Content-Type: multipart/mixed; boundary="-------------0003100358991" This is a multi-part message in MIME format. ---------------0003100358991 Content-Type: text/plain; name="00-106.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-106.keywords" Chaos, Nonequilibrium, Fluid mechanics, Irreversibility, Entropy ---------------0003100358991 Content-Type: application/x-tex; name="paris2000.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paris2000.tex" \documentstyle[twocolumn,prb,aps]{revtex} %\documentstyle[preprint,prb,aps]{revtex} %\documentclass[12pt]{article} \def\revtex{R\raise2pt\hbox{E}VT\lower2pt\hbox{E}X} \let\a=\alpha\let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon \let\z=\zeta \let\h=\eta \let\th=\vartheta\let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau \let\iu=\upsilon \let\f=\varphi\let\ch=\chi \let\ps=\psi \let\o=\omega \let\y=\upsilon \let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda\let\X=\Xi \let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi \let\O=\Omega \let\U=\Upsilon \def\V#1{{\,\underline#1\,}} \let\dpr=\partial \let\io=\infty \let\ig=\int \def\fra#1#2{{#1\over#2}}\def\media#1{\langle{#1}\rangle} \let\0=\noindent \def\tende#1{\ \vtop{\ialign{##\crcr\rightarrowfill\crcr \noalign{\kern-1pt\nointerlineskip} \hglue3.pt${\scriptstyle% #1}$\hglue3.pt\crcr}}\,} \def\otto{\ {\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\ } \def\*{\vskip0.3truecm} \def\lis#1{{\overline #1}} \def\ap{\hbox{\it a priori\ }} \def\ie{\hbox{\it i.e.\ }} \def\={{ \; \equiv \; }} \def\defi{\,{\buildrel def \over =}\,} \def\W#1{#1_{\kern-3pt\lower6.6truept\hbox to 1.1truemm {$\widetilde{}$\hfill}}\kern0pt} \def\uu{{\V u}}\def\kk{{\V k}}\def\xx{{\V x}} \def\cfr{{\it cf.\ }}\def\DD{{\cal D}} \def\*{\vskip0.3cm} \def\FINE{ \0{\it Preprints at: {\tt http://ipparco.roma1.infn.it}\\ \0\sl e-mail: {\tt giovanni.gallavotti@roma1.infn.it} }} \def\Eq(#1){\label{{#1}}}\def\eq(#1){\label{{#1}}} \def\equ(#1){(\ref{{#1}})} \def\CH{chaotic hypothesis\ } \title{Entropy driven intermitency} \author{G. Gallavotti\\ Fisica, Universit\`a di Roma La Sapienza,\\ P.le Moro 2, 00185 Roma, Italia} \vskip1cm \begin{document} \maketitle \begin{abstract} This note reviews some physical aspects of the chaotic hypothesis in nonequilibrium statistical mechanics and attempts at the physical interpretation of the fluctu\-ation theorem as a quantitative intermittency property. \end{abstract} \*\* \0The main assumption, for the foundation of a nonequilibrium statistical mechanics or a theory of developed turbulence, will be: \* \0{\bf Chaotic hypothesis: \it Asymptotic motions of a chaotic system, be it a system of $N$ particles or a viscous fluid, can be regarded as motions of a mixing Anosov system, for the purposes of computing time averages.} \* The point of view goes back to Ruelle, \cite{[Ru78a]}, \cite{[Ru78b]}, and in this specific form has been proposed in \cite{[GC95]}. I shall not define here ``mixing Anosov system'', see \cite{[Ga99a]}. It will suffice to say that Anosov systems are very well understood dynamical systems in spite of being in a sense the most chaotic: they are so well understood that they can be regarded as the {\it paradigm} of chaotic systems much as the harmonic oscillators are the paradigm of regular and orderly motions. This immediately implies the existence and uniqueness of an invariant probability distribution $\m$ which gives the {\it statistics} if the motions of the system in the sense \begin{equation} \lim_{T\to\io} T^{-1}\ig_0^T F(S_t x)\,dt=\ig \m(dy)\,F(y)\Eq(1)\end{equation} % for {\it almost all} initial data, \ie outside a set of $0$ volume in phase space (volume being measured in the ordinary sense of the Lebesgue measure). Here $S_t$ denotes the time evolution map, solution of the differential equations of motion, and $\m$ is called the {\it SRB distribution}. \* Particular interest will be reserved to {\it time reversible} systems: \ie systems for which there is an isometry $I$ of phase space such that \begin{equation} I^2=1,\quad and\quad I S_{-t}=S_t I\Eq(2) \end{equation} Denoting $\dot x= f(x)$ the equations of motion and $\s(x)$ the divergence ${-\rm div\,} f(x)=-\sum_j \dpr_{x_j} f_j(x)$ a key quantity to study will be \begin{equation} \s_+={\rm time\ average\ of\ } \s\Eq(3)\end{equation} % that shall be called {\it average entropy production rate} or {\it average phase space contraction rate}; it will be assumed that $\s_+>0$ (this quantity is, in general, non negative, \cite{[Ru96]}). The main result on $\s(x)$ concerns the probability distribution in the stationary state $\m$ of the quantity (an observable, \ie a function of the phase space point $x$) \begin{equation} p= T^{-1}\ig_{T/2}^{T/2}\fra{\s(S_t x)}{\s_+}\,dt\Eq(4)\end{equation} % which will be called the {\it $T$--average dimensionless entropy creation rate}. The result is a symmetry property of its probability distribution $\p_T(p)$: which can be written in the form $\p_T(p)\defi \,const\, e^{\z(p)T+o(T)}$, defining implicitly $\z(p)$, as it follows, on general grounds, from the theory of mixing Anosov systems, \cfr \cite{[Si77]}, see\cite{1} for a more mathematical statement. The function $\z(p)$ is called the {\it large deviation rate} for the observable $p$. Then (see \cite{[Ge98]}: the delicate continuous time extension of a similar result\cite{[GC95]} for discrete time) \* \0{\bf Fluctuation theorem: \it The ``rate function'' $\z(p)$ verifies \begin{equation} \z(-p)=\z(p)-p\,\s_+\Eq(5)\end{equation} % which is a parameterless relation, valid under the above hypotheses of chaoticity and of time reversibility. It is a ``mechanical'' identity valid for systems with arbitrarily many particles (\ie $N=1,2,\ldots 10^{23}, \ldots$).} \* Mathematically the above result holds for mixing Anosov flows which are time reversible. The chaotic hypothesis extends the result to rather general systems: of course for such systems it is no longer a theorem much in the same way as the consequences of the ergodic hypothesis in equilibrium statistical mechanics are not theorems for most systems to which they are applied. Therefore we can say that the chaotic hypothesis implies {\it in equilibrium} (when the equations of motion are Hamiltonian and therefore $\s(x)\=0$) the ergodic hypothesis (quite clearly): hence it implies classical statistical mechanics, starting with Boltzmann's heat theorem which says that $(dU+p dV)/T$ is an exact differential (with $U,p,V,T$ defined as time averages of suitable mechanical quantities, see \cite{[Ga99a]}). Likewise, out of equilibrium and in reversible systems, the chaotic hypothesis implies a general relation that is given by the fluctuation theorem, valid for systems with arbitrarily large numbers of particles. Both the heat theorem and the fluctuation theorem are ``universal'', \ie parameterless, system independent relations. They could perhaps be considered a curiosity for $N$ small, but certainly the first, at least, is an important property for $N=10^{23}$. \* It becomes, at this point, clear that one should attempt at an interpretation of the fluctuation theorem \equ(5). It is however convenient to analyze it first in more detail to get some familiarity with the physical questions that it is necessary to address to grasp its meaning. We begin with an attempt at justifying the name ``{\it entropy creation rate}'' for $\s_+$. Without too many comments we quote here the simple but relevant remark, \cite{[An82]}, that the so called ``{\it Gibbs entropy}'' of an evolving probability distribution which starts, at $t=0$, as absolutely continuous with density $\r(x)$ on phase space has the following property \begin{equation} -\fra{d}{dt}\ig \r_t(x)\log \r_t(x) \,dx=\ig\s(x)\,\r_t(x)\,dx\Eq(6)\end{equation} % where $\r_t(x)=\r(S_{-t}x)\det\fra{\dpr S_{-t}x}{\dpr x}$ is