%
%%% MF95-7, mp_arc@math.utexas.edu # , chao-dyn@xyz.lanl.gov #
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FORMATO
\newcount\mgnf\newcount\tipi\newcount\tipoformule
\newcount\aux\newcount\casa
\mgnf=0 %ingrandimento
\aux=0 %=0 non produce .aux per le formule, =1 si (\aux=1
%impone il formato di BOZZA)
\def\9#1{\ifnum\aux=1#1\else\relax\fi}
\tipoformule=1 %=0 da` numeroparagrafo.numeroformula; se no numero
%assoluto.
\casa=0 %=1 caratteri casarecci (semplificati), =0 no
%%%% INCIPIT
\ifnum\mgnf=0
\magnification=\magstep0
%\hsize=11.5truecm\vsize=19.5truecm%stampa
\hsize=14.5truecm\vsize=21.5truecm%preprint
\parindent=4.pt\fi
\ifnum\mgnf=1
\magnification=\magstep1
\hsize=16.0truecm\vsize=22.5truecm\baselineskip14pt\vglue6.3truecm%mpej
%\hsize=11.5truecm\vsize=19.5truecm\baselineskip14pt%stampa
%\hsize=15.5truecm\vsize=22.truecm\baselineskip=15pt%preprint
\parindent=4.pt\fi
%%%%%%%%%%%
%\overfullrule=10pt
%
%%%%%GRECO%
\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
%%%%%%%%%%%%%%%%%%%%% NUMERAZIONE PAGINE
{\count255=\time\divide\count255 by 60 \xdef\oramin{\number\count255}
\multiply\count255 by-60\advance\count255 by\time
\xdef\oramin{\oramin:\ifnum\count255<10 0\fi\the\count255}}
\def\ora{\oramin }
\def\data{\number\day/\ifcase\month\or gennaio \or febbraio \or marzo \or
aprile \or maggio \or giugno \or luglio \or agosto \or settembre
\or ottobre \or novembre \or dicembre \fi/\number\year;\ \ora}
\setbox200\hbox{$\scriptscriptstyle \data $}
\newcount\pgn \pgn=1
\def\foglio{\number\numsec:\number\pgn
\global\advance\pgn by 1}
\def\foglioa{A\number\numsec:\number\pgn
\global\advance\pgn by 1}
%%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI
%%%
%%% per assegnare un nome simbolico ad una equazione basta
%%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o,
%%% nelle appendici, \Eqa(...) o \eqa(...):
%%% dentro le parentesi e al posto dei ...
%%% si puo' scrivere qualsiasi commento;
%%% per assegnare un nome simbolico ad una figura, basta scrivere
%%% \geq(...); per avere i nomi
%%% simbolici segnati a sinistra delle formule e delle figure si deve
%%% dichiarare il documento come bozza, iniziando il testo con
%%% \BOZZA.
%%% All' inizio di ogni paragrafo si devono definire il
%%% numero del paragrafo e della prima formula dichiarando
%%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro
%%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione.
%%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi
%%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che
%%% viene letto all'inizio, se gia' presente. E' possibile citare anche
%%% formule o figure che appaiono in altri file, purche' sia presente il
%%% corrispondente file .aux.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\global\newcount\numsec\global\newcount\numfor
\global\newcount\numfig
\gdef\profonditastruttura{\dp\strutbox}
\def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax}
\def\SIA #1,#2,#3 {\senondefinito{#1#2}
\expandafter\xdef\csname #1#2\endcsname{#3} \else
\write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi}
\def\etichetta(#1){(\veroparagrafo.\veraformula)
\SIA e,#1,(\veroparagrafo.\veraformula)
\global\advance\numfor by 1
\9{\write15{\string\FU (#1){\equ(#1)}}}
\9{ \write16{ EQ \equ(#1) == #1 }}}
\def \FU(#1)#2{\SIA fu,#1,#2 }
\def\etichettaa(#1){(A\veroparagrafo.\veraformula)
\SIA e,#1,(A\veroparagrafo.\veraformula)
\global\advance\numfor by 1
\9{\write15{\string\FU (#1){\equ(#1)}}}
\9{ \write16{ EQ \equ(#1) == #1 }}}
\def\getichetta(#1){Fig. \verafigura
\SIA e,#1,{\verafigura}
\global\advance\numfig by 1
\9{\write15{\string\FU (#1){\equ(#1)}}}
\9{ \write16{ Fig. \equ(#1) ha simbolo #1 }}}
\newdimen\gwidth
\def\BOZZA{
\def\alato(##1){
{\vtop to \profonditastruttura{\baselineskip
\profonditastruttura\vss
\rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}}
\def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2
{\vtop to \profonditastruttura{\baselineskip
\profonditastruttura\vss
\rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}}
\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm
\foglio\hss}}
\def\alato(#1){}
\def\galato(#1){}
\def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor}
\def\verafigura{\number\numfig}
\def\geq(#1){\getichetta(#1)\galato(#1)}
\def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}}
\def\eq(#1){\etichetta(#1)\alato(#1)}
\def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}}
\def\eqa(#1){\etichettaa(#1)\alato(#1)}
\def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\write16{No translation for #1}%
\else\csname fu#1\endcsname\fi}
\def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi}
\openin13=#1.aux \ifeof13 \relax \else
\input #1.aux \closein13\fi
\openin14=\jobname.aux \ifeof14 \relax \else
\input \jobname.aux \closein14 \fi
\9{\openout15=\jobname.aux}
%%%%%%%%%%%%%%%%% CARATTERI
\newskip\ttglue
\ifnum\casa=1
\font\dodicirm=cmr12\font\dodicibf=cmr12\font\dodiciit=cmr12
\font\titolo=cmr12
\else
\font\dodicirm=cmr12\font\dodicibf=cmbx12\font\dodiciit=cmti12
\font\titolo=cmbx12 scaled \magstep2
\fi
%%%%%%%
\font\ottorm=cmr8\font\ottoi=cmmi8\font\ottosy=cmsy8
