Content-Type: multipart/mixed; boundary="-------------9903300001295" This is a multi-part message in MIME format. ---------------9903300001295 Content-Type: text/plain; name="99-88.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-88.comments" FM 99-2 ---------------9903300001295 Content-Type: text/plain; name="99-88.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-88.keywords" Chaos, chaotic hypothesis, large deviations, fluctuations, time reversal ---------------9903300001295 Content-Type: application/x-tex; name="fc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="fc.tex" \documentstyle[twocolumn,pre,aps]{revtex} \def\btt#1{{\tt$\backslash$#1}} %\let\bf=\mathbf \let\cal=\mathcal \begin{document} \relax \draft \preprint{RRU/3-cg} \title{Note on Two Theorems in Nonequilibrium Statistical Mechanics} \author{E. G. D. Cohen${}^1$, G. Gallavotti${}^2$} \address{ ${}^1$The Rockefeller University, New York, NY 10021, USA\\ ${}^2$Fisica, Universit\'{a} di Roma, ``La Sapienza'', 00185 Roma, Italia} \date{\today} \maketitle \begin{abstract} An attempt is made to clarify the difference between a theorem derived by Evans and Searles in 1994 on the statistics of trajectories in phase space and a theorem proved by the authors in 1995 on the statistics of f\/luctuations on phase space trajectory segments in a nonequilibrium stationary state. \end{abstract} \pacs{47.52, 05.45, 47.70, 05.70.L, 05.20, 03.20} \narrowtext Recently a Fluctuation Theorem (FT) has been proved by the authors (GC), \cite{[GC95]}, for f\/luctuations in nonequilibrium stationary states. Considerable confusion has been generated about the connection of this theorem and an earlier one by Evans and Searles (ES), \cite{[ES94]}, so that it seemed worthwhile to try to clarify the present situation with regards to these two theorems. In a paper in 1993 by Evans, Cohen and Morriss \cite{[ECM93]}, theoretical considerations lead them to a computer experiment about the statistical properties of the f\/luctuations of a shear stress model (viscous current) - or the related entropy production rate - in a thermostatted sheared viscous f\/luid (plane Couette f\/low) in a nonequilibrium stationary state. The Fluctuation Relation found in the simulation \cite{[ECM93]}, reads in current notation, \cite{[GC95]}: % \begin{equation} \frac{\pi_\tau(p)}{\pi_\tau(-p)} \simeq e^{\tau \sigma_+p} \label{1} \end{equation} % Here $\pi_\tau(p)$ is the probability of observing an average phase space contraction rate (which in the models considered has the interpretation of average entropy production rate) of size $p\, \sigma_+$ on one of many segments of duration $\tau$ on a long phase space trajectory of the dynamical system modeling the shearing f\/luid in a nonequilibrium stationary state; here $\sigma(x)$ will denote the phase space contraction rate near a phase point $x$ ({\it i.e.} the divergence of the equations of motion) and $\sigma_+$ is the average phase space contraction rate over positive infinite times so that $p$ is a dimensionless characterization of the phase space contraction (with time average $1$). The approximation within which eq. (\ref{1}) was observed was very convincing \cite{[ECM93]}. Under suitable assumptions, see below, a more precise formulation of eq. (\ref{1}) was derived in \cite{[GC95]}: % \begin{equation} \lim_{\tau \rightarrow \infty} \frac{1}{\tau \sigma_+} {\rm{ln}} \frac{\pi_\tau(p)}{\pi_\tau(-p)} = p \label{2} \end{equation} % where the validity of the f\/luctuation relation eq. (\ref{1}) for asymptotically long times $\tau$ is more clearly expressed. Later several other computer experiments have confirmed the relation eq. (\ref{2}), \cite{[BGG97]}, \cite{[BCL98]}, \cite{[LLP98]}. The original computer experiment, \cite{[ECM93]}, was inspired by a theoretical argument for the relative probabilities to find a phase space trajectory segment of length $\tau$ in a state $x$ with phase space contraction rate $p$ and in a state $x'$ with rate $-p$. These theoretical considerations lead to the correct prediction eq. (\ref{1}), which was confirmed by the - independently of the theory - carried out computer experiment. Evans and Searles \cite{[ES94]} gave a derivation of a theorem which had a similar form as eq. (\ref{1}). More