the evolving phase space density (here ${\dpr S_{-t}x}/{\dpr x}$ is the matrix of the derivatives of the time evolution map $S_t$). Since the r.h.s. of \equ(6) formally tends as $t\to\io$ to the average of $\s$ with respect to the distribution $\m$, \ie to $\s_+$, we realize that \equ(6) is a possible justification of the name used for $\s_+$. \* Before proceeding it is also necessary to understand the role of the reversibility assumption in order to see whether it is a serious limitation in view of a possible physical interpretation and physical interest of the fluctuation theorem. Here one can argue that in many cases an irreversible system is ``{\it equivalent}'' to a reversible one: in fact several reversibility conjectures have been proposed, see \cite{[Ga95]}, \cite{[GC95]}, \cite{[Ga96a]} (Sect. 2 and 5), \cite{[Ga96b]} (Sec. 8), \cite{[Ga97]}, \cite{[Ga99a]} (Sec. 9.11), \cite{[Ru99b]}. Rather than trying to be general I shall consider an example, and analyze a Navier--Stokes fluid, in a periodic container of side $L$, subject to a force with intensity $F$ and with viscosity $\n$. If $R=F L^3 \n^{-2}$ is the ``{\it Reynolds number}'', $p$ is the pressure field, the density is $\r=1$, and $\V g$ is a force field of intensity $1$ the equations are \begin{eqnarray} &\dot{{\V u}}=- R\, \W u\cdot\W\dpr \,\V u+\D\,\V u+\V g- \V\dpr p\nonumber\\ &\V\dpr\cdot\V u=0 \Eq(7)\end{eqnarray} % which are {\it irreversible} equations. According to the K41 theory (Kolmogorov theory, see \cite{[LL71]}) the above equations can be truncated retaining only a few harmonics of the field $\V u$: \ie replacing $\V u(\xx)= \sum_\kk \uu_\kk \, e^{i\kk\cdot\xx}$ by $\V u(\xx)= \sum_{|\kk|0$: this should happen with a frequency $e^{-\lis\s_+\,V_0\,T}$ giving us access to $\lis\s_+$. Given the special role that entropy generation plays it is very tempting to think that there might be many currents $J$ associated with the system: for each of them one could define $p= J_T/J_+$; then the new quantity $p$ {\it has the same probability distribution as the variable with the same name that we have associated with the entropy production}. This is true at least for the special case $p=-1$ as just noted: if true in general then we could have easily access to the function $\z(p)$ for several values of $p$. Hence analysizing this ``universality'' property in special models seems to be an interesting problem. For a general review on recent developments in non\-equilibrium statistical mechanics see \cite{[Ru99a]}. \* \0{\bf Acknowledgements: \it This is a contribution to the Proceedings of ``Inhomogeneous random systems'' January 25-26, 2000, (Universit\'e de Cergy-Pontoise, Paris); supported partially by ``Cofinanziamento 1999''.} \begin{thebibliography}{} \bibitem{1} This means, precisely, that the probability $P$ that $p$ is in an interval of size $\d$ around the value $p$ is such that $\lim_{T\to\io} T^{-1}\log P= \sup_{p}\z(p)$ with the supremum in the interval $\d$, for all $p\in (-p_{\max},p_{\max})$ and $-\io$ if $p\not \in [-p_{\max},p_{\max}]$. However I shall briefly say that the probability density of the variable $p$ is $\p_T(p)={\,const\,} e^{\z(p)T}$. \bibitem{[An82]} Andrej, L.: {\it The rate of entropy change in non--Hamiltonian systems}, Physics Letters, {\bf 111A}, 45--46, 1982. \bibitem{[BGG97]} Bonetto, F., Gallavotti, G., Garrido, P: {\it Chaotic principle: an experimental test}, Physica D, {\bf 105}, 226--252, 1997. \bibitem{[CL98]} Ciliberto, S., Laroche, C.: {\it An experimental verification of the Gallavotti--Cohen fluctuation theorem}, Journal de Physique, {\bf8}, 215--222, 1998. \bibitem{[Ga95]} Gallavotti, G.: {\it Ergodicity, ensembles, irreversibility in Boltzmann and beyond}, Journal of Statistical Physics, {\bf 78}, 1571--1589, 1995. And {\it Topics in chaotic dynamics}, Lectures at the Granada school, ed. 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