\font\ottobf=cmbx8\font\ottott=cmtt8\font\ottosl=cmsl8\font\ottoit=cmti8
\font\sixrm=cmr6\font\sixbf=cmbx6\font\sixi=cmmi6\font\sixsy=cmsy6
\font\fiverm=cmr5\font\fivesy=cmsy5\font\fivei=cmmi5\font\fivebf=cmbx5
\def\ottopunti{\def\rm{\fam0\ottorm}
\textfont0=\ottorm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
\textfont1=\ottoi \scriptfont1=\sixi \scriptscriptfont1=\fivei
\textfont2=\ottosy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
\textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
\textfont\itfam=\ottoit \def\it{\fam\itfam\ottoit}%
\textfont\slfam=\ottosl \def\sl{\fam\slfam\ottosl}%
\textfont\ttfam=\ottott \def\tt{\fam\ttfam\ottott}%
\textfont\bffam=\ottobf \scriptfont\bffam=\sixbf
\scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ottobf}%
\tt \ttglue=.5em plus.25em minus.15em
\setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}%
\normalbaselineskip=9pt\let\sc=\sixrm \normalbaselines\rm}
%\input tipi
\catcode`@=11 %%% ridefinizione di \footnote per note piccole
\def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf
\insert\footins\bgroup\ottopunti
\interlinepenalty100 \let\par=\endgraf
\leftskip=0pt \rightskip=0pt
\splittopskip=10pt plus 1pt minus 1pt \floatingpenalty=20000
\smallskip\item{#1}\bgroup\strut\aftergroup\@foot\let\next}
\skip\footins=12pt plus 2pt minus 4pt
\dimen\footins=30pc
\catcode`@=12
%
\let\nota=\ottopunti
%%%%%GRAFICA
%
%TO PRINT THE POSTCRIPT FIGURES THE DRIVER NUMBER MIGHT HAVE TO BE
%ADJUSTED. IF the 4 choices 0,1,2,3 do not work set in the following line
%the \driver variable to =5. Setting it =0 works with dvilaser setting it
%=1 works with dvips, =2 with psprint (hopefully).
%Using =5 prints incomplete figures (but still understandable from the
%text). The value MUST be set =5 if the printer is not a postscript one.
%%% the values =0,1 have been tested. The figures are automatically
%%% generated.
%
% Inizializza le macro postscript e il tipo di driver di stampa.
% Attualmente le istruzioni postscript vengono utilizzate solo se il driver
% e' DVILASER ( \driver=0 ), DVIPS ( \driver=1) o PSPRINT ( \driver=2);
% qualunque altro valore di \driver produce un output in cui le figure
% contengono solo i caratteri inseriti con istruzioni TEX (vedi avanti).
%
% 1) comando \ins#1#2#3
% Inserisce una scatola contenente #3 in modo che l'angolo superiore sinistro
% occupi la posizione (#1,#2)
% 2) comando \eqfig#1#2#3#4#5
% Crea una scatola di dimensioni #1x#2 contenente il disegno descritto in
% #4.ps; in questo disegno si possono introdurre delle stringhe usando \ins
% e mettendo le istruzioni relative in #3 (che puo' anche mancare);
% Il file esterno #4.ps contiene le istruzioni postscript, che devono
% essere scritte
% presupponendo che l'origine sia nell'angolo inferiore sinistro della
% scatola, mentre per il resto l'ambiente grafico e' quello standard.
% Infine #5 viene scritto a destra della figura e e' un testo tex,
% tipicamente fig.x oppure \equ(x.y) se la si considera una formula.
% Se \driver=2, e' necessario dilatare la figura in accordo al valore di
% \magnification, correggendo i parametri P1 e P2 nell'istruzione
% \special{#4.ps P1 P2 scale}
%
\newdimen\xshift \newdimen\xwidth \newdimen\yshift
\def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip}
\def\eqfig#1#2#3#4#5{
\par\xwidth=#1 \xshift=\hsize \advance\xshift
by-\xwidth \divide\xshift by 2
\yshift=#2 \divide\yshift by 2
\line{\hglue\xshift \vbox to #2{\vfil
#3 \special{psfile=#4.ps}
}\hfill\raise\yshift\hbox{#5}}}
%%%%% DISEGNO
% permette di scrivere file postscript in un test tex:
% il file postcript va scritto nel file tex riga per riga nella forma
% \8< .......... >
% dove i puntini stanno per una riga del file. A queste righe va
% premesso \figini#1 e posposto \figfin
% Il file postscript riceve il nome #1.ps
%%%%%
\def\8{\write13}
\def\figini#1{
\catcode`\%=12\catcode`\{=12\catcode`\}=12
\catcode`\<=1\catcode`\>=2
\openout13=#1.ps}
\def\figfin{
\closeout13
\catcode`\%=14\catcode`\{=1
\catcode`\}=2\catcode`\<=12\catcode`\>=12}
%%%%%% DEFINIZIONI VARIE
\def\didascalia#1{\vbox{\nota\0#1\hfill}\vskip0.3truecm}
\def\V#1{{\,\underline#1\,}}
\def\T#1{#1\kern-4pt\lower9pt\hbox{$\widetilde{}$}\kern4pt{}}
\let\dpr=\partial\def\Dpr{{\V\dpr}}
\let\io=\infty\let\ig=\int
\def\fra#1#2{{#1\over#2}}\def\media#1{\langle{#1}\rangle}\let\0=\noindent
\def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill}
\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}}
\let\implica=\Rightarrow\def\tto{{\Rightarrow}}
\def\pagina{\vfill\eject}\def\acapo{\hfill\break}
\let\ciao=\bye
\def\dt{\displaystyle}\def\txt{\textstyle}
\def\tst{\textstyle}\def\st{\scriptscriptstyle}
\def\*{\vskip0.3truecm}
\def\lis#1{{\overline #1}}
%%%%%%%LATINORUM
\def\etc{\hbox{\it etc}}\def\eg{\hbox{\it e.g.\ }}
\def\ap{\hbox{\it a priori\ }}\def\aps{\hbox{\it a posteriori\ }}
\def\ie{\hbox{\it i.e.\ }}
\def\fiat{{}}
%%%%%%%DEFINIZIONI LOCALI
\def\AA{{\V A}}\def\aa{{\V\a}}
\def\NN{{\cal N}}\def\FF{{\cal F}}\def\CC{{\cal C}}
\def\OO{{\cal O}}\def\LL{{\cal L}}\def\EE{{\cal E}}
\def\TT{{\cal T}}
\def\\{\hfill\break}
\def\={{ \; \equiv \; }}
