\n|f(x,s,\\xi)|\\leq \\sigma(x)|+a A'(|\\xi|)|\\xi|\\quad \\hbox{for}\\ x\\in \\Omega,\\ s\\in\\,[-k,0],\\ \\xi\\in\\rn\\,.\n\\end{equation*}\nBy Theorem \\ref{main}, problem \\eqref{Apq} has a nontrivial solution $u\\in [-k,0]$.\n\\end{example}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nIn the last few decades the detrended fluctuation analysis (DFA) method has become a\none of the standard techniques for the computation of the fractal scaling properties\n\\cite{PenBul1985}. It also serves as a great tool for the detection of long-range correlations \nnonstationary time series. Another very celebrated method is\nbased on the wavelet transformation of the signals \\cite{MuzBac1993,PhiLim2011} which together\nconstitutes widely accepted methods of the fluctuation analysis.\nIn 1941 Kolmogorov introduced multifractal formalism \\cite{Kolmogorov1941} in the context of \nanalysis of the turbulent data. It was extensively developed in the last\ndecade of the last century \\cite{Mandelbrot1999,StaMae1988} and still attracts considerable\nattention which includes such a distant fields like the biological systems \n\\cite{IvaAma1998}, financial markets\n\\cite{OswKwa2005,OhEom2012}, econophysics \\cite{Takayasu2004}, turbulence\n\\cite{Sreenivasan1991}, space data analysis \\cite{MacWaw2011}, physiology\n\\cite{StaAma1988} or medicine \\cite{MakRyn2011,Geras2013} to mention but a few.\n\\begin{figure}[ht!]\n\\centering\n\\subfigure[Signals obtained from muscle at complex work. The task was to tie the\nstandard surgery knots using the laparoscopic tools.]{\n\\includegraphics[width=.48\\linewidth]{fig1_1a}\n\\includegraphics[width=.48\\linewidth]{fig1_1b}\n\\label{fig11ab}}\n\\subfigure[Signals obtained from muscle in the relaxed state.]{\n\\includegraphics[width=.48\\linewidth]{fig1_2a}\n\\includegraphics[width=.48\\linewidth]{fig1_2b}\n\\label{fig12ab}}\n\\subfigure[Signals obtained from muscle at maximum contraction.]{\n\\includegraphics[width=.48\\linewidth]{fig1_3a}\n\\includegraphics[width=.48\\linewidth]{fig1_3b}\n\\label{fig13ab}}\n\\caption{Data collected from the channel 1 (in the vicinity of the trapezius ridge) for\namateur (left column, black) and professional (right column, red) at three\ndifferent states.}\n\\label{fig1}\n\\end{figure}\n\nThe electromyography (EMG) together with the electrocardiography, electroretinography and\nelectroencephalography are nowadays not only the diagnostic instruments in medicine but\nconstitute the proper and powerful scientific tool based on the electro-physiological\nactivities of our body \\cite{Aminoff1986}. In particular the surface EMG (sEMG) has\nbecome a promising apparatus for the non-invasive analysis of muscles\n\\cite{MerPar2004,BotCor2006}. The standard analysis of the sEMG signal covers usually three\naspects: the activation level of the muscle membrane potential, impact of the forces exerted\non the muscle and the degree of muscle fatigue. In this work on the contrary we will\nemploy the multifractal analysis of the electromyogram in order to extend the typical\nanalysis of the sEMG time series. The classical (mono-) fractal aspects \n\\cite{NaiKum2011,PhiLim2009,PhiPhu2012,XiaZhi2005} has also been \nextensively analysed for example in the context of force of contraction of different muscles \n\\cite{ArjKum2014}. \n\nElectromyography itself implies several challenges. It concerns an inappropriate location \nof the electrodes over the group of muscles \\cite{MesMer2009}, variation of the distance \nbetween the electrodes during the measurement and finally modification of the source position \nin relation to each electrodes. These in turn will influence the morphology of the series and\ncan manifest as a change of the shape, but most of all can affect the signal's amplitude.\nAnother important aspect is the cross talk, which is defined as the influence from the \nactivity of the neighbouring muscles. The last mentioned problem which can have quite a large\ncontribution to the systematic error is the inter--individual variability. This has a particular\nmeaning in the situation when we want to compare two or more individuals. This in \ngeneral is caused by the different tissue characteristics, which include\nthe thickness of adipose tissue, skin electrical resistance, sweating, hydration and \nthe thickness of epidermis.\n\nThe paper is organized as follows. In the next section we will describe the experimental\nsetup. Next we will report the details of the multifractal analysis. We will end with\nthe conclusions and summary.\n\n\\section{Set-up}\n\\label{setup}\nWe have performed the measurements on two individuals - a highly skilled one and\na complete beginner. They were given the complex\ntask which should be performed on the laparoscopic trainer. The task was to tie the\nstandard surgery knots using the laparoscopic tools. Usual recording time took around\n60 minutes and the signal was collected from four groups of muscless,\ntrapezius ridge (channels 1 and 5), deltoids (ch. 2 and 6), long palmar muscle and\nulnar wrist flexor (ch. 3 and 7) and abductor muscle of thumb and flexor brevis (ch. 4 and 8).\nChannels 1--4 and 5--8 were linked to the left and the right upper extremity, respectively. \nThe measurements were conducted with the portable 8 channel surface EMG recorder \n(OT Bioelettronica, Torino, IT) with the bipolar surface circular AgCl electrodes of size 15x15 mm. \nThe inter-electrode distance was set to 10 mm.\n\nThe system automatically records the maximum value of the signal from each of the active channels. \nThe Amplitude Rectified Value (ARV) measured in $\\mu$V was collected on MVC (maximum voluntary \ncontraction) recording mode. The ARV is mean value of the rectified EMG over a time interval \nT (125ms). The range of bandwidth was from 34 to 340 Hz. \nThe signal amplitudes were in the range of a few $\\mu$V to\neven more than 1000 $\\mu$V, see figure \\ref{fig11ab} for details. Additionally, we recorded the\nsignals from the relaxed muscles -- the recorded person was asked to sit down and reduce the \nmobility for a few minutes. This record consists of around 3500 data points (about seven minutes), \nsee figure \\ref{fig12ab}. The third measurement corresponds to the muscle in the strong contraction --\nboth subjects were supposed to keep 1 kg with arms bent and kept in parallel to the ground.\nAs it is quite hard to maintain such a posture for a long time, this data are the shortest \nand consists of only few hundreds measurement points (a little over a minute), see figure \\ref{fig13ab}.\n\nThere are visible differences in the electric potential generated by the muscle cells for all three different \nstates presented in figure \\ref{fig1}: the complex task (a, top row), the rest state (b, middle row) \nand the strong constant contraction (c, bottom row). \nThere is also a significant difference in the level of the muscle activity between a skilled person \nand an amateur. The amateur has about five times greater amplitude of the signal. The most probable\ncause for this lies in the different characteristics of the tissue, like the thickness of the skin \nand the adipose tissue which in turn leads to the different skin impedance for both cases.\n\n\\section{Multifractal analysis}\n\\label{mfdfa}\nIn the following we will present the typical Multifractal Detrended Fluctuation Analysis\n(MFDFA) as presented in \\cite{KanSte2002} and \\cite{Ihlen2012}. \nIn short, the analysis requires\nthe following stages. Suppose that we have time series with $N$ data points\n$\\{x_i\\}$, we perform than four consecutive steps\n\\begin{itemize}\n\\item[(i)] Calculate the profile $y_i$ as the cumulative sum from the data\nwith the subtracted mean\n\\begin{equation}\\label{eq1}\ny_i = \\sum_{k=1}^i [x_i - \\langle x \\rangle].\n\\end{equation}\n\n\\item[(ii)] The cumulative signal is split in $N_s$ equal non-overlapping segments of\nsize $s$. Here, for the width of the segments we use the power of two, $s = 2^r$,\nwhere $r = 4,\\dots,\\left \\lfloor \\log_2(N\/10) \\right \\rfloor$. Larger segment sizes \nwill result with rather weak statistics.\nTypically the length of the data will not be accordant\nwith the power of two and some data would have to be dropped from the analysis.\nTherefore the same procedure should be performed starting from the last index, and\nin turn the $2 N_s$ segments will be taken into account.\n\n\\item[(iii)] Calculate the local trend $y^m_{v,i}$ for $v^{th}$ segment by means of the\nleast--square fit of order $m$. Then determine the variance\n\\begin{equation} \nF^2(s,v) \\equiv {1 \\over s} \\sum_{i=1}^{s} \\left( y^m_{v,i} - y_{v,i} \\right)^2 \\label{fsdef} \n\\end{equation}\nfor each segment $v = 1, \\ldots, N_s$. The same procedure has to be \nrepeated in the reversed order (starting from the last index).\nNext determine the fluctuation function being\nthe $q^{th}$ statistical moment of the calculated variance.\n\\begin{eqnarray}\n\\label{eq2}\nF_q(s) &=& \\left(\\frac{1}{2N_s}\\sum_{v=1}^{2N_s} [F^{2}(s,v)] \\right)^\\frac{1}{q},\n\\quad q \\ne 0,\\\\\nF_0(s) &=& \\exp \\left\\{ {1 \\over 4 N_s}\n\\sum_{\\nu=1}^{2 N_s} \\ln \\left[F^2(s,\\nu)\\right] \\right\\},\n\\quad q = 0.\n\\end{eqnarray}\nThe above function needs to be calculated for all segment sizes $s=2^r$.\nWe have exploited several different orders of the fitted polynomials and end up with no\nstatistical difference between the results. Here we will present the analysis with\nthe quadratic fit.\n\n\\item[(iv)] In the last step the determination of the scaling\nlaw of the fluctuation function (\\ref{eq2}) is performed by means of the log--log\nplots of $F_q(s)$ versus segment sizes $s$ for all values of $q$. The function\n$F_q(s) \\sim s^{h(q)}$ is naturally smaller for the smaller fluctuations, which\nresults in the increasing function with the increasing segment size.\nFrom the calculated Hurst exponent $h(q)$ we are able to determine several quantifiers.\nFirstly, we work out the mass exponent using the formula\n\\begin{equation}\\label{mass}\n\\tau(q) = q h(q) - 1.\n\\end{equation}\nSecondly we can obtain the singularity exponent $\\alpha(q)$ by applying the Legendre\ntransform. The last quantifier and the main result of the MFDFA method is the singularity\nspectrum, given by\n\\begin{equation}\\label{spectrum}\nD[\\alpha(q)] = q \\alpha(q) - \\tau(q).\n\\end{equation}\n\\end{itemize}\nThe detailed information on how to read the singularity spectrum can be found in\nthe literature \\cite{MakRyn2011,Kantelhardt2008,MakFul2010}.\nMultifractality is an indication of the complex dynamics where the single exponent (like\nthe fractal dimension) will not be enough to describe the phenomenon. In the case\nwhen the data exhibits not just one individual exponent the continuous spectrum of exponents\nshould be taken into account.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.98\\linewidth]{fig2a}\n\\includegraphics[width=.98\\linewidth]{fig2aa}\\\\\n\\centering\n\\includegraphics[width=.985\\linewidth]{fig2b}\n\\includegraphics[width=.985\\linewidth]{fig2bb}\n\\caption{(color online) The multifractal spectrum for channel 1 (in the vicinity of the trapezius ridge) for\nprofessional (a,b) and amateur (c,d). On each panel three curves depict the multifractal\nspectrum corresponding to the three states of the performance of the the muscle group:\nexecuting complex task (red), at rest (green) and at the maximum contraction (blue).\nThe panels (a) and (c) present the spectra for the original data, while the panels\n(c) and (d) show the spectra for the shuffled data.\nAll figures have the same width for the sake of comparison.}\n\\label{fig2}\n\\end{figure}\n\n\\section{Multifractal spectra at complex work}\n\nThe central result of this work is presented in figure \\ref{fig2}. The plot presents the \nmultifractal spectrum. This relation is defined as the singularity spectrum \nas the function of the singularity exponent $D = D(\\alpha(q))$. \nThe multifractal spectrum describes how often the irregularity of certain degree occurs in the signal. \n$D(\\alpha)$ represent $q$-order singularity dimension and \n$\\alpha(q)$ stands for the $q$-order singularity exponent (\\ref{spectrum}). \nIt illustrates the variability in \nthe fractal structure of the time series. The monofractal time series has dense mf--spectrum \naround the single point $(H,1)$, where $H$ is the global Hurst exponent \n\\cite{Hur1951}.\nTo investigate the scaling properties of our data the analysis of the the position of the \nmaximum value of the mf--spectrum, the global Hurst exponent itself, the half--width and the \nextremes of the mf--spectrum have to be taken into consideration \\cite{MakRyn2011}. \nThe whole analysis was performed for all 8 channels. We have calculated the $q$-th order fluctuation\nfunction $F_q(s)$ for 100 values of the invert power $q \\in [-5,5]$ with the step equal to $0.1$\nas suggested in the literature \\cite{Ihlen2012}.