precisely: let $E_p$ be the set of initial conditions of a dynamical system for phase space trajectories along which the phase space contraction is $e^{-p\sigma_+ T}$ in a time $T$. We denote by $\mu_L(E_p)$ its Liouville measure. Similarly, let $\mu_L(E_{-p})$ be the Liouville measure of the corresponding set of phase space trajectories along which the phase space contraction in time $T$ is $e^{p\sigma_+ T}$. Quite generally, and in all models considered in the literature relevant here, $E_{-p}=I S_TE_p$, if $S_t$ is the time evolution (Liouville) operator of the system, and if $I$ denotes the time reversal operation, so that $t \rightarrow S_t x$ is the phase space trajectory over time $t$ starting at $x$ at $t=0$. Hence $E_{-p}$, the set of points around which phase space contracts at rate $-p \sigma_+T$, is obtained by evolving forward over a time $T$ those in $E_p$ (which would contract by $p \sigma_+T$) and then inverting the velocities by the time reversal operator $I$. In fact, the sets $E_p$ and $E_{-p}$ are those considered by Evans and Searles in \cite{[ES94]}. Then the proof in \cite{[ES94]} is the following: % \begin{eqnarray} \frac{\mu_L(E_p)}{\mu_L(E_{-p})} & = & \frac{\mu_L(E_p)}{\mu_L(I S_TE_p)} = \cr = \frac{\mu_L(E_p)}{\mu_L(S_TE_p)} & = & \frac{\mu_L(E_p)}{\mu_L(E_p) e^{-p \sigma_+T}} = e^{p \sigma_+T} \label{3} \end{eqnarray} % where one has used that the Liouville distribution is time reversal invariant, {\it i.e.} $\mu_L(E)\equiv \mu_L(I E)$ (although it is not stationary) to get the second equality as well as the definition of phase space contraction in the third equality. The arbitrary time interval $T$ includes the short times referring to the transient behavior of the system before reaching the nonequilibrium stationary state. In the derivation of eq. (\ref{3}) only time reversal symmetry is used. Later, \cite{[ES98]}, it was argued that under this assumption alone, eq. (\ref{3}) also holds in the nonequilibrium stationary state $\mu_\infty$, since eq. (\ref{3}) is valid for any $T$ and the Liouville distribution $\mu_L$ would evolve in a sufficiently long time $T$ into a distribution $S_T\mu_L$ arbitrarily close to a nonequilibrium stationary state $\mu_\infty$. Therefore the eq. (\ref{3}) was reinterpreted in \cite{[ES98]} and asserted to be identical to eq. (\ref{2}), which, however, refers to the statistics of trajectory segments, along a trajectory in a chaotic nonequilibrium stationary state $\mu_\infty$, {\it not to the statistics of independent trajectory histories emanating from the initial Liouville distribution} $\mu_L$ under the time reversibility assumption. In 1995 the authors proved eq. (\ref{2}) based on a dynamical assumption, called {\it Chaotic Hypothesis} (CH), which assured strong chaoticity (``Anosov system-like behavior'') for the systems for which eq. (\ref{2}) held. In that work the name Fluctuation Theorem (FT) was first introduced for eq. (\ref{2}), and was proposed as an explanation for the experimental result eq. (\ref{1}). We will call this the GCFT. It is worthwhile to emphasize again that, while the right hand sides of the eqs. (\ref{1}) and (\ref{3}) look very similar, they, as well as the left hand sides of these equations, really refer to entirely different physical situations. The eq. (\ref{3}), \cite{[ES94]}, holds for any $T$ on trajectories with initial data sampled from the Liouville distribution at $t=0$ and it can be considered as a simple, but interesting, consequence, for reversible systems, of the very definition of phase space contraction. We will call it here the ESI, where the I refers to ``identity''. The ESI is much more general than the FT in eq. (\ref{2}), which needs, {\it in addition} to phase space contraction ($\sigma_+>0$) and time reversal symmetry, {\it also} the Chaotic Hypothesis. The proof of the ESI, fully described in eq. (\ref{3}) above, is identical in essence to the proof in \cite{[ES94]} which is much more involved. In order to illustrate the fundamental difference between the two theorems we first give an example of a case where the more general ESI eq. (\ref{3}) holds, while the GCFT eq. (\ref{2}) does not. To that end we consider a single charged particle in a periodic box, with charge $e$ moving in an electric