\def\Im{{\rm\,Im\,}}\def\Re{{\rm\,Re\,}}
\def\sign{{\rm sign\,}}\def\atan{{\,\rm arctg\,}}
\fiat
\def\kk{{\V k}}
\def\pp{{\V p}}\def\qq{{\V q}}\def\ff{{\V f}}\def\uu{{\V u}}
\def\xx{{\V x}}\def\BB{{\cal B}}\def\RR{{\cal R}}
\def\CH{chaotic hypothesis}
\def\kk{{\V k}}
\def\annota#1#2{\footnote{${}^#1$}{{\parindent=0pt%\nota
\0#2\vfill}}}
\ifnum\aux=1\BOZZA\else\relax\fi
\ifnum\tipoformule=1\let\Eq=\eqno\def\eq{}\let\Eqa=\eqno\def\eqa{}
\def\equ{{}}\fi
%\def\equ{{}}\let\Eq=\eqno\let\eq=\eqno
%\headline{\nota\hss DRAFT 7: NOT FOR CIRCULATION}
%%%***
%\footline={\rlap{\hbox{\copy200}\ $\st[\number\pageno]$}\hss\tenrm
%\folio\hss}
\vglue1.5cm
\centerline{\titolo Chaotic hypothesis: Onsager
reciprocity and}
\vskip.5cm
\centerline{\titolo
fluctuation--dissipation theorem\/\rm\annota{*}{Archived in
{\it mp$\_$arc@math.utexas.edu}, \#95-????, and in {\it chao-dyn@
xyz.lanl.gov},\#95??????.}}
\vskip1.cm
\centerline{\dodiciit Giovanni Gallavotti}
\centerline{Fisica, Universit\`a
di Roma La Sapienza, P.le Moro 2, 00185, Roma, Italia.}
\vskip1cm
\0{\it Abstract: It is shown that the "chaoticity hypothesis",
analogous to Ruelle's principle for turbulence and recently introduced
in statistical mechanics, implies the Onsager reciprocity and the
fluctuation dissipation theorem in various reversible models for
coexisting transport phenomena.}
\vskip1cm
\*
\0{\it\S1: Introduction.}
\numsec=1\numfor=1\pgn=1
\*
In reference [GC2] we introduced, as a principle holding when a system
motions have an empirically chaotic nature, that:
\*
{\it Chaotic hypothesis: A many particle system in a stationary state
can be regarded as a smooth dynamical system with a transitive axiom A
global attractor for the purpose of computing macroscopic properties. In
the reversible case it can be regarded, for the same purposes, as a
smooth transitive Anosov system.}
\*
See [AA],[S],[R1],[Bo] for the notion and properties of Anosov systems
and [Bo,R2] for the more general systems with Axiom A attractors. I do
not know examples of reversible systems with an Axiom A transitive
global attractor which are not transitive Anosov systems: in this sense
I regard the second part of the hypothesis as a likely consequence of
the first (the viceversa is trivial). In this paper only the latter part
of the hypothesis will be needed and used.
The flow $x\to V_t x$ solving in phase space the differential equations
of motion generates an evolution $t\to F(V_tx)$ on the observables
$F(x)$. It can be naturally interpreted as a random process when the
initial data $x$ are chosen randomly with a given distribution $\m_0$.
The averages over the time variable $t$ give the {\it stationary} state
for the evolution $V_t$, {\it provided they are uniquely defined}, \ie
provided for almost all choices of $x$ with distribution $\m_0$ the
averages exist and are $x$--independent. In this case the stationary
state is a stationary probability distribution $\m$: to stress that it
is {\it dependent on $\m_0$} we call it the {\it statistics $\m$ of
$\m_0$}.
In many applications it is more convenient to regard the evolution as a
discrete trasformation defined on a restricted phase space $\CC$ of
{\it observed events}, also called {\it timing events}, (which could
be, for instance, the occurrence of a microscopic binary "collision").
The time evolution, or the {\it dynamics}, is a map $S$ of $\CC$ into
itself. The map $S$ is derived from the flow $V_t$ solving the
differential equations of motion of the system: the timing events $\CC$
have to be thought as a surface transversal to the flow and if $t(x)$
is the time between the timing event $x$ and the successive one $Sx$ it
is $V_{t(x)}x=Sx$. Note that the points $V_tx$ {\it are not} timing
events (\ie they are not in $\CC$) for the intermediate times $0< t<
t(x)$.
The notions of {\it statistics} $\m$ of $\m_0$ carry over unchanged to
the discrete notions of phase space and of evolution and the above
chaotic hypothesis is assumed in such a context. When referring to
phase space, unless stated otherwise, we think of a phase space $\CC$
consisting of timing events and of a map $S$ on $\CC$ defining the time
evolution. The {\it smoothness} of the Anosov system is intended in
all the coordinates and parameters on which the system equations
depend.
The chaotic hypothesis implies, as a mathematical consequence, that
{\it for most random distributions $\m_0$ for the choice of the initial
data $x$} the distribution $\m$ exists.
However the choice of the initial data with distribution $\m_0$ {\it
proportional to the volume measure on $\CC$} plays a special role,
because in the case of hamiltonian systems such distribution
is generated naturally via the microcanonical ensemble.
For instance one can read, translating symbols into the present
notations: "the appropriate objects of study of a statistical mechanics
of time dependent phenomena are the random processes $F(V_tx)$ {\it
with initial distribution of $x$, $\r_E(x)$ (the microcanonical
distribution)}, for all energies of interest and for all gross variables
$F$ of interest", (italics added), see the nice paper [GCDR].