\n\nThere are two general sources of multifractality which can affect the shape of the mf-spectrum:\n(i) one is due to the broad probability density function which lies behind the data (or its\nfluctuations); (ii) second is driven by the different behaviour of the (auto)correlation function for\nlarge and small fluctuations; (iii) both situations simultaneously.\nSimple data shuffling can test the possible source of multifractality. In the case (i)\nshuffling will not change the mf-spectrum, for (ii) will destroy the effect completely\nas the shuffling will destroyed the possible correlations; in the last case (iii) the spectrum\nwill differ from the original one-shuffled series will exhibit somehow weaker multifractality. \nFor the all analysed cases the correlation for large and small fluctuations seem to be\nthe main factor which causes the strong multifractality \\cite{Horvatic2011}\n-- please compare pairwise panels (a) -- (b) and (c) -- (d) in the figure \\ref{fig2}. \nFor the presented analysis this is the usual effect for all of the spectra except\nfor the working state for the professional and the maximal contraction state for the amateur.\nThe Hurst $H$ exponent for all of the cases behaves however in a similar way for the \nshuffled data. In the working state after shuffling the estimated values are very close\nto $0.5$ which suggest that in this very case we deal with the white uncorrelated white noise. \nThis means that the correlations are the only source of multifractality (case ii).\nThe other states show a little bit higher (contraction) and lower (rest) values of $H$ than\nin the work state, which may suggest some sort of the monofractal behaviour (again after\nshuffling) -- see panels (b) and (d).\n\nThe shape and the width of the multifractal spectrum provide the information about the local changes \nof the Hurst exponent. We can see that the value of the spectral width is different for different states \nof muscular tension for both individuals. \nA large difference between periods when small and large fluctuations takes place increases in turn the width of the spectrum.\nThe analysis of the signal where neither weak nor strong local fluctuations dominate will result in the symmetric \nshape of the mf-spectrum. This aspect is visible for the sequence of the nonprofessional performing full task \n-- see the red curve in figure \\ref{fig2}(c) for details. On the contrary the study of the corresponding signal \nbut for the professional exhibits the dominance of the low local fluctuations. This feature is also clearly visible \nin the raw signal, see figure \\ref{fig11ab} on the r.h.s. This situation is a manifestation of the influence\nof the training for the resulting electrical potential generated by group of the examined muscle cells.\nThe trained person will use his locomotor system very effectively, allowing only simple and necessary\nmovements. Therefore the resultant spectrum will in general express close similarity to the spectrum\nfor activity system at rest -- compare green curve in figure \\ref{fig2}(a). Both just described states\nwill show the dominance of the low fluctuations, which is an indicator of the weak excitations in muscle\ncell membranes. \nOn the contrary the spectrum for the untrained person at actual work will show rather broad and symmetric\nmultifractal spectrum. The rare events are as distant from the maximum value as low fluctuations and as \nresult neither weak nor strong fluctuations dominate in the signal. Even at the rest state the\nspectrum is rather wide and symmetric, which indicate constant excitations from the electrical \nactivity of the muscle cell membrane -- i.e. even at rest the amateur will unnecessarily exploit\nthe energy.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.9\\linewidth]{fig3}\n\\caption{$q$-order Hurst exponent $h(q)$ for the series with the dominance of the \nsmall (green, solid) and large (blue, dashed) local fluctuation for professional.}\n\\label{fig3}\n\\end{figure}\n\nThe time series, or rather its fluctuations, exhibit similarity for both individuals only in the state of \nthe full contraction. The natural\ntendency to strong excitation of the cell membranes will result in the predominance of the high \nfluctuations. This is visible as the high variability in the signal (figure \\ref{fig13ab}), \nas a mf--spectrum with the left truncation or as the leveling of the $q$-order Hurst exponents positive $q$'s,\nsee blue dashed curve in figure \\ref{fig3} for details.\n\nThe difference of the multifractal structure in the local fluctuation with small and large variation is visible \nalso on $h(q)$ and $\\alpha(q)$ dependence, which respectively presents Hurst and singularity exponent as a function of $q$.\nFor the $h(q)$ dependence see figure \\ref{fig3} for details. The $\\alpha(q)$ is calculated as the tangent slope of the mass exponent \n$\\tau(q)$ (\\ref{mass}). For monofractal signals (or similarly for the white noise) \nthe $h(q)$ and $\\alpha(q)$ would be independent of $q$. On the other hand, for the time series which exhibit \nmultifractality, the discussed quantifiers will typically be monotonically decreasing functions.\nNontheless, for certain behaviors of the series, for instance if only large (small) local fluctuations\nprevail the $h(q)$ dependence will show plateau for positive (negative) values of $q$. This feature\ncause the truncation of the left (right) branch of the corresponding mf--spectrum shown in \nthe figure \\ref{fig2}.\n\n\\begin{table}[htbp]\n\\caption{The parameters of the mf--spectrum, calculated for three states of muscle activity respectively for the raw and integrated data.}\n\\begin{center}\n\\begin{tabular}[H]{|c|c|c|c|} \\hline\nObject & maximal contraction & task & relaxation \\\\\n\\hline\n\\multicolumn{4}{|c|}{Hurst exponent $H^{int} =h^{int}(2)$}\\\\\n\\hline\nProfessional & $0.714$ & $0.31$ & $0.664$ \\\\\nAmateur & $1.01$ & $0.67$ & $0.772$\\\\ \\hline\n\\multicolumn{4}{|c|}{Hurst exponent $H = h(2)$}\\\\ \\hline\nProfessional & $0.062$ & $-0.147$ & $0.031$ \\\\\nAmateur & $0.094$ & $-0.017$ & $0.025$\\\\ \\hline\n\\multicolumn{4}{|c|}{$h_{max}^{int} =h^{int}(0)$ }\\\\ \\hline\nProfessional & $0.759$ & $1.06$ &$0.755$ \\\\\nAmateur & $0.981$ & $1.14$ & $1.05$\\\\ \\hline\n\\multicolumn{4}{|c|}{$h_{max} = h(0)$ }\\\\ \\hline\nProfessional & $0.111$ & $0.166$ &$0.027$ \\\\\nAmateur & $0.149$ & $0.239$ & $0.175$\\\\ \\hline\n\\multicolumn{4}{|c|}{$\\Delta_{1\/2}^{int}$ }\\\\ \\hline\nProfessional & $0.045$ & $0.756$ &$0.092$ \\\\\nAmateur & $0.031$ & $0.467$ & $0.282$\\\\ \\hline\n\\multicolumn{4}{|c|}{$\\Delta_{1\/2}$ }\\\\ \\hline\nProfessional & $0.0485$ & $0.312$ &$0.0044$ \\\\\nAmateur & $0.0556$ & $0.256$ & $0.150$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab1}\n\\end{table}\n\nThere are several parameters we can use to effectively describe mf--spectrum and, consequently, the signal which lies behind it.\nIn the table \\ref{tab1} we have collected the typical quantifiers -- the values of the typical Hurst exponent $H=h(2)$, \nsingularity exponent located at the maximum of the spectrum $h_{max}$, and\nspectrum half--width $\\Delta_{1\/2}$ defined as the absolute value of the difference between Hurst exponent and $h_{max}$, \nall calculated for both row and integrated data. \nThe integrated data are calculated as cumulative sums of the original (raw) signal. \nIn the case of the monofractal signal the spectrum of the integrated signal is usually the same as for the raw \none, but shifted by 1 to the right. Lower values of this difference suggest that the obtained \nmf--spectra have other than monofractal scaling.\nAgain, for almost all factors there is a significant difference between amateur \nand professional for two states -- at the relaxation and during the assumed task. For maximum contraction the only factor\nwhich distinguishes the professional and amateur is the Hurst exponent calculated for integrated data. \n\n\\section{Summary and conclusions}\nThe comparison of the kinesiological electromyographic signal between a professional (highly trained) and an amateur \nwas presented for the three typical states of work of the human musculo--skeletal system. Based on the multifractal \ndetrended analysis we have shown the differences and similarities for the data fluctuations. The main message\nwhich can be drawn from the analysis is that the locomotor aparatus for the trained person would require\nmuch less energy to perform tasks as it's work would produce much smaller local fluctuations. The multifractal spectrum\nwould in turn look more similar to the one at the rest state. On the other hand there is much less chance to distinguish \nthe depth of training between two persons if one would look at the sEMG data assembled at the strong muscle tension. \nThe muscle cell membranes will in this case tend to react with much higher voltage of the electrical potential as these \ncells will be much stronger activated neurologically.\n\nIn conclusions we would like to suggest the possible application for automatic verification of abilities for \nperforming complex tasks based on the fluctuation analysis. If the person's multifractal spectrum would be wide with\nno truncation on the right side (at higher singularity exponents $\\alpha(q)$) than the low and high local fluctuation\nwould be equally probable. This means that the examined person still operates with too much stress and struggles with\nthe task, therefore some more training would still be needed.\n\n\\section*{Acknowledgements}\nAuthors would like to thank Danuta Makowiec and Jan \u017bebrowski for valuable discussions. \nThis work was partially supported by the Polish Ministry of Science and Higher Education\n(Grant K\/ZDS\/003962).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThis sequel of our paper on the Onsager-Machlup theory \\cite{Jurisch1} is inspired by a serious mistake we noticed in a paper by Taniguchi and Cohen \\cite{Taniguchi}. As it turned out, this mistake has already been made by Onsager and Machlup in their seminal paper \\cite{Machlup}, and has been applied ever since. In the attempt to include inertial effects into the Langevin-equation Onsager and Machlup proposed a second order variational-principle based on a generalization of the Onsager-Machlup Lagrangian. This but violates Ostrogradsky's theorem \\cite{Ostrogradsky}. In the following we elucidate the source of the mistake and suggest a solution of the problem.\n\nThe equation of motion of interest is given by\n\\begin{equation}\nm\\,\\ddot{q}(t)\\,=\\,F[q(t)]\\,+\\,\\mu_{1}\\,\\dot{q}(t)\\,+\\,\\sqrt{\\sigma_{2}}\\,\\eta(t)\\quad,\n\\label{Introduction1}\\end{equation}\nwhere $F[q(t)]$ is a conservative force-field, the second term describes Stokes-friction, and the last term is a Gaussian stochastic process. Thus, it is assumed that already the acceleration is distorted by noise. In \\cite{Taniguchi} as in \\cite{Machlup} friction has been introduced by purely phenomenological arguments.\n\nBy the assumption that the motion takes place in the over-damped regime, $\\ddot{q}(t)\\,=\\,0$, Eq. (\\ref{Introduction1}) can be cast into\n\\begin{equation}\n\\dot{q}(t)\\,=\\,-\\,\\mu_{1}^{-1}\\,\\left(F[q(t)]\\,+\\,\\sqrt{\\sigma_{2}}\\,\\eta(t)\\right)\\quad,\n\\label{Introduction2}\\end{equation}\nwhich may be identified as a Langevin-equation, from which the corresponding Onsager-Machlup Lagrangian can be derived. We remark that such a reinterpretation already becomes impossible when Newtonian friction $\\sim\\,\\dot{q}^{2}$ is present, see \\cite{Jurisch1}.\n\nIn a next step, following Onsager and Machlup \\cite{Machlup}, the acceleration is included as an additional drift-field\n\\begin{equation}\n\\dot{q}(t)\\,=\\,\\frac{m}{\\mu_{1}}\\,\\ddot{q}(t)\\,-\\,\\mu_{1}^{-1}\\,\\left(F[q(t)]\\,+\\,\\sqrt{\\sigma_{2}}\\,\\eta(t)\\right)\\quad,\n\\label{Introduction3}\\end{equation}\nwhich is just a rearrangement of Eq. (\\ref{Introduction1}). Along with \\cite{Machlup} Eq. (\\ref{Introduction3}) is still interpreted as a Langevin-equation, which is a highly doubtful interpretation. Furthermore, again following Onsager and Machlup \\cite{Machlup}, the Onsager-Machlup Lagrangian is set up by\n\\begin{equation}\n\\mathcal{L}(\\ddot{q},\\,\\dot{q},\\,q)\\,=\\,\\frac{\\mu_{1}^{2}}{2\\,\\sigma_{2}}\\,\\left(\\dot{q}(t)\\,-\\,\\frac{m}{\\mu_{1}}\\,\\ddot{q}(t)\\,+\\,\\mu_{1}^{-1}\\,F[q(t)]\\right)^{2}\\quad.