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. Garrido--Marro, Lecture Notes in Physics, Springer Verlag, {\bf
448}, p. 271--311, 1995.
\bibitem{[Ga96a]} Gallavotti, G.: {\it New methods in nonequilibrium gases and
fluids}, Open Systems and Information Dynamics, Vol. {\bf 6}, 101--136, 1999
(original in chao-dyn \#9610018).
\bibitem{[Ga96b]} Gallavotti, G. {\it Chaotic hypothesis: Onsager reciprocity
and
fluctuation dissipation theorem},
Journal of Statistical Phys., {\bf 84}, 899--926, 1996.
\bibitem{[Ga97]} Gallavotti, G.:
{\it Dynamical ensembles equivalence in fluid mechanics},
Physica D, {\bf 105}, 163--184, 1997.
\bibitem{[Ga99a]} Gallavotti, G.:
{\it Statistical mechanics. A short treatise}, p. 1--345,
Springer Verglag, 1999.
\bibitem{[Ga99b]} Gallavotti, G.:
{\it Fluctuation patterns and conditional reversibility in
nonequilibrium systems}, Annales de l' Institut H. Poincar\'e,
{\bf70}, 429--443, 1999.
\bibitem{[Ga99c]} Gallavotti, G.:
{\it A local fluctuation theorem}, Physica A,
{\bf 263}, 39--50, 1999. And: {\it Chaotic Hypothesis and
Universal Large Deviations Properties}, Documenta Mathematica,
extra volume ICM98, vol. I, p. 205--233, 1998, also in chao-dyn
9808004.
\bibitem{[GC95]} Gallavotti, G., Cohen, E.G.D.: {\it Dynamical
ensembles in non-equilibrium statistical mechanics}, Physical Review
Letters, {\bf74}, 2694--2697, 1995. Gallavotti, G., Cohen,
E.G.D.: {\it Dynamical ensembles in stationary states}, Journal of
Statistical Physics, {\bf 80}, 931--970, 1995.
\bibitem{[GP99]} Gallavotti, G., Perroni, F.: {\it An experimental test of the
local fluctuation theorem in chains of weakly interacting Anosov
systems}, mp$\_$arc \#99-???.
\bibitem{[Ge98]} Gentile, G.: {\it Large deviation rule for Anosov flows},
Forum Mathematicum, {\bf10}, 89--118, 1998.
\bibitem{[LL71]} Landau, L., Lifchitz, E.: {\sl M\'ecanique
des fluides}, MIR, Mosca, 1971.
\bibitem{[OM53]}
Onsager, L., Machlup, S.: {\it Fluctuations and irreversible
processes}, Physical Review, {\bf91}, 1505--1512, 1953. And Machlup,
S., Onsager, L.: {\it Fluctuations and irreversible processes},
Physical Review, {\bf91}, 1512--1515, 1953.
\bibitem{[RS99]} Rondoni, L., Segre, E.: {\it Fluctuations in two dimensional
reversibly damped turbulence}, Nonlinearity, {\bf12}, 1471--1487, 1999.
\bibitem{[Ru78a]} Ruelle, D.: {\it Sensitive dependence on initial conditions
and turbulent behavior of dynamical systems}, Annals of the New York
Academy of Sciences, {\bf356}, 408--416, 1978.
\bibitem{[Ru78b]} Ruelle, D.: {\it What are the measures describing
turbulence?}, Progress of Theoretical Physics, (Supplement) {\bf64},
339--345, 1978.
\bibitem{[Ru96]} Ruelle, D.: {\it Positivity of entropy production in
nonequilibrium statistical mechanics}, Journal of Statistical Physics,
{\bf 85}, 1--25, 1996. And Ruelle, D.: {\it Entropy production in
nonequilibrium statistical mechanics}, Communications in Mathematical
Physics, {\bf189}, 365--371, 1997.
\bibitem{[Ru99a]} Ruelle, D.: {\it Smooth dynamics and new theoretical
ideas in nonequilibrium statistical mechanics}, Journal of
Statistical Physics, {\bf 95}, 393--468, 1999.
\bibitem{[Ru99b]} Ruelle, D.: {\it A remark on the equivalence of isokinetic
and isoenergetic thermostats in the thermodynamic limit},
IHES 1999, to appear in Journal of Statistical Physics.
\bibitem{[Si77]} Sinai, Y.G.: {\sl Lectures in ergodic
theory}, Lecture notes in Mathematics, Prin\-ce\-ton U. Press,
Princeton, 1977.
\end{thebibliography}
\0\revtex\\
\FINE
\end{document}
---------------0003100358991--