I ``strongly disagree'' with such a philosophical position: but I will
adopt it here because it is obvious that, whether one agrees or not, it
is {\it of fundamental interest} to understand the statistics of initial
data chosen at random with a distribution proportional to the phase
space volume. Other random choices may have to wait until the latter is
properly understood.
In the Physics literature the existence of the distribution $\m$ is, in
fact, assumed in general as stated by the following (extension) of the
zeroth law, [UF], giving a global property of the motions generated by
initial data chosen randomly with distribution $\m_0$ proportional to
the volume measure on $\CC$:
\*
\0{\it Extended zero-th law: A dynamical system $(\CC,S)$ describing a
many particle system (or a continuum such as a fluid) generates motions
that admit a statistics $\m$ in the sense that, given any (smooth)
macroscopic observable $F$ defined on the points $x$ of the phase space
$\CC$, the time average of $F$ exists for all $\m_0$--randomly--chosen
initial data $x$ and is given by:
%
$$\lim_{M\to\io}\fra1M\sum_{k=0}^{M-1}
F(S^jx)=\ig_\CC\m(dx')F(x')\Eq(1.1)$$
%
where $\m$ is a $S$--invariant probability distribution on $\CC$.}
\*
The chaotic hypothesis was proposed by Ruelle in the case of fluid
turbulence, and it is extended to non equilibrium many particle systems
in [GC1]. If one assumes it, then it {\it follows} that the zeroth law
holds, [S,Bo,R]; however it is convenient to regard the two statements
as distinct because the hypothesis we make is "{\it only}" that one can
suppose that the system is Axiom A or Anosov for "practical purposes":
this leaves the possibility that it is not strictly speaking such and
some ("neglegible in the thermodynamic limit") corrections may be
needed on the predictions obtained by using the hypothesis.
>From now on only reversible systems will be considered in this paper:
they are dynamical systems such that there is an isometric map $i$ of
phase space such that $i^2=1$ and $iS=S^{-1}i$. In [GC2] the generality
of the hypothesis is discussed and in [GC1], [GC2] we derived, as a
rather general consequence, predictions testable by numerical experiment
in systems with few degrees of freedom. The most relevant feature of
the prediction, which is a {\it large deviation theorem} or {\it
fluctuation theorem} for systems with reversible dynamics, is that it is
{\it parameter free}.
A drawback is that it {\it is not} testable directly in really large
systems.
The question of whether the "chaotic hypothesis" could be tested on real
experiments, \ie for really macroscopic systems, remained open. In this
paper we show, through examples, that the \CH\ implies quite generally,
in systems with reversible dynamics,
the Onsager reciprocity and the fluctuation--dissipation theorem: see
[DGM] for a classical general discussion of the reciprocity relations,
see [C] (Appendix A, p.187-200) for a kinetic derivation, see also
[E,ELS, GJL] for recent developments. A very nice introduction and an
exposition of the basic ideas can be found in [D].
The ideas of the present paper can
be applied also to systems relevant for the theory of developed
turbulence (see [G4]).
A puzzling aspect of the \CH\ is that it implies that the system has a
positive gap separating from $0$ the Lyapunov exponents and one may
have serious doubts on the validity of such a strong property: so that
a discussion is in order.
This is discussed in [GC2], \S8, and [G4] by suggesting that there may
well be many vanishing exponents (or exponents of order $O(N^{-1})$) if
$N$ is the particle number: such exponents should be "ignored" as they
should correspond to macroscopic evolution laws which microscopically
will be effectively described by local conservation laws [EMY,KV,LOY].
They have an approximate character, unless $N\to\io$, but one can think
of imposing them as {\it exact} conservation rules by adding to the
forces acting on the system other suitable "auxiliary" forces {\it
minimally} required to achieve the purpose of turning the slow
macroscopic observables responsible for the existence of the $0$
Lyapunov exponents into exact local conservation rules. For instance
one can find the auxiliary forces by applying the Gauss' {\it principle
of least constraint}, see appendix. The dynamical system obtained in
this way should be one to which the \CH\ should apply.
In \S2,\S3 we introduce some models which will be used to illustrate our
general ideas; in \S4 we discuss the relevant mathematical facts about
reversible Anosov systems; in \S5 we give a proof, whose full
mathematical rigor still rests on a mathematical conjecture (\S5) on
Anosov systems (that I hope to address elsewhere), of the validity of
the fluctuation dissipation relation and of Onsager reciprocity in the
models introduced in \S2,\S3: but the generality of the argument, which
seems largely model independent, will also emerge. Comments and a
critical comparison with the classical derivations (that apply to our
models as well) are presented in \S6.
\*
\0{\it\S2: A diffusion problem.} \numsec=2\numfor=1\pgn=1 \*
We consider a mixture of two chemically inert gases whose $N=N_1+N_2$
molecules (with equal masses $m$ and respective numbers $N_1=N_2$)
are contained in a box $\BB=[-\fra12 L,\fra12L]^2$ with periodic boundary
conditions and are subject to a respective force field $\V E^1=E^1\V i$
and $\V E^2 =E^2\V i$. The molecules interact via a pair interaction
with a short range potential (\eg a Lennard Jones type of potential).
Furthermore the motion is subject to the constraint that the total
energy is constant, via a constraint force law which is {\it ideal} in
the sense that it verifies Gauss' principle of minimal constraint. This
means that, if $E_j$ is the field on the $j$-th particle (equal either
to $E^1$ or to $E^2$) the equations of motion are:
%
$$\dot{\V q}_j=\fra1m \V p_j,\qquad
\dot{\V p}_j=\V F_j+ E_j\V i-\a\V p_j\Eq(2.1)$$
%
with $\a=\fra{\sum_j E_j\V i\cdot\V p_j}{\sum_j \V p_j^2}$,
and we expect that the future time average of the total momentum $\V P$
and of the dissipation $\a$ will
be some $\media{\V P}_+>0$ and $\media{\a}_+>0$ if $(E^1,E^2)\ne0$.