\n\\label{Introduction4}\\end{equation}\nHowever, this Lagrangian is completely meaningless. The fundamental, but unfortunately scarcely known theorem of Ostrogradsky \\cite{Ostrogradsky} proves that Lagrangians, which depend on higher than first order derivatives in a non-degenerate way do not describe any meaningful dynamics. Non-degeneracy here means that $\\partial\\mathcal{L}\/\\partial\\ddot{q}$ is a function of $\\ddot{q}$, to which the Lagrangian Eq. (\\ref{Introduction4}) applies for. Ostrogradsky's theorem holds in general, it is one of the two hard criteria every Lagrangian, which claims physical content must obey. The second hard criterion are the Helmholtz-conditions \\cite{Helmholtz}, see also Nigam and Banerjee \\cite{Nigam}, Nucci and Leach \\cite{Nucci}, and references therein. A violation of one of these criteria is sufficient to render a Lagrangian meaningless. Consequently, the Lagrangian under consideration here does not exist. This clarifies that the interpretation of Eq. (\\ref{Introduction3}) as a Langevin-equation is ineligible. Furthermore, this rules out the second order variational-principle given by Onsager and Machlup in \\cite{Machlup}. As a consequence, inertial effects cannot be included in a standard way. We have not checked if the Lagrangian Eq. (\\ref{Introduction4}) also violates the Helmholtz-conditions, since the violation of Ostrogradsky's theorem is sufficient to rule this ansatz out.\n\nSystems described by Lagrangians with higher than first order derivatives suffer from the Ostrogradsky-instability. In short, the phase-space of such systems either explodes or decays almost instantly. This comes from pathological excitations and states of unbounded positive and negative energy. We emphasize that this must not be confused with the Dirac-theory. The Dirac-Lagrangian is sound with respect to it's derivatives, the spinors are relativistic fields and the unbounded scale of energy has a precise physical meaning.\n\nTo illustrate the origin of the Ostrogradsky-instability, we calculate the Hamiltonian of the Lagrangian Eq. (\\ref{Introduction4}). The canonic coordinates are given by\n\\begin{equation}\nQ_{1}\\,=\\,q,\\quad Q_{2}\\,=\\,\\dot{q},\\quad P_{1}\\,=\\,\\frac{\\partial\\,\\mathcal{L}}{\\partial\\,\\dot{q}}\\,-\\,\\frac{d}{d\\,t}\\,\\frac{\\partial\\,\\mathcal{L}}{\\partial\\,\\ddot{q}},\\quad P_{2}\\,=\\,\\frac{\\partial\\,\\mathcal{L}}{\\partial\\,\\ddot{q}}\\quad.\n\\label{Introduction5}\\end{equation}\nIt is easy to see that this choice is the natural generalization of the first order case. For our present system, this gives\n\\begin{equation}\nP_{2}\\,=\\,\\frac{m}{\\sigma_{2}}\\,\\left(m\\,\\ddot{q}(t)\\,-\\,\\mu_{1}\\,\\dot{q}(t)\\,-\\,F[q(t)]\\right)\\,\\rightarrow\\,\\ddot{q}(t)\\,=\\,\\mathcal{Q}(P_{2},\\,Q_{1},\\,Q_{2})\\,=\\,\\frac{\\sigma_{2}}{m^{2}}\\,P_{2}\\,+\\,\\frac{\\mu_{1}}{m}\\,Q_{2}\\,+\\,m^{-1}\\,F[Q_{1}]\\quad.\n\\end{equation}\nThe Hamiltonian follows on the standard route\n\\begin{eqnarray}\n\\mathcal{H}(P_{2},\\,P_{1},\\,Q_{2},\\,Q_{1})&=&P_{1}\\,Q_{2}\\,+\\,P_{2}\\,\\mathcal{Q}(P_{2},\\,Q_{1},\\,Q_{2})\\,-\\,\\mathcal{L}\\left(\\mathcal{Q}(P_{2},\\,Q_{1},\\,Q_{2}),\\,Q_{2},\\,Q_{1}\\right)\\nonumber\\\\\n&=&P_{1}\\,Q_{2}\\,+\\,\\frac{\\sigma_{2}}{2\\,m^{2}}\\,P_{2}^{2}\\,+\\,P_{2}\\,\\left(\\frac{\\mu_{1}}{m}\\,Q_{2}\\,+\\,m^{-1}\\,F[Q_{1}]\\right)\\quad.\n\\label{Introduction6}\\end{eqnarray}\nThe term linear in $P_{2}$ goes without problems, it can be removed by quadratic completion and acts like a gyroscopic potential. The term linear in $P_{1}$ but is non-trivial, since it does not resolve by Legendre-transform. The momentum $P_{1}$ is not subject to any constraints, it is completely unbounded and can take any value. By coupling to $Q_{2}$ this lack of boundary tends to disrupt the whole system. It is now easy to see that still higher order derivatives in the Lagrangian make things only worse. This is the origin of the Ostrogradsky-instability. By calculating the canonic equations and going back to the Euler-Lagrange equations one can convince oneself that Ostrogradsky's choice for the canonic coordinates indeed is correct.\n\nFor the whole impact of Ostrogradsky's theorem, please see e.g. Motomashi \\cite{Motomashi}, especially Woodard \\cite{Woodard}, and references therein. For completeness we add that Eq. (\\ref{Introduction4}) must not be confused with the Gaussian action-principle, see e.g. Lanczos \\cite{Lanczos}. The Gaussian action is no Lagrangian, but an application of the method of least squares.\n\\newline\n\n\\emph{Short historical note}: Because Ostrogradsky's theorem is unbeknownst to a broader audience, we think some words about it's history are in order. Ostrogradsky discovered his theorem in 1850, but it got almost unnoticed. At this time the understanding of energy and stability was still in it's childhood, and nobody could grasp what Ostrogradsky's theorem really says. Obviously even Landau and Lifshitz have not known about Ostrogradsky's theorem, nothing else could explain why it is not to be found in their textbook about mechanics. Thus, there is no wonder about that also Onsager and Machlup have not been aware of Ostrogradsky's theorem.\n\nOstrogradsky's theorem was rediscovered in particle physics, general relativity and cosmology. As far as we know, the first who proposed higher order Lagrangians were Pais and Uhlenbeck \\cite{Pais}, ironically in 1950. Skeptics asked if this is allowed. On this route Ostrogradsky's theorem was rediscovered and is a nuisance in particle physics and cosmology ever since.\n\n\n\\section{Suggested solution of the problem}\nIn this section we shall elucidate our suggestion for a solution of the problem posed by Eqs. (\\ref{Introduction1}). Since the Newtonian equation of motion with noise is not tractable by the Onsager-Machlup theory, we need two first-order equations with independent variables to introduce Langevin-equations. Thus, we employ the canonical formalism.\n\nThe equation of motion with Stokes-friction\n\\begin{equation}\n\\ddot{q}(t)\\,=\\,-\\,\\frac{1}{m}\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,+\\,\\mu_{1}\\,\\dot{q}(t)\\quad,\n\\label{Solution7}\\end{equation}\nis a consequence of the Onsager-Machlup Lagrangian, see \\cite{Jurisch1},\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,b(t)^{-2}\\,\\left(\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{I}}[q(t)]\\right)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\left(\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{I}}[q(t)]\\right)\\quad.\n\\label{Solution8}\\end{equation}\nThe function $b(t)=\\exp[\\mu_{1}\/2\\,t]$ is a Helmholtz-factor or, likewise, a Jacobi-multiplier. The Hamiltonian thus reads\n\\begin{equation}\n\\mathcal{H}(p,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p^{2}(t)}{2\\,m}\\,+\\,\\exp[-\\,\\mu_{1}\\,t]\\,U_{\\rm{I}}[q(t)]\\quad,\n\\label{Solution9}\\end{equation}\nwhich also is known as the Caldirola-Kanai Hamiltonian \\cite{Caldirola, Kanai}.\nThe canonic equations follow by\n\\begin{eqnarray}\ndq(t)&=&\\partial_{p}\\mathcal{H}(p,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p(t)}{m}\\,dt\\quad,\\nonumber\\\\\ndp(t)&=&-\\,\\partial_{q}\\mathcal{H}(p,\\,q)\\,=\\,-\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,dt\\,+\\,\\sqrt{\\sigma_{2}}\\,dW(t)\\quad,\n\\label{Solution10}\\end{eqnarray}\nwhere we already have introduced the stochastic momentum-process with variance $\\sigma_{2}$. The Onsager-Machlup Lagrangian of the momentum-process then is\n\\begin{equation}\n\\mathcal{L}(\\dot{p},\\,q)\\,=\\,\\frac{1}{2\\,\\sigma_{2}}\\,\\left(\\dot{p}(t)\\,+\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\right)^{2}\\quad.\n\\label{Solution11}\\end{equation}\nThe new Hamiltonian of the system can now safely be written by\n\\begin{equation}\n\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p(t)^{2}}{2\\,m}\\,+\\,\\frac{\\sigma_{2}}{2}\\,P(t)^{2}\\,-\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,P(t)\\quad,\n\\label{Solution13}\\end{equation}\nwhere the second and third terms follow by a Legendre-transform of the Lagrangian Eq. (\\ref{Solution11}. The unified canonic equations finally yield\n\\begin{eqnarray}\n\\dot{q}(t)&=&\\partial_{p}\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p(t)}{m}\\quad,\\nonumber\\\\\n\\dot{p}(t)&=&\\partial_{P}\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\sigma_{2}\\,P(t)\\,-\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\quad,\\nonumber\\\\\n\\dot{P}(t)&=&-\\,\\partial_{q}\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}^{2}U_{\\rm{I}}[q(t)]\\,P(t)\\quad.\n\\label{Solution14}\\end{eqnarray}\nFor $\\sigma_{2}=0$ the canonic equations decouple, and we recover Eq. (\\ref{Solution10}), as it must. As a last step, we eliminate the momenta and calculate the Newtonian equation of motion for the the most probable path $q(t)$. The calculation is easily done by differentiating the first line in Eq. (\\ref{Solution14}) and insert the other equations. This yields\n\\begin{equation}\n\\ddot{q}(t)\\,=\\,-\\,\\frac{1}{m}\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,+\\,\\mu_{1}\\,\\dot{q}(t)\\,+\\,\\frac{\\sigma_{2}}{m}\\,\\exp[\\mu_{1}\\,t]\\,\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\exp[-\\,\\mu_{1}\\,\\tau]\\,\\partial_{q}^{2}U_{\\rm{I}}[q(\\tau)]\\right]\\quad.\n\\label{Solution15}\\end{equation}\nThis Newtonian equation of motion elucidates that the noisy distortion influences the trajectory of the most probable path as additional force, which but depends on the interaction $U_{\\rm{I}}[q(t)]$. Note that this force still remains finite for $\\partial_{q}^{2}U_{\\rm{I}}[q(\\tau)]\\,=\\,0$. Furthermore, the additional force grows or decays as time evoles. If the force grows, this means that the stochastic noise tends to destroy the action of the potential and the friction, while a decay leaves the action of the potential and the friction sound or even enhances it.\n\nWe can integrate Eq. (\\ref{Solution15}) and obtain the Onsager-Machlup Lagrangian for the combined system, reading\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\left(\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{I}}[q(t)]\\right)\\,+\\,\\sigma_{2}\\,q(t)\\,\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\exp[-\\,\\mu_{1}\\,\\tau]\\,\\partial_{q}^{2}U_{\\rm{I}}[q(\\tau)]\\right]\\quad.\n\\label{Solution16}\\end{equation}\nIn principal, it is always possible to derive the Newtonian equation of motion from the canonic equations, but a reconstruction of the primal Lagrangian of the system may not always be possible. We chose the word \\emph{primal} here to emphasize that this Lagrangian combines the environmental effects all in one, and thus is the Lagrangian to start with for all further examinations. We remark that the time-dependent exponential, last term in Eq. (\\ref{Solution16}), shows the general structure of a Helmholtz-factor.\n\n\n\\section{Notes on the hierarchy of immersions and actions of environments}\nFollowing our classification of interactions in \\cite{Jurisch1}, we see that the additional potential in Eq. (\\ref{Solution16}) acts like an external potential $U_{\\rm{E}}[q(t),\\,t]$. The first term in the Lagrangian Eq. (\\ref{Solution16}) describes an ideal harmonic oscillator, that is immersed in environment (I), which creates Stokes-friction. The stochastic process disturbs the environment (I) from the outside, and consequently may be understood as environment (II), that acts upon environment (I). This leads to a hierarchy, that can be written by sets\n\\begin{equation}\n\\{\\rm{harmonic\\,oscillator}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{environment (II)}\\quad.\n\\label{Solution17}\\end{equation}\nWe chose to write $\\subset$ instead of $\\in$, since we understand the ideal system not as to be an element of environment (I), but as a system immersed into environment (I). Thus, it is an additional subset of environment (I), which can be taken out again without reducing environment (I). If the ideal system would be an element, then a take-out would reduce environment (I).\n\nSome more remarks have to be made about the system described in \\cite{Taniguchi}. There, the potential $U[q(t)]$ is used as an external harmonic potential, that traps Brownian particles. To our regards, the correct initial ansatz for this system must be written by\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\quad,\n\\label{Solution18}\\end{equation}\nsince the harmonic trap acts from the outside. The Lagrangian Eq. (\\ref{Solution18}) generates a Newtonian equation of motion with Stokes-friction. The system of Brownian particles itself is thus solely described by the kinetic energy plus the noise, while the harmonic trap surrounds this system, and consequently acts from the outside. According to our results in \\cite{Jurisch1}, this modifies the equation of motion, and thus also leads to a different primal Lagrangian. In this interpretation, the hierarchy of systems as described in \\cite{Taniguchi} is given by\n\\begin{equation}\n\\left\\{\\{\\rm{free\\,particles}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{environment (II)}\\right\\}\\,\\leftarrow\\,\\rm{harmonic\\,trap}\\quad.