field ${\bf{E}}$, i.e. a Lorentz gas without scatterers, and subject to a Gaussian thermostat (to obtain a nonequilibrium stationary state): % \begin{equation} {\bf{\dot{q}}} = {\bf{p}}, \qquad {\bf{\dot{p}}} = e {\bf{E}} - \alpha {\bf{p}} \label{4} \end{equation} % where the ``thermostat'' force $-\alpha {\bf p}$, with $\alpha = \frac{ e {\bf{E}}\cdot{\bf p}}{|\bf p|^2}$, assures the reaching of a nonequilibrium stationary state of this system. In this case one can solve explicitly the trivial equations of motion eq. (\ref{4}) and check that the Liouville distribution $\mu_L$ indeed evolves towards a stationary state $\mu_\infty$, which is simply a state in which the particle moves with constant speed parallel to $\bf E$. The ESI eq. (\ref{3}) will (of course) hold for the phase space trajectories of this system sampled with the initial Liouville distribution $\mu_L$, but it will not be a f\/luctuation theorem, since there are no f\/luctuations. Also, GCFT's eq. (\ref{2}) will not hold for the phase space trajectory segment f\/luctuations of this system, which is not a contradiction because the system is not chaotic. {}From this simple example follows that the two theorems cannot be equivalent, and the validity of eq. (\ref{3}) cannot imply much, {\it without extra assumptions}, about the f\/luctuations (absent in this case) in the stationary state. Note that eq. (\ref{3}) is an identity which is always valid in the systems considered. The system of eq. (\ref{4}) is therefore a counterexample to the statement, that eq. (\ref{3}) implies eq. (\ref{2}), {\it i.e.} to the statement, \cite{[ES98]}, that ESI implies GCFT. Second, and more interestingly, one can try to derive the GCFT eq. (\ref{2}) from the more general ESI eq. (\ref{3}). One could then try to proceed as follows. First one would need to show that on a subsequent trajectory segment of length $\tau$, {\it after} time $T$, the ratio of the probabilities of finding a phase space contraction of $+p\sigma_+\tau$ to that of finding $-p\sigma_+\tau$ over this segment of length $\tau$, would be given by $e^{p\sigma_+\tau}$. Here $p\sigma_+\tau$ is any preassigned value of the phase space contraction. However, eq. (\ref{3}) gives no information whatsoever about those points which after a time $T$ evolve into points which in the next $\tau$ units of time show a phase space contraction $\pm p\sigma_+\tau$. In other words, the ESI does not contain the detailed information needed to derive the GCFT. If one adds the Chaotic Hypothesis to the time reversal symmetry assumptions made about the dynamical system in the ESI, one could use Sinai's theorem, \cite{[Si72]}, to assert that such a system, starting from the initial Liouville distribution $\mu_L$, will indeed approach a chaotic nonequilibrium stationary distribution $\mu_\infty$ supported (however) on a fractal attractor $A$ with $0$ Liouville measure $\mu_L(A)=0$. This is, in this case, the SRB distribution, $\mu_{SRB}$, of the system, which was used in \cite{[GC95]}. However, for a proof of the GCFT, details of the SRB distribution are needed, which contain just the details considered in \cite{[GC95]}. That is, one has to make an appropriate (Markov) partition of the phase space and assign weights to the cells of increasingly finer partitions leading, to the SRB distribution. This then allows one to assign appropriate weights to those regions ({\it of $0$ Liouville measure}) in phase space that will give rise, to phase space contractions on trajectory segments $\tau$ of $\pm p\sigma_+\tau$ leading to the GCFT. A more advanced comparison between the ESI and GCFT requires a more quantitative statement of the latter result. Namely, \cite{[GC95]}, eq. (\ref{2}) can be derived from the stronger relation: % \begin{equation} \frac{\pi_\tau(p)}{\pi_\tau(-p)} = e^{(p\,\sigma_++ O(\frac{T_\infty }{\tau}))\,\tau} \label{5} \end{equation} % where $T_\infty $ is a time scale of the order of magnitude of the time necessary in order that the distribution $S_T\mu_L$, into which the Liouville distribution $\mu_L$ evolves in time $T$, be ``practically'' indistinguishable from the stationary state that we denote by $\mu_\infty$. Here the validity of the f\/luctuation relation (\ref{1}) for asymptotically long times $\tau$ is more clearly expressed. The existence of the time $T_\infty $ and its role in bounding the error term in eq. (\ref{5}) are among the {\it main results of} \cite{[GC95]}. The time $T_{\infty}$ appears in \cite{[GC95]} as the range of the potential that generates the representation of $\mu_\infty$ as a Gibbs state using a symbolic dynamic representation of the SRB distribution on the Markov partition, \cite{[Si72]}. Examining the ESI derivation above, one easily sees that the following % \begin{equation} \frac{(S_T\mu_L)(E_p)}{(S_T)\mu_L(E_{-p})}= \frac{\mu_L(S_{-T}E_p)}{\mu_L(S_{-T}E_{-p})}= e^{(p\,\sigma_++ O(\frac{T}\tau))\,\tau} \label{6} \end{equation} % can also be derived, \cite{[Le]}; it is important to note that the argument leading to eq. (\ref{3}) {\it cannot say more than this}: in particular one must justify why $T$, which in principle should be large, {\it strictly speaking infinite}, so that $S_T\mu_L$ be identifiable with $\mu_{SRB}$, {\it can in fact be taken smaller than $\tau$}. Justifying this requires assumptions (as the above counterexample indicates) like the mentioned Gibbs property of the SRB distribution which is even stronger than requiring exponential decay of correlations (also implied by the CH). Otherwise, without extra assumptions, one has to say that $T$ has to be taken infinite first with the result that eq. (\ref{6}), hence eq. (\ref{3}), {\it becomes empty} in content. Of course one can argue that ``on physical grounds'' $T$ needs not to be taken infinite but just as large as some characteristic time scale for the approach to the attractor: but the precise meaning of this, and the assumptions under which it can be stated, is precisely what needs to be determined particularly because in nonequilibrium systems the attractor $A$ is {\it fractal} with $\mu_L(A)=0$ and one can very well doubt that the $S_T\mu_L$ distribution is {\it ever close enough to the $\mu_\infty$ distribution to allow comparing} eq. (\ref{5}) and (\ref{6}). This requires a convincing argument, were it only because $(S_T\mu_L)(A)=0$. \vspace{7mm} \noindent {\small{\it Acknowledgements:} E.G.D.C. is very much indebted to L. Rondoni for many very clarifying discussions. The authors are also indebted to C. Liverani for suggestions on the proof of eq. (\ref{3}) and to F. Bonetto, J. L. Lebowitz, and L. Rondoni for helpful discussions. E.G.D.C. gratefully acknowledges support from DOE grant DE-FG02-88-ER13847, while G.G. that from Rutgers University's CNR-GNFM, MPI.} \begin{references} \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{[ES94]} Evans, D., Searles, D.: {\it Equilibrium microstates which generate the second law violating steady states}, Physical Review E, {\bf 50E}, 1645--1648, 1994. \bibitem{[ECM93]} Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Probability of second law violations in shearing steady f\/lows}, Physical Review Letters, {\bf 71}, 2401--2404, 1993. \bibitem{[BGG97]} Bonetto, F., Gallavotti, G., Garrido, P.: {\it Chaotic principle: an experimental test}, Physica D, {\bf 105}, 226--252, 1997. \bibitem{[BCL98]} Bonetto, F., Chernov, N., Lebowitz, J.: {\it (Global and local) f\/luctuations of phase space contraction in deterministic stationary nonequilibrium}, Chaos, {\bf8}, 823--833, 1998. \bibitem{[LLP98]} Lepri, S., Livi, R., Politi, P.: Physica D, {\bf{119}}, 140, 1998. \bibitem{[ES98]} Evans, D., Searles, D.: {\it The conjugate f\/luctuation theorem and Green Kubo relations}, preprint, 1998. \bibitem{[Si72]} Sinai, Y.G.: {\it Gibbs measures in ergodic theory}, Russian Mathematical Surveys, {\bf 27}, 21--69, 1972. And {\sl Lectures in ergodic theory}, Lecture notes in Mathematics, Prin\-ce\-ton U. Press, Princeton, 1977. \bibitem{[R98]} Ruelle, D.: {\it Smooth dynamics and new theoretical ideas in non-equilibrium statistical mechanics}, Lecture notes, Rutgers University, mp$\_$arc \$98-770, 1998, in print on Nonlinearity. \bibitem{[Ga95]} Gallavotti, G.: {\it Reversible Anosov maps and large deviations}, Mathematical Physics Electronic Journal, MPEJ, (http:// mpej.unige.ch), {\bf 1}, 1--12, 1995. \bibitem{[Le]} Lebowitz, J.L.: private communication. \end{references} \def\revtex{R\raise2pt\hbox{E}VT\lower2pt\hbox{E}X} \revtex \end{document} ---------------9903300001295--