The average is expected to be attained exponentially fast
while the "conjugate" variable $\V C$, the center of
mass position, will be expected to evolve with zero Lyapunov exponent
(and to behave asymptotically as $\media{\V C}_+=\fra1{(N_1+N_2)m}
\media{\V P}_+t+ cost$).
According to the analysis of [GC2],\S8, and [G4], we expect that we can
freely add to the equations of motion a minimal constraint force
imposing the constraint $\V P=P(E^1,E^2){\V i}$ if $P(E^1,E^2)$
is the average horizontal momentum, \ie the (unknown) "equation of
state",
(and the implied $\dot {\V C}=cost$). This means that considering
a modified equation of
motion:
%
$$\dot{\V q}_j=\fra1m \V p_j,\qquad
\dot{\V p}_j=\V F_j+ E_j\V i-\a\V p_j-\V\b\Eq(2.2)$$
%
with $\V\b=\fra{E_1N_1+E_2N_2}{N_1+N_2}\V i-\a\fra{\V P}{N_1+N_2}$,
should not lead to appreciably different qualitative behaviour if the
initial data are consistent with the equation of state
$\V P=P(E^1,E^2)\V i,\,\V C=\V0$.
The phase space contraction per unit time is:
%
$$\g=(2(N_1+N_2)(1-\fra1{N_1+N_2})-1)\,\a=(2(N_1+N_2)-3)\,\a\Eq(2.3)$$
%
Note that without imposing the extra constraint the phase spave
contraction would have been $(2(N_1+N_2)-1)\,\a$: with a relative
difference of $O(N^{-1})$, which should become
neglegible in the thermodynamic limit $L\to\io, N_1 L^{-2}=\r_1,
N_2 L^{-2}=\r_2$.
We call $Q$ the work per unit time performed by the forces $\a \V p_j$
so that the phase space contraction rate in the configuration
$x$ is (to leading order in $N$) $2N\a(x)$, \ie $\fra{E^1N_1\dot
a_1(x)+E^2 N_2\dot a_2(x)}{k_B T(x)}$ $=\fra{Q}{k_B T}$
if $k_B$ is Boltzmann's constant, $k_B T(x)$ is the
kinetic energy per particle and $a_1(x), a_2(x)$ are the horizontal
coordinates of the centers of mass $\V C_1,\V C_2$
of the two species, and $\dot
a_1(x),\dot a_2(x)$ the corresponding velocities. Thus the phase space
contraction rate can be interpreted as $\fra1{k_B}$ times the {\it
entropy creation rate} (at least if $E^1,E^2$ are so small that the
centers of mass horizontal velocities are small compared to the root
mean square velocities).
The above model is closely related to the {\it color diffusion} model
considered in [BEC]. A {\it very} important feature of the model is its
{\it time reversibility}: the map $i(\pp,\qq)=(-\pp,\qq)$ has the
property that it is an isometry of phase space such that $iS=S^{-1}i$
and $i^2=1$.
\*
\0{\it\S3: A heat conduction--electrical conduction model.}
\numsec=3\numfor=1\pgn=1
\*
As a second model we consider a modification of model 4 of [CG2],
inspired by [HHP], see also [PH] for a more general perspective on
constrained systems. This is a model for a heat conducting and
electrically conducting gas. In a box $\BB=[-2L- H,
2L+H]\times[-\fra12L,\fra12L]$ are enclosed $N$ particles with mass $m$,
interacting via a rather general pair potential, like a hard core
potential with a tail or via a Lennard Jones potential, and they are
subject to a constant force field ({\it electric field}) $E\V i$ in the
$x$ direction. The boundary conditions are periodic in the horizontal
direction and reflecting in the vertical direction.
Adjacent to the box $\BB$ there are two boxes $\RR_+,\RR_-$ containing
$N_+=N_-$ particles interacting with each other via a hard core
interaction, and with the particles in $\BB$ via a pair interaction (\eg
with potential equal to the one between the particles in $\BB$),
but are separated from the latter by a reflecting wall.
\figini{tmpuvw1}
\8
\8<0.75 0.75 scale>
\8
\8
\8
\8
\8
\8
\8< 0 0 lineto stroke} def>
\8< /R {newpath 0 0 moveto 0 L lineto L L lineto L 0 lineto 0 0>
\8< lineto closepath stroke} def>
\8
\8
\8
\8
\figfin
\vskip3mm
\eqfig{187.5pt}{97.5pt}{
\ins{9pt}{75pt}{$\scriptstyle \RR_-$}
\ins{120pt}{75pt}{$\scriptstyle \RR_+$}
\ins{-15pt}{22.5pt}{$\scriptstyle \BB$}
\ins{37.5pt}{78.75pt}{$\scriptstyle N_-$}
\ins{142.5pt}{78.75pt}{$\scriptstyle N_+$}
\ins{75pt}{37.5pt}{$\scriptstyle N$}}{tmpuvw1}{}
\vskip5mm
The model name is motivated because we imagine other forces to act on
the system: they are the minimal forces (in the sense of Gauss' minimal
constraint principle, see appendix A1) necessary to enforce the
following constraints:
1) the total kinetic energy in the "hot plate" $\RR_+$ and in the "cold
plate" $\RR_-$ are constrained to be $N_+k_B T_+$ and, respectively,
$N_-k_B T_-$ where $T_-$ and $T_+= T_-+\d T, \, \d T\ge0$ are the {\it
temperatures of the plates}.
2) the total energy $U$ in the box $\BB$ is constrained to stay constant.