\n\\label{Solution19}\\end{equation}\nThis also means that the external harmonic potential $U_{\\rm{E}}[q(t)]$ must only be added to the primal Onsager-Machlup Lagrangian as an external potential. This hierarchy then leads to\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\left\\{\\left\\{\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\right\\}\\,+\\,\\sigma_{2}\\,q(t)\\right\\}\\,-\\,U_{\\rm{E}}[q(t)]\\quad.\n\\label{Solution20}\\end{equation}\n\nIf, and only if environment (II) shall also disturb the external harmonic trap, then the initial Lagrangian can be taken by\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{E}}[q(t)]\\quad.\n\\label{Solution21}\\end{equation}\nThe set-relation in this case yields\n\\begin{equation}\n\\left\\{\\{\\rm{free\\,particles}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{harmonic\\,trap}\\right\\}\\,\\leftarrow\\,\\rm{environment (II)}\\quad,\n\\label{Solution22}\\end{equation}\nand the primal Lagrangian follows by\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\left\\{\\left\\{\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\right\\}\\,-\\,U_{\\rm{E}}[q(t)]\\right\\}\\,+\\,\\sigma_{2}\\,q(t)\\,\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\,\\partial_{q}^{2}U_{\\rm{E}}[q(\\tau)]\\right]\\quad.\n\\label{Solution23}\\end{equation}\n\nThe primal Lagrangians Eqs. (\\ref{Solution16}, \\ref{Solution20}, \\ref{Solution23}) elucidate how sensitive the dynamics of a system depends on the hierarchy of immersions and actions of several environments. Just for completeness, we illustrate the case, where the system Eq. (\\ref{Solution23}) is immersed into an environment (III), described by the function $b[q(t),\\,t]$. The set-relation then reads\n\\begin{equation}\n\\left\\{\\left\\{\\{\\rm{free\\,particles}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{harmonic\\,trap}\\right\\}\\,\\leftarrow\\,\\rm{environment (II)}\\right\\}\\,\\subset\\,\\rm{environment\\,(III)}\\quad,\n\\label{Solution24}\\end{equation}\nand the primal Lagrangian becomes\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)=b^{-2}[q(t),\\,t]\\left(\\left\\{\\left\\{\\left\\{\\exp[-\\,\\mu_{1}\\,t]\\frac{m}{2}\\,\\dot{q}(t)^{2}\\right\\}-U_{\\rm{E}}[q(t)]\\right\\}+\\sigma_{2}\\,q(t)\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\partial_{q}^{2}U_{\\rm{E}}[q(\\tau)]\\right]\\right\\}\\right)-\\mathcal{V}[q(t),\\,t]\\,,\n\\label{Solution25}\\end{equation}\nwhere the potential $\\mathcal{V}[q(t),\\,t]$ subsumes all additional potentials, that depend on $b[q(t),\\,t]$, see \\cite{Jurisch1}.\n\nBy our discussion we understand that immersions and actions of environments mostly can be implemented directly into the Lagrangian without complications. Only the action of an additional stochastic process on the Newtonian equation of motion requires to go through the formalism we have developed above.\n\n\n\\section{Remark}\nAs a last point, we shall remark that in exotic field-theory methods have been developed, which, under certain circumstances, can remove the Ostrogradsky-instability from the Hamiltonian, see e.g. Chen et. al. \\cite{Chen}. The method of choice is the Dirac-constraint \\cite{Dirac}. If the Ostrogradsky-instability is removable, this leads to a reduction of the phase-space, where the system then evolves on a restricted hyper-surface only. Chen et. al. \\cite{Chen} successfully probe the method of Dirac-constraints on a variety of elementary problems, including the Pais-Uhlenbeck oscillator \\cite{Pais}. However, if a reduction of the phase-space is possible, then this comes at the cost that the resulting Lagrangian is completely remote from the initial problem. Furthermore, in order to be successful, artificial auxiliary terms must be added to the initial ill-defined Lagrangian by Lagrange-multipliers, which are constructed in a way that the Dirac-constraints can act as wished. From a sane bottom-up view, however, such approaches are more than questionable. To enforce something by tinkering, which is just not there ignores a distinct caveat.\n\nOur suggestions is built upon the contrary, in that we enlarge the dimension of the phase-space. This allows us not only to resolve the second order derivative of the Newtonian equation of motion, but also the inclusion of the stochastic process by applying the Onsager-Machlup theory straight.\n\n\n\\section{Conclusion}\nBy using Ostrogradsky's theorem, we ruled out the possibility to reinterpret Newton's equation of motion with Stokes-friction and stochastic noise as a Langevin-equation, since the corresponding Onsager-Machlup Lagrangian contains second order derivatives in a non-degenerate way. Such a Lagrangian is ill-defined and there is no physics in it. This shows that Newton's equation of motion and the Langevin-equation are something completely different. \n\nFurthermore, we suggested a method of how to treat a system where Newton's equation of motion is distorted by noise. Our ansatz is built upon the canonical formalism, and the results we achieved seem to be sound. We were able to derive an Onsager-Machlup Lagrangian, that describes the effects of the stochastic distortion by a force, which additionally influences of the most probable path a particle takes through a disordered environment.\n\nLast, we gave arguments of how to understand the hierarchy of environments, that prove useful for the analysis of systems of higher complexity. This has shown that the structure of the primal Lagrangian sensitively depends on the hierarchy of immersions and actions of the environments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{A Deep Unsolved Problem}\n\nThis article is proposing a simplistic model to account for the behavior of the viscosity of liquids as a function of temperature. The best illustration of this behavior is the data from Angell's review paper \\cite{Ang95}, that can be found below in Fig.~\\ref{visc.fig-ang}. It shows on one graph the behavior of the logarithm of the viscosity $\\eta$ as a function of inverse temperature, in order to reveal whether or not an Arrhenius law is valid $\\eta \\sim e^{W\/\\kB T}$. The excitation energy $W$ is represented by the slope of the curve in such coordinate axis. The graph actually shows that the Arrhenius law is valid for a category of materials called {\\em ``strong liquids''}. For most other materials, called {\\em ``fragile''}, the slope changes as the temperature approach the glass transition.\n\n\\begin{figure}[ht]\n \\centering\n\\includegraphics[width=9cm]{angell.jpg}\n\\caption{\\label{visc.fig-ang} Viscosity of the liquid phase of various glassy materials. The logarithm of the viscosity is maps as a function of the normalized inverse temperature, to exhibit Arrhenius law. The normalization is provided by the glass transition temperature $T_g$. (Taken from \\cite{Ang95}).}\n\\end{figure}\n\nThe glassy and liquid states of a material share many features. It is understood that the difference between them cannot be account for by using equilibrium Thermodynamics: the glass transition is not a phase transition, it is a dynamical process in which the viscosity becomes so large that the typical time scales involved in the material become macroscopic. The viscosity changes by 15 orders of magnitude over a very small temperature range near the transition. The reason why viscosity changes so rapidly, while the atomic structure does not show signs of this change, has been a mystery for a long time. It has been considered as one of the deepest unsolved problems in science \\cite{An95,La07} by the most prominent experts.\n\n The present model is based upon several decades of experimental observations, numerical simulations and attempts to built a theory liable to explain the origin of viscosity. The author is especially indebted to T.~Egami, from Oak Ridge National Laboratory and to J.~Langer from UC Santa Barbara, for having patiently explained their understanding and knowledge of this difficult subject.\n\n The work presented here is a small brick added to the knowledge, hoping that it will open the door to a more accurate theory liable to explain this mystery. However, several evidences point toward this model providing some universal results: (a) it introduces a parameter that permits to distinguish between strong and fragile glasses, (b) it exhibits a {\\em ``cross-over temperature''} $T_{co}$, below which the viscosity varies very quickly {\\em w.r.t.} the temperature. In addition, while being first motivated by the study of bulk metallic glasses, it seems to be relevant in other areas of Physics and Material Sciences, such as colloids \\cite{Ega17}, usual glassy materials in their liquid phase, or even quantum fluids \\cite{Hel17}.\n\n\\subsection{Anankeons}\n \\label{visc.ssect-ananke}\n\nThe fundamental concept on which this work is based will be called {\\em anankeon}. The name came from a long discussion between the present author and T.~Egami, during which the later formulated his view on what were the most fundamental degrees of freedom in the liquid phase \\cite{EB12}. More precisely, in his view, it is the cutting and forming of the atomic bond which matters, an action changing the local topology of atomic connectivity network. To explain quickly what the term anankeon means, it suffices to understand that a liquid, like a solid, is a condensed material. Namely the atoms are jam packed together in a local structure made of well ordered clusters, with not much empty places available around. The local stress felt by atoms leads them to jump or to locally reorder themselves to find a more comfortable position, trying to minimize their energy. These events are fast. They correspond to crossing a saddle point in the energy landscape, namely a passage through an unstable equilibrium positions \\cite{Ega14}. In practice they happen at unpredictable times, so that their dynamic is more efficiently described in terms of a Markov process. Hence the corresponding stress tensor felt by each atom is constantly varying under the {\\em stress of circumstances}, a concept that can be translated by the word $\\alpha\\nu\\alpha \\gamma\\kappa\\epsilon\\iota\\alpha$ (anagkeia) in Greek. Associated with this concept was the Greek goddess {\\em Ananke}, expressing the fate or destiny, due to uncontrollable forces, constraints, necessity. Hence the name anankeon associated with these stress induced unpredictable atomic moves.\n\n \\subsection{Local Structure, Local Dynamics}\n \\label{visc.ssect-loc}\n\n If there is only one atomic species in a condensed material, the local clusters are likely to have the shape of a regular icosahedron. Since regular icosahedra cannot tile the $3D$-space, there is either a geometrical frustration or the local cluster deform in the solid phase to lead to a crystal. The most likely symmetry of such a crystal is {\\em fcc}. To get a glass, frustration is necessary. In the case of metallic glasses, frustration is obtained by mixing several species with very different atomic radius in order to prevent the systems to crystallize in a periodic order. However, Egami insisted for a long time that this geometrical description of the atomic distribution, while useful, is insufficient at describing the material. He insisted that an atom is not a hard sphere, but exhibits a bit of elasticity. Consequently, an atom can be seen as having a local stress tensor, in a way similar with particle having spins. This representation is powerful enough to describe the liquid phase in the high temperature regime \\cite{ES82}. In this view, a liquid appears as a free gas of objects represented by non interacting atomic stress tensors. Numerical simulations \\cite{ES82} suggest that this atomic stress tensor is a random Gaussian variable with zero average and with a covariance matrix given by a Maxwell-Bolztman factor using the expression of the elastic energy. It is even quantitatively better to use an effective elastic energy by taking into account the long distance effect of stress propagation using the mean field theory developed by Eshelby \\cite{Esh57}, in analogy with the Clausius-Mossoty theory of effective electric permittivity in dielectric. The equilibrium state of such a free gas can then be explicitly computed using Statistical Mechanics leading to a law of Dulong and Petit for the heat capacity, an observation made for a long time, which found an explanation with this theory.