The equations of motion are:
$$\eqalign{
\dot{\V q}_j=&\fra{\V p_j}m\cr
\dot{\V p}_j=&\V F_j+ E \chi(\V
q_j)\V i-\a_+\chi_{+}(\V q_j)\pp_j-\a_-\chi_{-}(\V q_j)\pp_j-\a
\chi(\V q_j)\pp_j\cr}\Eq(3.1)$$
\0where $\chi,\chi_{\pm}$ are the characteristic functions of
the regions $\BB,\RR_\pm$ and $\a_+,\a_-,\a$ are multipliers defined so
that for some $T_\pm,U$:
$$\sum_{j=1}^{N_\pm}\chi_{\pm}(\qq_j)\fra{\pp_j^2}{2m}=N_\pm k_B T_\pm,
\qquad \sum_{j=1}^N \chi_{\BB}(\qq_j)\fra{\pp_j^2}{2m} +V(\qq)=U\Eq(3.2)$$
%
\0are exact constants of motion.
We suppose for simplicity (see however comment (1) in \S6 below) that
the system in $\BB$ is kept at a constant total energy $U$, and at
constant reservoirs temperatures $T_\pm$. In this case we call
$Q_+,Q_-,Q_0$ the work per unit time performed by the forces
$\a_+\chi_{+}\pp_j,\a_-\ch_{-}\pp_j,\a\ch\pp_j$ respectively.
Let $\LL_E,\LL_+^0,\LL_-^0,\LL_+,\LL_-$ denote, respectively, the work
per unit time performed by the field $E$ or by the particles in the
thermostats $\RR_+,\RR_-$ on the gas in $\BB$, or by the gas in $\BB$ on
the ther\-mostats $\RR_+,\RR_-$. Then the imposed conservation laws
give:
%
$$\eqalign{ &-Q_0+\LL_E+\LL_++\LL_-=0\qquad\otto\qquad(U=const)\cr
&-Q_++\LL^0_+=0\kern1.7truecm\qquad\otto\qquad
(\sum_{\qq_j\in\RR_+}\fra{\pp^2_j}{2m}=N_+k_B T_+)\cr
&-Q_+-\LL^0_-=0\kern1.7truecm\qquad\otto\qquad
(\sum_{\qq_j\in\RR_-}\fra{\pp^2_j}{2m}=N_-k_B T_-)\cr}\Eq(3.3)$$
%
so that one easily finds (by differentiating \equ(3.2) with respect to
time and by applying the equations of motion, \equ(3.1)) expressions
for $\a_+,\a_-,\a$:
%
$$\a=\fra{\LL_E+\LL_++\LL_-}{\sum_{\qq_j\in\BB}\pp_j^2/m}=
\fra{Q_0}{\sum_{\qq_j\in\BB}\pp_j^2/m},\qquad
\a_\pm=\fra{\LL^0_\pm}{2N_\pm k_BT_\pm}=
\fra{Q_\pm}{2N_\pm k_B T_\pm}\Eq(3.4)$$
%
and the phase space contraction per unit time, or $\fra1{k_B}$ times the
{\it entropy creation per unit time}, is:
%
$$\g(x)=(2N_\pm-1)(\a_++\a_-)+(2N-1)\a\Eq(3.5)$$
%
in the configuration $x$.
The above model also shares the feature that it is {\it time reversible},
and the isometric map $i(\pp,\qq)=(-\pp,\qq)$ is again such that
$i\,S=S^{-1}i$, and $i^2=1$.
\*
\0{\bf\S4: The SRB distribution.}
\numsec=4\numfor=1\pgn=1
\*
The chaotic hypothesis of \S2 allows us to represent the SRB
distribution in a simple form, by using {\it Markov partitions}, [S].
We consider {\it only} transitive reversible Anosov systems, although
many concepts make sense for more general systems with chaotic
attractors.
The notion of Markov partition is a mathematically precise version of
the intuitive idea of {\it coarse graining}. We just recall here the
main properties of Markov partitions. For a discussion of the intuitive
meaning and the connection with the coarse graining see [G3].
1) A parallelogram will be a small set with a boundary consisting of
pieces of the stable and unstable manifolds of the map $S$ joined
together as described below. The smallness has to be such that the
parts of the manifolds involved look essentially ``flat'': \ie the sizes
of the sides have to be small compared to the smallest radii of
curvature of the {\it unstable manifolds} $W^u_x$ or of the {\it stable
manifolds} $W^s_x$, as $x$ varies in $\CC$.
Therefore let $\d$ be a length scale small compared to the minimal
(among all $x$) curvature radii of the stable and unstable manifolds.
Let $\D^u$ and $\D^s$ be small (and ``small'' means of size $\ll\d$)
connected surface elements on $W^u_x$ and $W^s_x$ containing $x$. We
define a {\it parallelogram} $E$ in the phase space $\CC$, to be denoted
by $\D^u\times\D^s$, with center $x$ and axes $\D^u$, $\D^s$
as follows.
Consider $\x\in\D^u$ and $\h\in\D^s$ and suppose that the point $z$,
denoted $\x\times\h$, such that the shortest path joining $\x$ with $\h$
formed by a path on the stable manifold $W^s_\x$ joining $\x$ to $z$
and by a path on the unstable manifold $W^u_\h$ joining $z$ to $\h$, is
uniquely defined. This will be so if $\d$ is small enough and if
$\D^u$, $\D^s$ are small enough compared to $\d$ as we assume, (because
the stable and unstable manifolds are ``smooth'' and transversal).
The set $E=\D^u\times\D^s$ of all the points generated in this way when
$\x,\h$ vary arbitrarily in $\D^u,\D^s$ is called a parallelogram
(or rectangle), provided the boundaries $\dpr\D^u,\dpr\D^s$ of $\D^u$ and
$\D^s$ as subsets of $W^u_x$ and $W^s_x$, respectively, have zero
surface area on the manifolds on which they lie. The sets $\dpr_u
E\=\D^u\times\dpr\D^s$ and $\dpr_s E=\dpr\D^u\times\D^s$ will be called
the {\it unstable} or {\it horizontal} and {\it stable} or {\it
vertical} sides of the parallelogram $E$.
Consider a partition $\EE=(E_1,\ldots,E_\NN)$ of $\CC$ into $\NN$
rectangles $E_j$ with pairwise disjoint interiors. We call
$\dpr_u\EE\=\cup_j\dpr_u E_j$ and $\dpr_s\EE\=\cup_j\dpr_s E_j$: these
are called respectively the {\it unstable boundary} of $\EE$ and the
{\it stable boundary} of $\EE$, or also the horizontal and vertical
boundaries of $\EE$, respectively.
2) We say that $\EE$ is a {\it Markov partition} if the transformation
$S$ acting on the stable boundary of $\EE$ maps it into itself (this
means $S \dpr_s\EE\subset \dpr_s\EE$) and if, likewise, the map $S^{-1}$
acting on the unstable boundary maps it into itself ($S^{-1}\dpr_u
\EE\subset \dpr_u\EE$).
The actual construction of the SRB distribution then proceeds from the
important result of the theory of Anosov systems expressed by
a remarkable theorem (Sinai, [S]):
\*
{\it Theorem: Every Anosov system admits a Markov partition} $\EE$.
\*
\0{\it Comments:} \\
\0a) If the reversibility property holds it is clear that $i\EE$ is also a
a Markov partition. This follows from the definition of Markov
partition and from the fact that reversibility implies:
$$W^s_x=i W^u_{ix}\Eq(4.1)$$
\0b) The definition of a Markov partition also implies that the intersection
of two Markov partitions is a Markov partition, hence it is clear that
if a transitive Anosov system is reversible (\ie there is an isometry
$i$ such that $iS=S^{-1}i$ and $i^2=1$) there are Markov partitions $\EE$
that are reversible in the sense that $\EE=i\EE$, \ie if $E_j\in \EE$
there is $j'$ such that $i E_j=E_{j'}\in\EE$.
\0c) The usefulness of the Markov partitions comes from the possibility
that they provide of representing the points of $\CC$ as {\it infinite
strings} of symbols (in a more useful way than representing them, for
instance, as the strings of digits that give the value of their
coordinates).
This is simply achieved by associating with $x\in\CC$ the string $\V
j=\{j_k\}_{k=-\io}^\io$, such that $S^kx\in E_{j_k}$. The invertibility
of the map between $x\in \CC$ and the {\it compatible} or {\it allowed}
sequences, \ie the sequences $\V j$ such that the interior of $S
E_{j_k}$ intersects the interior of $E_{j_{k+1}}$ is a well known (and
easy) consequence of the definition of Markov partition. The
correspondence is in fact one to one with some obvious exceptions:
namely to each sequence $\V j$ with the above compatibility property
corresponds one $x$; viceversa, if $x$ is not on a boundary
of some of the $E\in \EE$ nor on the image of a boundary under a power of
$S$ then $x$ admits only one symbolic representation. The points on the
boundaries or visiting, in their evolution, the boundaries of course
have several (but finitely many) symbolic representations. Just in the
same way as the decimal representation of a number is unique for most
numbers: the ones which end with infinitely many successive $9$ admit
two representations.
The correspondence $\V j\otto x$ between points $x\in\CC$ and their
history, or symbolic representation, $\V j$ as a compatible sequence will
be denoted $x=x(\V j)$ ({\it symbolic code}).
\0d) If we define the {\it compatibily matrix}, or {\it intersection
matrix}, $C_{ij}$ by setting $C_{ij}=1$ if the interior of $E_j$
intersects the interior of $S E_i$ and $C_{ij}=0$ otherwise, then the
assumed transitivity implies that there is a iterate $q$ of $C$ such
that all elements of $C^q$ are positive (\ie $S^q E_j$ has interior
intersecting $E_k$ for all pairs $j,k$, simoultaneosly).
\0e) Consider the partition $\EE_M=\cap_{-M}^M S^{-j}\EE$ obtained by
intersecting the images under $S^k$, $k=-M,\ldots,M$, of $\EE$. Then
$\EE_M$ is still a Markov partition and it is time reversal invariant
if $\EE$ is (see (b) above). Note that the parallelograms of $\EE_M$
can be labeled by the strings of symbols $j_{-M},\ldots,j_M$ and they
consist of the points $x$ such that $S^k x\in E_{j_k}$ for $-M\le k\le
M$. In other words the parallelograms consist of those points $x$
which, in their time evolution, visit at time $k$ the parallelogram
$j_k$.
\0f) If $F(x)$ is a function on phase space ({\it observable}) then we
can regard it as a function $F(x(\V j))$ on symbolic sequences. An
observable $F$ is {\it local} if it ``depends exponentially little'' from the
history symbols $j_k$ with large $k$: \ie if $F(x(\V j))-F(x(\V j'))$
tends to zero exponentially fast with the maximum number $k$ such that
$\V j$ and $\V j'$ agree on the sites with label $h$ with $|h|\le k$. A
simple condition on $F$ guaranteeing its locality is that $F$ is
H\"older continuous in $x$.
\*
We now construct a probability distribution on $\CC$ by defining it as a
probability distribution on the space of the compatible strings $\V j$
and then by interpreting it as a distribution on the phase space $\CC$.
\0(1) For this purpose we first pick a point, that we call the {\it
center}, $x_{j_{-M},\ldots,j_M}$ in each $E_{j_{-M},\ldots,j_M}$ with non
empty interior simply by considering the compatible string which is
obtained by continuing the string $j_{-M},\ldots,j_M$ "to the right"
into a string $j_M,j_{M+1},\ldots$ and to the "left" into a string
$\ldots, j_{-M-1}, j_{-M}$ in a way that the whole string $\V j$ is
compatible (\ie such that there is a point $x$ such that $S^kx\in
E_{j_k}$ for all $k$) and, {\it furthermore}, the entries of the
continuation strings {\it depend only} on the value of $j_M$ and $j_{-M}$
respectively.
In general given $j_{\pm M}$ there will be many choices of the
continuation strings: which one we actually take is irrelevant. We
impose however the further constraint that the continuation is made in a
"time reversible way", \ie we choose the continuations so that if $x$ is
the center of $E_j$ then $ix$ is the center of $i E_j$. A further
restriction (not necessay in the following but very nattural) that one
could consider is imposing that the continuation string to the right of
$j_M$ (or to the left of $j_{-M}$) agree identically after finitely many
steps. Note that the existence of the continuation strings and the
possibility of imposing on them the above restrictions is an immediate
consequence of the transitivity property of the compatibility matrix $C$.