\n\n \\subsection{A Toy Model}\n \\label{visc.ssect-toy}\n\n What happens at lower temperature~? The atomic vibrations, leading to acoustic waves, which will be called abusively {\\em phonons} here, whereas quantized or classical, are still existing in liquids as in solids. However the acoustic waves with short wave length, namely comparable to the inter-atomic distance, are strongly damped in the liquid phase at high temperature. Whenever the typical time scale characterizing the jump dynamics is comparable to the phonon period, the phonon has no time to oscillate and it is damped. The present model investigates this damping more quantitatively. To proceed, the same intuition will be followed as in the original paper of Drude on electronic transport in metals \\cite{Dr1900}. In order to make the model computable, it will be assumed that it describes the mechanical fate of a given atom: (i) as long as no jump occurs, this atom is located near the minimum of a local potential well, and it will oscillate, harmonically, with a frequency given by the local curvature of the well, (ii) then at random times (Poissonian distribution) the atom jumps, finding itself, after the jump in a new potential well, with a new local curvature and a new initial location relative to the new equilibrium position and a new initial velocity (phase-space position). Hence, the curvature of potential well (expressed in terms of the frequency of oscillation) will be considered as a random variable with a fixed average and a given covariance $\\sigma$ which will be shown to be the most important parameter. Similarly, the new relative initial phase-space position, after the jump, will also be chosen randomly according to a Maxwell-Boltzmann distribution. To make the model even more computable, it will be assumed that only one degree of freedom of oscillation is allowed. It will be seen that, even though this model is so simplistic, it exhibits a transition between two time-scales as $\\sigma$ decreases. The computation of correlation functions, like the viscosity, shows that this change of scale may affect the viscosity in an essential way.\n\n \\subsection{Anankeons in Glass: the STZ-Theory}\n \\label{visc.ssect-stz}\n\n For a very long times, engineers investigating the mechanical properties of materials measured the strain as the material is submitted to an external stress. The strain-stress curves shows usually a line at small stress, corresponding to the elastic response. This response is usually reversible: increasing then decreasing the stress follows the same curve. If the stress applied becomes large most materials just break. But some materials, like bulk metallic glasses, exhibit a plastic region in which the sample can be strained irreversibly, without much of additional stress. In a series of seminal papers \\cite{Fa98,FL98,LaPe03,Lan04,Lan06}, Langer and his collaborators developed a theory of what they called {\\em shear transformation zone} (STZ), to account for the plasticity region in bulk metallic glasses. The idea is that in the plastic region, the local clusters exhibit droplets in which the solid really behaves like a liquid, called an STZ. Considering the size of such droplets as a random variable, acting as a local lubricant, Langer {\\em et al.} proposed a system of effective equation at the macroscopic scale that are modeling the behavior of the material under stress with an amazing accuracy \\cite{RB12}. At the atomic scale though, numerical simulation show that these STZ's are the result of a cascade or avalanche of atomic swaps, namely of anankeons, \\cite{DA05}, starting at a random site, called the germ. This initial anankeon induces other in its neighborhood, because of the disruption created by the propagation of the stress induced by such swaps. The cascade looks like a local catastrophic event, occurring during a very short time, leading to the building of the local liquid-like droplets. The time scale representing the time necessary to create an STZ is however long in comparison with the typical time scale of the anankeon-swap. A model for such a cascade of event was proposed in \\cite{SDM01,FIE15}.\n\n\\section{Liquids at Large Temperature as a Perfect Anankeon Gas}\n\\label{visc.sect-perfect}\n\n The numerical simulation produced by Egami and Srolovitz \\cite{ES82} can be interpreted as a description of the liquid phase, far from the liquid-solid transition, in terms of a free gas of anankeon. The degree of freedom associated with each anankeon will be given in terms of an atomic stress tensor. Their Gaussian distribution observed in the numerical results can be interpreted in terms of a Gibbs state describing the thermal equilibrium in a statistical mechanical approach. This Section is dedicated to describe this in detail.\n\n \\subsection{The Delone Hypothesis and the Ergodic Paradox}\n \\label{visc.ssect-DelHyp}\n\n An instantaneous snapshot of the atomic arrangement can be described as a set of points ${\\mathcal L}$, representing the position of atomic nuclei. With a very good approximation ${\\mathcal L}$ is efficiently described as a {\\em Delone set}. Namely it is characterized by two length scales $00$, somewhere in the atomic configuration two atoms are at a distance shorter than $\\epsilon$. Similarly, with probability one, given any $\\epsilon >0$, somewhere in the atomic configuration there is a hole of size larger than $1\/\\epsilon$. However, such local arrangement are rare, so that dynamically their lifetime is so short that they are unobservable. Hence restricting the atomic configurations to make up a Delone set is a reasonable assumption called the {\\em Delone Hypothesis}\n\n \\subsection{Voronoi Tiling, Delone Graphs and Local Topology}\n \\label{visc.ssect-VDLT}\n\n Given a Delone set ${\\mathcal L}$ and a point $x\\in{\\mathcal L}$, the Voronoi cell $V(x)$ is defined as the set of points in space closer to $x$ than to any other point in ${\\mathcal L}$. $V(x)$ can be shown to be the interior of a convex polyhedron containing the open ball centered at $x$ of radius $r$ and contained in the ball centered at $x$ of radius $R$ \\cite{AK00,AKL13}. The closure $T(x)$ of the Voronoi cell is called a Voronoi tile. Two such tiles can only intersect along a common face. In particular two Voronoi cells centered at two distinct points of ${\\mathcal L}$ do not intersect. The entire space is covered by such tiles, so that the family of Voronoi tiles constitute a tiling, the {\\em Voronoi tiling}. The vertices of the Voronoi tiles will be called {\\em Voronoi points}. Their set ${\\mathcal L}^\\ast$ is usual called the {\\em dual lattice}. For a generic atomic configuration, a Voronoi point has not more than $d+1$ atomic neighbors in space-dimension $d$ ($3$ in the plane and $4$ in the space). In particular, the Voronoi points are the center of a ball with $d+1$ atoms on its boundary, a property called by Delone (who signed Delaunay) the {\\em ``empty sphere property''} \\cite{Del34}. It means that it is at the intersection of $d+1$ Voronoi tiles. The $d+1$ atoms located at the center of those tiles are the vertices of a simplex (triangle for $d=2$, tetrahedron for $d=3$) that generates the so-called {\\em Delaunay triangulation}.\n\n\\vspace{.1cm}\n\n Two points $x,y\\in{\\mathcal L}$ will be called nearest neighbors whenever their respective Voronoi tiles are intersecting along a facet, namely a face of codimension 1. The pair $\\{x,y\\}$ will be called an {\\em edge}. In practice, though, using physical criteria instead of a geometrical one, an edge might be replaced by the concept of {\\em bond}, to account for the fact that the atoms associates with $x$ and $y$ are strongly bound \\cite{Be59,Be60,Be64}. The term ``strongly'' is defined through a convention about the binding energy, so as to neglect bonds between atoms weakly bound to each other. Whatever the definition of an edge, geometrical or physical (bonds), this gives the Delone graph ${\\mathscr G}=({\\mathcal L}, {\\mathscr E})$, where ${\\mathcal L}$ plays the role of the vertices and ${\\mathscr E}$ denotes the set of edges or of bonds. At this point the Delone graph needs not being oriented. \n\n\\vspace{.1cm}\n\n Along a graph, a path is an ordered finite set of edges, each sharing a vertex with its successor or with its predecessor. The number of such edges is the length of the path. The graph-distance between two vertices is the length of the shortest paths joining them. A {\\em graph-ball} centered at $x$ of radius $n$ will be the set of all vertices at graph distance at most $n$ from $x$. Such graph balls correspond to local clusters. Two graph balls are called {\\em isomorphic} if there is a one-to-one surjective map between their set of vertices, that preserve the graph distance. As a result, if the atomic positions is slightly changed locally in space, the graph balls might still be isomorphic. Therefore graph ball, modulo isomorphism are encoding what physicists called the {\\em local topology} of the atomic configuration. \n\n\\vspace{.1cm}\n\n The transition between two graph balls can be described precisely through the concept of {\\em Pachner moves} \\cite{Pa91,AK00,AKL13}. This concept was created by experts of computational geometry, to describe the deformation of manifolds via a computer. A Pachner moves can only occur if at least one Voronoi point becomes degenerate, namely it admits at least $d+2$ atomic neighbor at some point during the move. Such moves correspond to a local change in a graph ball, namely an edge disappear another may reappear. A Pachner move corresponds exactly to the concept of anankeon. It can be described through using the Delaunay triangulation as explained in Fig.~\\ref{visc.fig-pachner}.\n\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{pachner2dD.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{pachner3d4V.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{figure}[ht]\n \\centering\n\\caption{\\label{visc.fig-pachner} Left: a Pachner move in the plane. \nRight: a Pachner move in the $3$-space}\n\\end{figure}\n\n In $3D$-space such a move involves at least five atoms moving quickly to deform the Delaunay triangulation, changing the nature of the Delone graph, namely the local topology. This number five has been seen in numerical simulations \\cite{Yue14} as can be seen in Fig.~\\ref{visc.fig-yue}.\n\n\\begin{figure}[ht]\n \\centering\n\\includegraphics[width=9cm]{STZstat.jpg}\n\\caption{\\label{visc.fig-yue} Number of atoms involved in a Pachner move \\cite{Yue14}}\n\\end{figure}\n\n \\subsection{Atomic Stress Tensor}\n \\label{visc.ssect-atomstress}\n\n The definition of stress tensor in solids \\cite{BH54,LPKL}, using the hypothesis of Continuum Mechanics, gives an intuition upon how to build a stress tensor at the atomic level \\cite{EMV80}. The present presentation will use a different route, but leads to a similar result. It will be assumed that interatomic forces are described by a two-body isotropic potential $W(r)$, where $r$ denotes the distance between the two atoms. In such a case, given a pair $e=(x,y)$ of atoms bound by an edge, let $n_e$ denotes the unit vector along the line generated by $x,y$ oriented from $x$ to $y$. By isotropy, the force acting upon the atom located at $x$ coming from $y$ is $F_y(x)= W'(r) n_e$. The total force acting on $x$ is therefore given by the sum $F(x)=\\sum_{y} n_e W'(r)$, where the sum includes the nearest neighbors of the atom at $x$. In Continuum Mechanics, the stress tensor is defined so as to satisfy $F=\\int_{\\partial \\Lambda}\\sigma(s)\\cdot n(s)ds$ for any volume $\\Lambda$ with boundary $\\partial \\Lambda$. At the atomic level though, the integral will be replaced by the previous sum. The role of the volume $\\Lambda$ is played by the Voronoi tile $T(x)$. Each nearest neighbor of $x$ corresponds to a unique facet of $T(x)$, and then the unit vector $n_e$ is nothing but the normalized unit vector perpendicular to the facet oriented outwardly {\\em w.r.t.} the tile. This suggests the following formula for the stress tensor: each bond $e$ gives it a value $\\sigma_e$ so that\n\n$$\\sigma_e=|n_e\\rangle\\langle n_e| \\;\\frac{W'(r_e)}{A_e}\\,,\n$$\n\n where $r_e$ is the Euclidean length of the bond and $A_e$ is the Euclidean area of the facet associated with $e$. It ought to be remarked that the ratio $\\tr(\\sigma_e)=p_e= W'(r)\/A_e$ represents the pressure felt by the atom coming from its neighbor in the direction of the edge $e$. The deviatory stress is the traceless part $\\sigma_e-p_e\/d$ where $d$ is the dimension of the space in which this material lies. The atomic stress at $x$ is just the sum of these contributions, namely for an atom located at the vertex $x\\in{\\mathcal L}$, \n\n\\begin{equation}\n\\label{visc.eq-atomstress}\n\\sigma(x)= \\sum_{\\partial_0e=x} \n |n_e\\rangle\\langle n_e| \\;\\frac{W'(r_e)}{A_e}\\,.\n\\end{equation}\n\n In this expression, $\\partial_0e=x$ means the edges with origin at $x$. The formula above is very similar to the one obtained in \\cite{EMV80}, using an argument from Continuum Mechanics, apart from a possible numerical constant close to $1$.\n\n\\vspace{.1cm}\n\n The numerical simulation in \\cite{ES82} suggests that in the liquid phase, far from the liquid-solid transition, the statistical distribution of the atomic stress tensor follows a Maxwell-Boltzmann-Gibbs law of the form\n\n\\begin{equation}\n\\label{visc.eq-ES}\n\\Pro\\{\\sigma\\in {\\widehat{\\Lambda}}\\}=\\frac{1}{Z_1}\n \\int_{{\\widehat{\\Lambda}}} e^{-\\beta(p^2\/2B+\\tau^2\/2G)}\\; d\\sigma\\,.\n\\hspace{2cm}\n \\mbox{\\rm\\bf (Egami-Srolovitz Principle)}\n\\end{equation}\n\n In this expression, $\\beta=1\/\\kB T$, ${\\widehat{\\Lambda}}$ represents a volume in the space of stress tensors, $Z_1$ is a normalization factor, $p$ is the pressure given by the normalized trace of the stress tensor, and $\\tau$ is called the {\\em von Mises stress}. The latter is defined such that $\\tau^2$ is the trace of the square of the deviatory stress (traceless part of $\\sigma$). The coefficients $B,G$ are respectively the {\\em bulk modulus} and the {\\em shear modulus}. The expression in the exponential represents the elastic energy associated with the atomic stress.\n\n\\begin{figure}[ht]\n \\centering\n\\includegraphics[width=11cm]{heatCap.jpg}\n\\caption{\\label{visc.fig-heatCap} Specific heat as a function of temperature for an alloy made of Gold, Germanium and Silicon signaling a glass-liquid transition \\cite{CT68}}\n\\end{figure}\n\n\\vspace{.1cm}\n\n \\subsection{Equilibrium for Liquid at High Temperature}\n \\label{visc.ssect-liqeq}\n\n Under the Egami-Srolovitz principle, if the atomic stress are uncorrelated, it becomes possible to compute the partition function $Z(T,N)$ of a volume of liquid containing $N$ atoms. Namely $Z(T,N)=Z_1(T)^N$, where\n\n$$Z_1(T)= \\int e^{-\\beta(p^2\/2B+\\tau^2\/2G)}\\; d\\sigma\\,,\n \\hspace{2cm}\n \\beta=\\frac{1}{\\kB T}\\,.\n$$\n\n The domain of integration is the space of all real symmetric $d\\times d$ matrices, representing a possible stress tensor. This Gaussian integral is easy to compute giving $Z_1(T)= {\\mathscr Z}\\, T^{d(d+1)\/2}$ where ${\\mathscr Z}$ is a constant depending on $B,G$, but independent of the temperature. The Clausius entropy is given by $S=\\kB \\ln(Z(N,T))$ leading to the following expression of the heat capacity $C_v= c_d N\\kB$ where $c_d=d(d+1)\/2$, which is $6$ in $3D$. In particular, the system follows a {\\em Law of Dulong-Petit} like the contribution of phonons in crystal. However, the physical origin is different in liquid as the main degree of freedom associated with atoms are the atomic stress. Hence a liquid can be seen as a {\\em perfect gas of anankeons}. This prediction gave a satisfactory answer to the experimental observation \\cite{CT68} of the saturation of the heat capacity at high temperature (see Fig.~\\ref{visc.fig-heatCap}).\n\n\n\\section{Phonon-Anankeon Interaction}\n\\label{visc.sect-phan}\n\n What happens at lower temperature, near the solid-liquid transition~? The following model will show that as the temperature decreases, the vibration modes are constraining the shape of local clusters, so as to change the viscosity in a non trivial way.\n\n \\subsection{Time Scales}\n \\label{visc.ssect-tsc}\n\n Since the dynamics plays such an important role in explaining the difference between solids and liquids, the various relevant time scales ought to be discussed. The most elementary is $\\tau_{\\mbox{\\em \\tiny LC}}$, where $LC$ stands for {\\em local configuration}, representing the time required between two consecutive Pachner moves. It corresponds to the anankeon relaxation time. The next time scale is called {\\em Maxwell's relation time} $\\tau_{\\mbox{\\em \\tiny M}}$ \\cite{HMcD06} is defined by the ratio\n\n\\begin{equation}\n\\label{visc.eq-maxwell}\n\\tau_{\\mbox{\\em \\tiny M}}=\\frac{\\eta}{G_\\infty}\\,,\n\\end{equation}\n\n where $\\eta$ is the viscosity and $G_\\infty$ is the high frequency shear modulus. The Maxwell relaxation time represents the time scale below which the system behaves like a solid and beyond which it can be considered as a liquid. At high temperature $\\tau_M=\\tlc$. Besides these two time scales, there is the $\\alpha$-relaxation time $\\tau_\\alpha$ \\cite{FKST}, defined by the scattering function at the peak of the structure function $S(Q)$. Finally there is the {\\em bond lifetime} $\\tau_B$ \\cite{YO98}. The value of these time scales has been discussed thoroughly in \\cite{IE12,INE13} (see Fig.~\\ref{visc.fig-relTime}). This was done through classical as well as {\\em ab initio} molecular dynamics simulation on various metallic liquids like liquid iron, $Cu_{56}Zr_{44}$ or $Zr_{50}Cu_{40}Al_{10}$. \n\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{relaxTimes.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{timeRatio.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{figure}[ht]\n \\centering\n\\caption{\\label{visc.fig-relTime} Various relaxation times in the liquid phase as function of the temperature. Left: $T_g$ is the temperature at the liquid-solid transition. The Arrhenius law is visible on this graph: the slope of the lines give an estimate of $W$. Right: $T_A$ is the crossover temperature below which the Maxwell time differs from the $\\tau_{\\mbox{\\em \\tiny LC}}$. (\\cite{INE13})}\n\\end{figure}\n\n All these time scales are proportional in the liquid phase, far enough from the liquid-solid transition as shown in Fig.~\\ref{visc.fig-relTime} and are given by an Arrhenius law $\\tau \\sim e^{W\/\\kB T}$.\n\n However, near the liquid-glass transition point, there is a significant discrepancy between the Maxwell time and the local configuration time as can be seen on the right of Fig.~\\ref{visc.fig-relTime}. This is precisely the discrepancy that the present model is going to describe. It seems natural, then, to choose the following basic time scale for the purpose of the modeling:\n\n\\begin{itemize}\n \\item\\label{visc.Time-scale} {\\bf Time-scale assumption: } In the toy model $\\tau=\\tau_{\\mbox{\\em \\tiny LC}}$.\n\\end{itemize}\n\n \\subsection{Phonon Dynamic: the main approximations}\n \\label{visc.ssect-phdynapp}\n\n As explained in Section~\\ref{visc.ssect-toy}, the model will describe the situation of one atom in the liquid. Most of the time, this atom is located near the bottom of a potential well, created by the interaction with other atoms around. Since the potential is smooth near its minimum, there is no loss of generality in assuming that it is well approximated by a local paraboloid, at least if the atom does not move too far from the minimum. Hence the atomic motion is well approximated by an harmonic oscillator. To simplify further, it will be assumed that {\\em only one of the three dimension really matter}. This looks drastic, but the method of solving the model is actually the same if all the dimensions are taken into account. Hence, the curvature of the well near the minimum defines the oscillator pulsation $\\omega=2\\pi\\nu$ where $\\nu$ is the frequency. Let then $q=q(t)$ denotes the $1D$ distance of the atom from the potential minimum at any given time $t$. It is convenient to introduce the variable $u=\\omega q$, which has the same dimension as the velocity $v=\\stackrel{\\cdot}{q}$. Hence the $2D$-vector $X$ with coordinate $u,v$ represents the phase-space position of the atom at any given time. The usual equation of motion is given by \n\n$$m\\frac{d^2q}{dt^2}+ kq^2=0\\,,\n \\hspace{2cm}\n \\omega= \\sqrt{\\frac{k}{m}}\\,,\n$$\n\n where $m$ is the mass of the atom and $k$ is the spring constant, expressible in terms of the second derivative of the potential energy. This equation can equivalently be written as\n\n\\begin{equation}\n\\label{visc.eq-harmosc}\n\\frac{dX}{dt}=\\omega J X\\,,\n \\hspace{2cm}\n J=\\left[\n\\begin{array}{cc}\n~0 & 1\\\\\n-1 & 0\n\\end{array}\n\\right]\\,.\n\\end{equation}\n\n The solution of this equation, with initial position $u(0)$ and initial velocity $v(0)$, is given by\n\n\\begin{equation}\n\\label{visc.eq-solharm}\nX(t)=e^{\\omega t J} X(0)= \n\\left[\n\\begin{array}{cc}\n\\cos{\\omega t} & \\sin{\\omega t}\\\\\n-\\sin{\\omega t} & \\cos{\\omega t}\n\\end{array}\n\\right]\\;\\left[\n\\begin{array}{c}\nu(0)\\\\\nv(0)\n\\end{array}\n\\right]\\,.\n\\end{equation}\n\n The mechanical energy of this oscillator is given by \n\n$$E= \\frac{m\\stackrel{\\cdot}{q}^2}{2}+\\frac{k q^2}{2}=\n \\frac{m|X|^2}{2}\\,.\n$$\n\n It ought to be stressed that the spring constant $k$ depends in an essential way upon the local environment seen by the test-atom. In particular, if the local configuration changes due to a Pachner move, or an anankeon, this constant will change as well. This will be described mathematically in the next paragraph.\n\n \\subsection{The toy model for anankeon-phonon interaction}\n \\label{visc.ssect-anandyn}\n\n When an anankeon strikes, the atom is ejected from its position to jump on a position nearby, in a new harmonic potential. To describe such a situation, following the strategy described in Section~\\ref{visc.ssect-toy}, it will be assumed that the anankeons strikes at {\\em random times} $\\{ \\cdots <\\tau_n <\\tau_{n+1}<\\cdots\\}$. Here the integer index $n$ varies from $-\\infty$ to $+\\infty$. To be more precise, the randomness of those times will be assumed to be {\\em Poissonian}, namely the random variables $\\tau_{n+1}-\\tau_n$ are {\\em i.i.d.}, with average\n\n$${\\mathbb E}\\{\\tau_{n+1}-\\tau_n\\}\\stackrel{def}{=}\n \\langle \\tau_{n+1}-\\tau_n\\rangle= \\tau =\\tlc\\,,\n$$\n\n and an exponential distribution\n\n$$\\Pro\\{(\\tau_{n+1}-\\tau_n)\\in A\\subset {\\mathbb R}\\}= \n \\int_A e^{-t\/\\tau} \\; \\frac{dt}{\\tau}\\,.\n$$\n\n The phase-space position $X(t)$ after the jump will also be relative to the minimum of the new potential well. In particular, there is a need to update this position right after the jump in terms of the position right before, namely\n\n$$X(\\tau_n+0)=X(\\tau_n-0)+\\xi_n\\,.\n$$\n\n The vector $\\xi_n$ represents the result of the anankeon kick \non the initial phase-space position, relative to the potential minimum. It is reasonable to assume that $\\xi_n$ is also a random variable and that the family $(\\xi_n)_{n\\in{\\mathbb Z}}$ is made of {\\em i.i.d.}'s. In addition it will be assumed that the common distribution is provided by the Gibbs state at the temperature $T$. Hence\n\n\\begin{equation}\n\\label{visc.eq-gibbs}\n\\Pro\\{\\xi_n \\in B\\subset {\\mathbb R}^2\\}= \n \\frac{\\kB T}{2\\pi m}\n \\int_B e^{-m|\\xi|^2\/\\kB T} \\; d^2\\xi\\,.\n\\end{equation}\n\n At last, the pulsation $\\omega$ will depend upon the potential well in which the atom fell. Hence between the times $\\tau_n$ and $\\tau_{n+1}$ it will be given by $\\omega_n$. Since the atom this model is describing should be typical, the $\\omega_n$'s should be considered as random variables, and again, the family $(\\omega_n)_{n\\in{\\mathbb Z}}$ will be made of {\\em i.i.d.}'s. At this point, it is unwise to fix the common distribution. It enough to fix the average $\\omega$ and the variance $\\sigma$\n\n\\begin{equation}\n\\label{visc.eq-omega}\n\\omega = {\\mathbb E}\\{\\omega_n\\}\\,,\n \\hspace{2cm}\n \\sigma=\\Var(\\omega_n)= {\\mathbb E}\\{(\\omega_n-\\omega)^2\\}^{1\/2}\\,.\n\\end{equation}\n\n At last, during the time interval $(\\tau_n,\\tau_{n+1})$, the evolution of $X(t)$ will satisfy eq.~(\\ref{visc.eq-harmosc}) and eq.~(\\ref{visc.eq-solharm}).\n\n \n \\subsection{Viscosity and correlation function}\n \\label{visc.ssect-visc}\n\n The viscosity is given by the fluctuation-dissipation theorem as\n\n\\begin{equation}\n\\label{visc.eq-visc}\n\\eta = \\frac{V}{\\kB T}\n \\int_0^\\infty C_f(t)\\, dt\\,,\n\\end{equation}\n\n where $V$ is the volume of the fluid, $T$ the temperature, and $C_f$ is a correlation functions defined by\n\n\\begin{equation}\n\\label{visc.eq-corr2}\nC_f(t) =\n {\\mathbb E}\\left\\{ f(X(t))\\;\\overline{f(X(0)} \\right\\}\\,,\n\\end{equation}\n\n where $f$ is a function on the phase space giving the expression of the stress tensor. For the purpose of this calculation, it is sufficient to consider $f$ as a scalar valued function, because it does not change the conclusion of the computation. The goal will be to show that, as $t\\to\\infty$, the correlation $C_f(t)\\sim e^{-t\/\\tau_M}$ defining the Maxwell time $\\tau_M$. And indeed the time integral in eq.~(\\ref{visc.eq-visc}) will be proportional to $\\tau_M$ as anticipated by eq.~(\\ref{visc.eq-maxwell}).\n\n\\vspace{.1cm}\n\n This correlation function can also be computed using the {\\em dissipative evolution} defined by the operator $P_t$ as the following conditional expectation\n\n\\begin{equation}\n\\label{visc.eq-sg}\nP_tf(x) = {\\mathbb E}\\left\\{ f(X(t))|X(0)=x\\right\\}\\,.\n\\end{equation}\n\n Then, the correlation function is given by the phase-space integral\n\n\\begin{equation}\n\\label{visc.eq-corrHilb}\nC_f(t) =\\int_{{\\mathbb R}^2} \\overline{f(x)}\\,P_tf(x)\\;d^2x\\,.\n\\end{equation}\n\n In order to compute $\\tau_M$ precisely, it is convenient to compute the Laplace transform of $C_f$ instead, namely, given a {\\em complex variable} $\\zeta$\n\n\\begin{equation}\n\\label{visc.eq-lapcorr}\n{\\mathscr L} C_f(\\zeta)=\n \\int_0^\\infty e^{-t\\zeta}\\, C_f(t)\\, dt\\,.\n\\end{equation}\n\n It turns out that the asymptotic behavior of the correlation at $t\\to \\infty$ can be interpreted as the position of the nearest singularity of ${\\mathscr L} C_f$ in the complex plane. Namely the Laplace transform is analytic in the half-plane $\\Re\\zeta >-1\/\\tau_M$. Hence, in order to compute the correlation function, it is sufficient to compute the value $f(X(t))$ taken by the function $f$ along the random phase-space trajectory of the atom under scrutiny.\n\n\n\n\\section{Computing the Laplace transform of a correlation function}\n\\label{visc.sect-corrcomp}\n\n\\noindent The formula~(\\ref{visc.eq-corrHilb}) suggests to compute the Laplace transform of the dissipative evolution operator $P_t$ instead, acting on some function space describing the relevant functions on the phase-space. In order to do so, it will be more efficient to compute the actions of the various part of the dynamic on functions rather than on the phase-space trajectory. As will be seen this calculus is very similar to the operator calculus used in Quantum Mechanics.\n\n \\subsection{Operator Calculus}\n \\label{visc.ssect-opcal}\n\n\\noindent The first part of the evolution is provided by the phase-space mouvement of the harmonic oscillator, as described in Section~\\ref{visc.ssect-phdynapp}. Namely\n\n\\begin{equation}\n\\label{visc.eq-angmom}\nf\\left(e^{\\omega t J}x\\right)=\n \\left(e^{-\\omega t{\\mathbb J}}f\\right)(x)\\,,\n \\hspace{2cm}\n -{\\mathbb J}=v\\partial_u-u\\partial_v\\,,\n\\end{equation}\n\n\\noindent where $x=(u,v)$ is the phase-space position and ${\\mathbb J}$ is very similar to the {\\em phase-space angular momentum} but for the missing $\\imath$ in front (namely ${\\mathbb J}$ is anti-Hermitian instead) and its $2D$ character in phase-space.\n\n\\vspace{.1cm}\n\n\\noindent Similarly, the impact of the anankeon on the atom is to translate the phase-space position of the atom after a kick, by a vector $\\xi=(a,b)$, leading to the {\\em translation operator}\n\n\\begin{equation}\n\\label{visc.eq-trans}\nf(x+\\xi)= \\left(e^{\\xi\\cdot \\nabla}f\\right)(x)\\,,\n \\hspace{2cm}\n \\xi\\cdot \\nabla= a\\partial_u+b\\partial_v\\,.\n\\end{equation}\n\n\\noindent Hence $\\nabla$ plays the role of a phase-space momentum operator apart for the missing $\\imath$ in front (namely $\\nabla$ is anti-Hermitian instead).\n\n\\vspace{.1cm}\n\n\\noindent Let $X(t)$ denote the phase-space trajectory of the atom at time $t$ with initial condition $X(0)=x$. From the description of the model, an harmonic motion takes place between time $\\tau_0=0$ to $\\tau_1$ at pulsation $\\omega_1$, then a phase-space translation by $\\xi_1$, then another harmonic motion between time $\\tau_1$ to $\\tau_2$ at pulsation $\\omega_2$, then a new phase-space translation by $\\xi_2$, and so on, until the time $t$ is reached. It becomes possible to describe it from the functional point of view as follows\n\n\\begin{equation}\n\\label{visc.eq-evol}\nP_tf(x)= {\\mathbb E}\\left(f(X(t))|X(0)=x\\right) =\n{\\mathbb E}\\left\\{\n \\prod_{j=1}^n\n \\left(e^{-(\\tau_j-\\tau_{j-1})\\omega_{j-1}{\\mathbb J}}\\, \n e^{\\xi_j\\cdot\\nabla}\n \\right)\n e^{-(t-\\tau_n)\\omega_n{\\mathbb J}}\\,f\n \\right\\}(x)\\,,\n\\end{equation}\n\n\\noindent where the product is ordered from left to right, with $j=1$ on the left and $j=n$ on the right, and where $n$ is the integer such that $\\tau_n\\leq t<\\tau_{n+1}$. From the assumptions made on the model, the $(\\tau_j-\\tau_{j-1})$ are {\\em i.i.d.}, as well as the $\\omega_j$'s and the $\\xi_j$'s. It follows that $n$ is also a random variable. Computing the distribution of $n$ though, is hard enough to try to avoid it. The Laplace transform will allow to avoid such a calculation.\n\n \\subsection{Laplace transformation}\n \\label{visc.ssect-laping}\n\n\\noindent After averaging and taking the Laplace transform the result is\n\n$${\\mathscr L} P_\\zeta f(x)=\n \\int_0^\\infty e^{-t\\zeta}\\; P_tf(x)\\;dt=\n {\\mathbb E}\\left\\{\\int_0^\\infty e^{-t\\zeta}\\;f(X(t)\\;dt \\Bigg| X(0)=x\\right\\}\\,,\n \\hspace{1cm}\n \\zeta\\in {\\mathbb C}\\,.\n$$\n\n\\noindent Thanks to the linearity of the averaging process, this integral over time can be decomposed, under the averaging, into the time intervals $[\\tau_n,\\tau_{n+1}]$, to give\n\n$${\\mathscr L} P_\\zeta f(x)=\\sum_{n=0}^\\infty\n {\\mathbb E}\\left\\{\\int_{\\tau_n}^{\\tau_{n+1}} \n e^{-t\\zeta}\\;f(X(t)\\;dt\\Bigg| X(0)=x\\right\\}\\,.\n$$\n\n\\noindent Using eq.~(\\ref{visc.eq-evol}), this gives\n\n\\begin{equation}\n\\label{visc.eq-ev}\n{\\mathscr L} P_\\zeta f(x)= \\sum_{n=0}^\\infty\n {\\mathbb E}\\left\\{\\int_{\\tau_n}^{\\tau_{n+1}} \n e^{-t\\zeta}\\; \\prod_{j=1}^n\n \\left(e^{-(\\tau_j-\\tau_{j-1})\\omega_{j-1}{\\mathbb J}}\\, \n e^{\\xi_j\\cdot\\nabla}\n \\right)\n e^{-(t-\\tau_n)\\omega_n{\\mathbb J}}\\,f(x)\\;dt\n \\right\\}\\,.\n\\end{equation}\n\n\\noindent Since $\\tau_0=0$, a telescopic sum give $t=(t-\\tau_n)+(\\tau_n-\\tau_{n-1})+\\cdots+ (\\tau_1-\\tau_0)$. This leads to distribute the first term $e^{-t\\zeta}$ inside each terms of the product leading to replace each $\\omega_{j-1}{\\mathbb J}$ by $\\omega_{j-1}{\\mathbb J} + \\zeta$, namely\n\n\\begin{equation}\n\\label{visc.eq-ev2}\n{\\mathscr L} P_\\zeta f(x)= \\sum_{n=0}^\\infty\n {\\mathbb E}\\left\\{\\int_{\\tau_n}^{\\tau_{n+1}} \n \\prod_{j=1}^n\n \\left(e^{-(\\tau_j-\\tau_{j-1})(\\omega_{j-1}{\\mathbb J}+\\zeta)}\\, \n e^{\\xi_j\\cdot\\nabla}\n \\right)\n e^{-(t-\\tau_n)(\\omega_n{\\mathbb J}+\\zeta)}\\,f(x)\\;dt\n \\right\\}\\,.\n\\end{equation}\n\n\\noindent The first operation consists in evaluating the integral over $t$. The change of variable $s=t-\\tau_n$ gives the following operator valued integral\n\n\\begin{equation}\n\\label{visc.eq-sint}\n\\int_0^{\\tau_{n+1}-\\tau_n}\n e^{-s(\\omega_n{\\mathbb J}+\\zeta)}\\;ds= \n \\frac{1-e^{-(\\tau_{n+1}-\\tau_n)(\\omega_n{\\mathbb J}+\\zeta)}}\n {\\zeta+ \\omega_n{\\mathbb J}}\\,.\n\\end{equation}\n\n\\noindent The next step consists in recognizing that the averaging ${\\mathbb E}$ can be decomposed into averaging over the time Poisson process, before averaging over the random phase-space positions $\\xi_j$'s or the random pulsations $\\omega_{j-1}$'s. In addition, the stochastic independence of the $(\\tau_j-\\tau_{j-1})$'s permits to reduce the averaging to each factor \nin the product in eq.~(\\ref{visc.eq-ev2}). Using the the {\\em time-scale assumption} in Section~\\ref{visc.Time-scale} on page~\\pageref{visc.Time-scale}, this averaging reduces to the following integral, where $A$ is an operator\n\n\\begin{equation}\n\\label{visc.eq-pint}\n{\\mathbb E}_\\tau\\left\\{e^{-(\\tau_j-\\tau_{j-1})A}\\right\\}=\n \\int_0^\\infty e^{-s\/\\tau-sA}\\;\\frac{ds}{\\tau} =\n \\frac{1}{1+\\tau A}\\,,\n\\end{equation}\n\n\\noindent where ${\\mathbb E}_\\tau$ denotes the average over the Poissonian times. The stochastic independence of each factor in the product, leads to a factorization of the averaging, since, whenever two random variable $X,Y$ are stochastically independent the average of their product is given by the product of their average, that is ${\\mathbb E}(XY)={\\mathbb E}(X){\\mathbb E}(Y)$. After a little algebra, using eq.~(\\ref{visc.eq-sint}, \\ref{visc.eq-pint}), the eq.~(\\ref{visc.eq-ev2}) leads to\n\n\\begin{equation}\n\\label{visc.eq-ev3}\n{\\mathscr L} P_\\zeta f(x)= \\tau \\sum_{n=0}^\\infty\n {\\mathbb E}\\left\\{ \n \\prod_{j=1}^n\n \\left(\n \\frac{1}{1+\\tau(\\zeta+\\omega_{j-1}{\\mathbb J})}\\, \n e^{\\xi_j\\cdot\\nabla}\n \\right)\n \\frac{1}{1+\\tau(\\zeta+\\omega_n{\\mathbb J})}\\,f(x)\n \\right\\}\\,.\n\\end{equation}\n\n\\noindent Similarly, since the $\\xi_j$'s and the $\\omega_{j-1}$'s are stochastically independent, it is enough to compute the following averages\n\n\\begin{equation}\n\\label{visc.eq-xiaver}\n{\\mathbb E}_\\xi\\left\\{e^{\\xi_j\\cdot\\nabla}\\right\\}=\n e^{\\kB T\\,\\Delta\/2m}\\,,\n \\hspace{2cm}\n \\Delta=\\nabla\\cdot\\nabla= \\partial_u^2+\\partial_v^2\\,,\n\\end{equation}\n\n\\noindent where ${\\mathbb E}_\\xi$ denotes the average over the phase-space position, using the Gaussian integral in eq.~(\\ref{visc.eq-gibbs}). Similarly, the average over the $\\omega_{j-1}$'s leads to the operator-valued function\n\n\\begin{equation}\n\\label{visc.eq-omaver}\n{\\mathbb A}(\\zeta)=\n {\\mathbb E}_\\omega\\left\\{\n \\frac{1}{1+\\tau(\\zeta+\\omega_{j-1}{\\mathbb J})}\n \\right\\}\\,.\n\\end{equation}\n\n\\noindent Evaluating this integral requires to know the distribution of the $\\omega_{j-1}$'s. At this point, no further assumption will be made yet. Using eq.~(\\ref{visc.eq-xiaver}, \\ref{visc.eq-omaver}), the eq.~(\\ref{visc.eq-ev3}) leads to\n\n\\begin{equation}\n\\label{visc.eq-ev4}\n{\\mathscr L} P_\\zeta f(x)= \\tau \\sum_{n=0}^\\infty\n \\left\\{ \n {\\mathbb A}(\\zeta)\\,e^{\\kB T\\,\\Delta\/2m}\n \\right\\}^n\n {\\mathbb A}(\\zeta)f(x)=\\tau\\,\n \\frac{1}{1-{\\mathbb A}(\\zeta)\\,e^{\\kB T\\,\\Delta\/2m}}\\,{\\mathbb A}(\\zeta)f(x)\\,.\n\\end{equation}\n\n\\noindent The last series converges only if $\\|{\\mathbb A}(\\zeta)\\,e^{\\kB T\\,\\Delta\/2m}\\|<1$ where $\\|\\cdot\\|$ denotes the operator norm. At this point though, no accurate description of the space on which these operators act has been done, so that the norm condition is still meaningless. It will be the purpose of the discussion of the next Section. With this cautionary statement in mind, the problem is reduced to analyzing the properties of one operator at the end, giving the Laplace transform as a function of $\\zeta$.\n\n \\subsection{Digression: choosing the function space}\n \\label{visc.ssect-fspace}\n\n\\noindent In order to make sense of the calculation made in the previous Section~\\ref{visc.ssect-laping}, as was remarked at the end, the function space on which these operators act must be made more precise. In order to do so, the first question is to know which type of function $f$ are physically relevant. In the calculation of viscosity, the ideal function should be the atomic deviatory stress tensor as given in eq.~(\\ref{visc.eq-atomstress}). Using the finite sum decomposition over the edges linking the atom at $x$ with its neighbors, it boils down to find a space containing the functions of the form\n\n\\begin{equation}\n\\label{visc.eq-function}\ng(x) = \\frac{|y-x\\rangle\\langle y-x|}{|y-x|^2}\\, W'(|y-x|)\\,,\n\\end{equation}\n\n\\noindent where $y$ is a fixed point representing the position of a neighboring atom. First this expression is matrix valued instead of being a scalar. This matrix is $d\\times d$ if the liquid is lying in a $d$-dimensional space. It is real symmetric by construction. Moreover, {\\em only the traceless part} of this matrix matters for the viscosity. In $d$-dimension the space of $d\\times d$ real symmetric matrices is invariant by the rotation group $SO(d)$, and it decomposes into the direct sum of two invariant subspaces, one corresponding to the trace (trivial representation of $SO(d)$) and the traceless part (unit representation of $SO(d)$). For $d=3$ it is well known that the unit representation of $SO(3)$, acting on this space of traceless real symmetric matrices, corresponds to the spin $2$ representation of $SU(2)$.\n\n\\vspace{.1cm}\n\n\\noindent On the other hand, the effective potential energy $W$ decays fast enough at infinity. The decay is mostly exponential due to screening effect of negatively charges electrons over the Coulomb potential of the positively charged nucleus. Moreover, it derivative is likely to share the same property. Hence, the function $g$ decays fast enough as the position $x$ of the atom diverges from its equilibrium position.\n\n\\vspace{.1cm}\n\n\\noindent The previous considerations motivated by the physical problem, have to be translated into the langage of the phase-space for the $1D$ harmonic oscillator replacing the atomic motion. In the previous Sections, the letter $x$ has been confusingly used also to describe the phase-space position $x=(u,v)$, where $u$ represents the $1D$ position of the harmonic oscillator relative to the origin chosen at the minimum of the potential well, while $v$ describes the corresponding velocity. The translation between the language used in $d$ dimensions and the phase-space language will be done in the following way:\n\n\\vspace{.1cm}\n\n(i) The space ${\\mathcal H}$ of relevant functions will be chosen so that any of them decay fast enough at infinity;\n\n(ii) the space ${\\mathcal H}$ needs to be mathematically convenient, in particular, choosing it to be a Hilbert space is a very convenient choice;\n\n(iii) the symmetry properties of the function $f$ used in the calculation of the viscosity must reflect as much as possible the ones of the realistic stress tensor given in eq.~(\\ref{visc.eq-atomstress}).\n\n\\vspace{.1cm}\n\n\\noindent A reasonable choice is ${\\mathcal H}=L^2({\\mathbb R}^2)$, which denotes the space of complex valued square integrable measurable functions over the phase-space ${\\mathbb R}^2$, namely which satisfy \n\n$$\\|f\\|_{\\mathcal H}^2= \\int_{{\\mathbb R}^2} |f(x)|^2\\,d^2x <\\infty\\,.\n$$\n\n\\noindent The symmetry properties might be reflected by the presence of the operator ${\\mathbb J}$, generating the phase-space rotations, namely the group $U(1)$ of rotations in the plane. It is well known that ${\\mathbb J}$ can be defined as an anti-selfadjoint operator on ${\\mathcal H}$ with eigenvalues $\\imath \\ell$ where $\\ell \\in{\\mathbb Z}$ takes on any nonnegative of nonpositive integer values. Moreover the eigenvectors are given by functions of the form (using polar coordinates)\n\n\\begin{equation}\n\\label{visc.eq-evell}\ng_\\ell(x) =e^{\\imath \\ell \\theta} g(r)\\,,\n \\hspace{2cm}\n x=(r\\cos{\\ell\\theta},r\\sin{\\ell\\theta})\\in{\\mathbb R}^2\\,,\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation}\n\\label{visc.eq-L2int}\n\\int_0^\\infty |g(r)|^2\\,rdr <\\infty\\,.\n\\end{equation}\n\n\\noindent In particular this eigenspace is infinite dimensional. Let $\\Pi_\\ell$ denote the orthogonal projection from ${\\mathcal H}$ onto this space. Moreover, these spaces are mutually orthogonal and then span the entire Hilbert space ${\\mathcal H}$. \n\n\\vspace{.1cm}\n\n\\noindent The analogy between the realistic situation and the phase-space modeling, suggests to choose the eigenspace of ${\\mathbb J}$ with angular momenta $\\ell=\\pm 2$. It will be seen though, that this restriction does not matter in terms of the qualitative features of the model.\n\n\\vspace{.1cm}\n\n\\noindent Similarly the operator $\\Delta$ becomes self-adjoint with nonpositive generalized eigenvalues. In particular, the operator $\\exp\\{\\kB T \\Delta\/2m\\}$ is selfadjoint with its spectrum given by the interval $[0,1]$, so that $0\\leq \\exp\\{\\kB T \\Delta\/2m\\}\\leq 1$ implying $\\|\\exp\\{\\kB T \\Delta\/2m\\}\\|\\leq 1$.\n\n\\vspace{.1cm}\n\n\\noindent The previous discussion shows that, while a choice has to be made for ${\\mathcal H}$, it will always be partly arbitrary. In particular, the discussion of symmetries might not be totally realistic. But the choice of ${\\mathcal H}$ is not innocent. The properties of the unbounded operators ${\\mathbb J}$ and $\\Delta$ depend crucially upon which choice is made. Hence, the main criterion is whether or not the model is effective at describing the physical properties of the system under study.\n\n \\subsection{Analyticity Domain}\n \\label{visc.ssect-holom}\n\n\\noindent Using the assumptions made in Section~\\ref{visc.ssect-fspace}, the operator $\\{1+\\tau(\\zeta+ \\omega_j {\\mathbb J})\\}^{-1}$ has a spectrum of isolated eigenvalues (each which infinite multiplicity) given by $\\{1+\\tau(\\zeta + \\imath\\omega_j\\ell)\\}^{-1}$ with $\\ell\\in{\\mathbb Z}$. In particular, it has poles at\n\n$$\\zeta_\\ell= -\\left(\\frac{1}{\\tau} + \\imath \\ell \\omega_j\\right)\\,.\n$$\n\n\\noindent As the real valued random variable $\\omega_j$ varies, though, the pole position describes a subset of the vertical line $\\Re{\\zeta}=-1\/\\tau$, the extend of which depends upon the support of the probability distribution of $\\omega_j$. Hence after taking the average over $\\omega_j$, this line becomes a cut in the $\\zeta$-complex plane. It is convenient to introduce the following functions\n\n\\begin{equation}\n\\label{visc.eq-aell}\na_\\ell(\\zeta) ={\\mathbb E}\n \\left\\{\n \\frac{1}{1+\\tau(\\zeta +\\imath\\omega_j\\ell)}\n \\right\\}\\stackrel{\\Re(\\zeta)\\uparrow +\\infty}{\\simeq}\n \\frac{1}{1+\\tau\\zeta}+ \n O\\left(\\frac{1}{(1+\\tau\\zeta)^2}\\right)\\,,\n \\hspace{1.5cm}\n \\ell\\in{\\mathbb Z}\\,.\n\\end{equation}\n\n\\noindent In view of the definition of the Laplace transform (see eq.~(\\ref{visc.eq-lapcorr})), it implies that only the part $\\Re(\\zeta)>-1\/\\tau$ is relevant for this analysis. Hence\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 1: } {\\em ${\\mathbb A}(\\zeta)$ and the $a_\\ell(\\zeta)$'s are complex analytic in the $\\zeta$-complex plane in the domain $\\Re(\\zeta)>-1\/\\tau$.}\n\n\\vspace{.1cm}\n\n\\noindent Since ${\\mathbb J}$ is anti-selfadjoint, the operator ${\\mathbb A}(\\zeta)$ is {\\em normal}, namely it commutes with its adjoint. It can actually be expressed in terms of the eigenprojections $\\Pi_\\ell$ using its {\\em spectral decomposition}\n\n\\begin{equation}\n\\label{visc.eq-spdecomp}\n{\\mathbb A}(\\zeta)= \\sum_{\\ell\\in{\\mathbb Z}} a_\\ell(\\zeta)\\;\\Pi_\\ell\\,.\n\\end{equation}\n\n\\noindent Consequently its norm is given by the maximum of the modulus of its eigenvalues, namely, using $|{\\mathbb E}(X)|\\leq {\\mathbb E}(|X|)$, \n\n\\begin{equation}\n\\label{visc.eq-normA}\n\\|{\\mathbb A}(\\zeta)\\|=\n \\sup_{\\ell\\in{\\mathbb Z}}\n \\left|\n {\\mathbb E}_\\omega\\left\\{\n \\frac{1}{1+\\tau(\\Re{\\zeta}+\\imath (\\Im{\\zeta}+\\omega_j\\ell))}\n \\right\\}\n \\right|\\leq \n \\frac{1}{|1+\\tau\\Re(\\zeta)|}\\,.\n\\end{equation}\n\n\\noindent Going back to eq.~(\\ref{visc.eq-ev4}), the Laplace transform of the stochastic evolution shows a singularity at values of $\\zeta$ for which the spectrum of ${\\mathbb A}(\\zeta)\\exp\\{\\kB T \\Delta\/2m\\}$ contains the number $1$. Since the product of two operators might not commute, the computation of the spectrum might be difficult. However, he Laplacian is always invariant by rotation in any dimension, so that\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 2: } {\\em the operators ${\\mathbb J}$ and $\\Delta$ commute.}\n\n\\vspace{.1cm}\n\n\\noindent More precisely, using the polar coordinates in $2D$ gives\n\n$$\\Delta= \\frac{1}{r}\\,\\frac{\\partial}{\\partial r}\n \\left(r\\frac{\\partial}{\\partial r}\\right)+ \\frac{{\\mathbb J}^2}{r^2}=\n -\\left({\\widehat{p}}_r^2+\\frac{-{\\mathbb J}^2}{r^2}\\right))\\,,\n$$\n\n\\noindent where ${\\widehat{p}}_r$ is the analog of the (selfadjoint) {\\em radial momentum}. On the eigensubspace $\\Pi_\\ell{\\mathcal H}$, the operator $-{\\mathbb J}^2$ takes on the value $\\ell^2$. So that the eq.~(\\ref{visc.eq-ev4}) becomes\n\n\\begin{equation}\n\\label{visc.eq-ev5}\n{\\mathscr L} P_\\zeta = \\tau\\sum_{\\ell\\in{\\mathbb Z}}\n \\frac{1}{1-a_\\ell(\\zeta) e^{-({\\widehat{p}}_r^2+\\ell^2\/r^2)}}\\,\n a_\\ell(\\zeta) \\Pi_\\ell\\,.\n\\end{equation}\n\n\\noindent Since the operator ${\\widehat{p}}_r^2+\\ell^2\/r^2$ is known to admit $[0,\\infty)$ as its spectrum, the {\\em l.h.s.} is analytic in $\\zeta$, in the subdomain of $\\Re(\\zeta)>-1\/\\tau$ made of values for which $a_\\ell(\\zeta)\\notin [1,+\\infty)$ for all the $\\ell$'s that are relevant for the viscosity formula. Without knowing more about the distribution of the $\\omega_n$'s it is not possible to go further.\n\n \\subsection{An explicit case}\n \\label{visc.ssect-expl}\n\n\\noindent There is a distribution of the pulsation that permits do get a closed formula for the $a_\\ell$'s. Namely we assume that the pulsation is uniformly distributed on some finite interval $I$. Since ${\\mathbb E}(\\omega_n)=\\omega$ and $\\Var(\\omega_n)=\\sigma$ (see eq.~(\\ref{visc.eq-omega})), it follows that $I=[\\omega-\\sigma\\sqrt{3}, \\omega+\\sigma\\sqrt{3}]$. This gives\n\n\\begin{equation}\n\\label{visc.eq-aell1}\na_\\ell(\\zeta)= \\frac{1}{2\\imath\\sqrt{3} \\ell \\sigma\\tau}\\,\n \\ln\\left\\{\n \\frac{1+\\tau(\\zeta+\\imath\\ell\\omega)+\\imath\\sqrt{3}\\ell\\sigma\\tau}\n {1+\\tau(\\zeta+\\imath\\ell\\omega)-\\imath\\sqrt{3}\\ell\\sigma\\tau}\n \\right\\}\\,.\n\\end{equation}\n\n\\noindent It follows that $a_0=0$. Changing $\\ell$ into $-\\ell$ and $\\omega$ into $-\\omega$ do not change $a_\\ell$. Hence there is no loss of generality in assuming that $\\ell \\geq 1$. As long as $\\Im(\\zeta)\\neq -\\ell \\omega$, $\\Im\\left(a_\\ell(\\zeta)\\right)\\neq 0$ so that $1-a_\\ell(\\zeta)e^{-p}\\neq 0$ for $p\\geq 0$, so the Laplace transform of $P_t$ is well defined. To find the positions of the singularities it is therefore necessary to assume that $(\\zeta+\\imath\\ell\\omega)=\\Re(\\zeta)\\in {\\mathbb R}$. From the definition of $a_\\ell$ in eq.~(\\ref{visc.eq-aell}), the branch of the logarithm in eq.~(\\ref{visc.eq-aell1}) is such that as $\\Re(\\zeta)\\to +\\infty$, it satisfies the asymptotics given in eq.~(\\ref{visc.eq-aell}). Hence, using polar coordinates in the complex plane, this gives\n\n$$1+\\tau\\Re(\\zeta)+\\imath\\sqrt{3}\\ell\\sigma\\tau=\n \\rho e^{\\imath \\theta}\\,,\n \\hspace{2cm}\n \\tan{\\theta}= \\frac{\\sqrt{3}\\ell\\sigma\\tau}{1+\\tau\\Re(\\zeta)}\\,.\n$$\n\n\\noindent It ought to be remarked that, in the domain of analyticity, $1+\\tau\\Re(\\zeta)>0$. With the asymptotics for $\\Re(\\zeta)$ large, since $\\ell\\geq 1$ the definition of $\\theta$ implies $0<\\theta<\\frac{\\pi}{2}$. To summarize,\n\n\\begin{equation}\n\\label{visc.eq-aell2}\na_\\ell(\\zeta)= \\frac{\\theta}{\\sqrt{3}\\ell\\sigma\\tau}\\,,\n \\hspace{2cm}\n \\tan{\\theta}= \\frac{\\sqrt{3}\\ell\\sigma\\tau}{1+\\tau\\Re(\\zeta)}\\,,\n \\hspace{2cm}\n 0<\\theta<\\frac{\\pi}{2}\\,.\n\\end{equation}\n\n\\noindent The analyticity condition $a_\\ell\\neq [1,+\\infty)$ imposed on the Laplace transform implies, whenever $\\zeta+\\imath\\ell\\omega=\\Re(\\zeta)$, that \n\n(i) if $\\sqrt{3}\\ell\\sigma\\tau\\geq \\pi\/2$, then $a_\\ell(\\zeta)\\neq [1,+\\infty)$ for $\\Re(\\zeta)>-1\/\\tau$;\n\n(ii) If $\\sqrt{3}\\ell\\sigma\\tau< \\pi\/2$, then in order that $a_\\ell(\\zeta)\\neq [1,+\\infty)$, it is necessary that $\\theta<\\sqrt{3}\\ell\\tau\\sigma<\\pi\/2$; since the function $\\theta\\mapsto \\tan{\\theta}$ is increasing on $[0,\\pi\/2)$ then\n\n\\begin{equation}\n\\label{visc.eq-maxtime}\n\\Re(\\zeta)>-\\frac{1}{\\tau_M}\\,,\n \\hspace{2cm}\n \\frac{\\tau_M}{\\tau} =\n \\left(\n 1-\\frac{\\sqrt{3}\\ell\\tau\\sigma}{\\tan{\\sqrt{3}\\ell\\tau\\sigma}}\n \\right)^{-1}\n\\end{equation}\n\n\\noindent In other words\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 3: } {\\em The Maxwell time is given, in terms of $\\tau=\\tlc$ by the following formulae}\n\n\\begin{equation}\n\\label{visc.eq-maxtime2}\n\\frac{\\tau_M}{\\tlc} =\n\\left\\{\n\\begin{array}{ll}\n1 & \\mbox{\\rm if}\\;\\;\\sqrt{3}\\ell\\tau\\sigma\\geq \\pi\/2\\\\\n\\left(\n 1-\\frac{\\sqrt{3}\\ell\\tau\\sigma}{\\tan{\\sqrt{3}\\ell\\tau\\sigma}}\n \\right)^{-1}>1 & \\mbox{\\rm otherwise}\n\\end{array}\\right.\n\\end{equation}\n\n\\vspace{.5cm}\n\n\\section{Interpreting the Results}\n\\label{visc.sect-results}\n\n\\noindent From the mathematical computations made in the previous Section~\\ref{visc.ssect-expl}, it follows that there are two regimes concerning the Maxwell time, both being controlled by the dimensionless parameter\n\n$$K = \\sqrt{3}\\tlc\\sigma\\,,\n$$\n\n\\noindent which is proportional to the variance of the distribution of the pulsations $\\omega_n$, representing the curvature of the local potential energy, near its mechanical equilibrium, where the atom is located. In order to draw practical predictions for a liquid material, some physical hypothesis are required. In particular, {\\em it is highly expected that the parameters $\\tlc$ and $\\sigma$ depend on the temperature of the liquid}. If so, the following conclusion can be made\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 4: } {\\em Depending upon the material, there may or may not exist a cross-over temperature $T_{co}$, such that (a) for $T\\geq T_{co}$, the anankeons dominate and the Maxwell time coincides with the local configuration time, while, (b) if $TW\\,, \n \\hspace{1cm}\n K_\\infty >\\frac{\\pi}{2}\\,,\n \\hspace{1cm}\\Rightarrow\\hspace{1cm}\n \\kB T_{co}= \\frac{W_v-W}{\\ln(2 K_\\infty\/\\pi)}\n\\end{equation}\n\n\\noindent The first consequence of these hypothesis is that the difference between strong and fragile glassy materials \\cite{Ang95} can be expressed by the conditions\n\n\\begin{equation}\n\\label{visc.eq-strfra}\n\\left\\{\n\\begin{array}{ll}\nW_v\\leq W & \\mbox{\\rm for strong liquids}\\\\\nW_v>W & \\mbox{\\rm for fragile liquids}\n\\end{array}\\right.\n\\end{equation}\n\n\\noindent The second consequence is that {\\em the ratio between the Maxwell and the local configuration times diverges exponentially fast below the cross-over temperature} (see eq.~(\\ref{visc.eq-maxtime2}))\n\n\\begin{equation}\n\\label{visc.eq-ratio}\n\\frac{\\tau_M}{\\tlc}\\;\\;\n \\stackrel{T\\ll T_{co}}{\\sim}\\;\\;\n \\frac{1}{\\ell^2K(T)^2} =\n \\frac{e^{2(W_v-W)\/\\kB T}}{\\ell^2 K_\\infty^2}\\,.\n\\end{equation}\n\n\\noindent Such a law would explain why the viscosity diverges so fast near the glass transition temperature.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}