%\ifnum\mgnf=0\pagina\fi
\0(2) We then define, given $\t>0$:
%
$$\lis\L_{u,\t}(x)=\prod_{j=-\t/2}^{\t/2-1}\L_u(S^jx)\Eq(4.2)$$
%
where $\L_u(x)$ is the {\it local expansion coefficient} of the surface
elements of the unstable manifold at $x$, \ie it is the jacobian
determinant of the transformation $S$ regarded as a map of $W^u_x$ into
$W^u_{S x}$. Likewise we define $\L_s(x)$ and $\lis\L_{s,\t}(x)$ as the
corresponding quantities obtained by regarding $S$ as a map of the
stable manifold $W^s_{x}$ to $W^s_{Sx}$.
\0(3) Finally we define a distribution $\m_{M,\t}$ on $\CC$ by
"giving" to each set $E_{j_{-M},\ldots,j_M}\in \EE_M$ a probability
proportional to
$\lis\L_{u,\t}^{-1}(x_{j_{-M},\ldots,j_M})\d_\t^{-1}(x_{j_{-M},\ldots,j_M})
$ where $\d(x)=\sin \th(x)=\d_0(x)$ is the sine of the angle between
the stable and the unstable manifolds at $x$ and
$\d_\t(x)=\d(S^{\t/2}x)$.
More precisely we define the distribution $\m_{M,\t}$ so that the
integral of a smooth function $F$ is:
%
$$\ig_\CC \m_{M,\t}(dx) F(x)\, {\buildrel def \over =}\, \fra{\sum_j
\lis\L^{-1}_{u,\t}(x_j)\d^{-1}_\t(x_j) F(x_j)}{\sum_j
\lis\L^{-1}_{u,\t}(x_j)\d^{-1}_\t(x_j)}\Eq(4.3)$$
%
where $j$ is a short hand notation for $j_{-M},\ldots,j_M$ and
$x_j=x_{j_{-M},\ldots,j_M}$ is the ``center'' chosen above in $E_j\in
\EE_M$. No relation is assumed here between $T$ and $\t$, although in
the applications we shall (naturally) take $T=\t/2$, as this simplifies
the discussion considerably.
The distribution $\m_{M,\t}$ is very interesting because it is an
approximation (a very good one) of the SRB distribution. In fact in
[G1,CG1,CG2] the following theorem is shown to be a reformulation
(convenient although trivially equivalent) of a basic theorem by Sinai:
\*\0{\it Theorem: If $(\CC,S)$ is a transitive Anosov system the SRB
distribution $\m$ exists and the $\m$ average of a local function (see
(f) above) $F$ is:
%
$$\ig_\CC \m(dx) F(x)=
\lim_{M\to\io,\t\to\io}\ig_\CC \m_{M,\t}(dx) F(x) \Eq(4.4)$$
%
Furthermore in \equ(4.3) the factor $\d_\t^{-1}(x_j)$ could be replaced
by $\d^{z}_\t(x_j)$ with $z$ any prefixed real number (\eg $z=0$). The
limits can be interchanged.}
\*
The original statement is that $\m$ exists and it is a Gibbs state with
potential $\log\L^{-1}_u(x)$: see [Bo],\-[R1],\-[R2],\-[S] for a
discussion of this form of the statement. In [R2] the latter statement
is extended to cover the case in which $(\CC,S)$ has a global transitive
Axiom A attractor: the discussion in [G1],[G2] shows that the above
theorem extends, unchanged, to such cases. The extra factor $\d_\t^z$
with $z=-1$ was absent in [GC1],[GC2] where $z$ was chosen equal to $0$
(an admissible alternative choice).
The possibility of fixing $z$ arbitrarily, in spite of the apparently
strong modification it introduces, is easily seen by examining the proof
of \equ(4.4), see [G1], [G2]. The proof is based on the interpretation
of \equ(4.3) as a probability distribution on the space of the
compatible strings. In this interpretation one immediately recognizes
that \equ(4.3) corresponds to a finite volume Gibbs distribution for a
suitable short range hamiltonian defined on the space of compatible
strings. An extra factor $\d^z_\t(x_j)$ corresponds to considering the
same Gibbs distribution {\it just with a different boundary condition}:
which becomes irrelevant in the limit as $\t\to\io$ because one
dimensional Gibbs states with short range interactions do not have phase
transitions and therefore are insensitive to changes in the boundary
conditions. Different choices of the center points also correspond to
different choices of boundary conditions.
The choice $z=-1$ is much better than $z=0$ because it leads to simpler
formulae and arguments: we shall call \equ(4.3) a {\it balanced}
approximation to the SRB distribution because as we shall see it is {\it
reminiscent} of a probability distribution verifying the detailed
balance (which however is {\it not} verified in our models, except in
$0$ forcing, \ie in equilibrium).
In \equ(4.4) with $T\gg\t/2$ the choice of the point $x_j$ in the
parallelograms of $\EE_M$ can be really arbitrary, and it does not
matter that $x_j$ is really chosen as said above or just {\it anywhere}
in $E_{j_{-M},\ldots,j_M}$ (because the variation of the weigths
\equ(4.2) is in this case neglegible, provided $M-\tau/2\to\io$ fast
enough).
The extra properties that we need are that $\EE_M$ is reversible (see
above), \ie $i E_j=E_{j'}\in\EE_M$ (for a suitable $j'$) and that, as a
consequence of the reversibility (via \equ(4.1) and the isometric nature
of the time reversal map $i$ and the validity of $\g_\t(x)=-\g_\t(ix)$
for the underlying differential equations):
%
$$
\lis\L_{u,\t}(ix)=\lis\L_{s,\t}^{-1}(x),\qquad \d_0(x)=\d_0(ix),\qquad
\d_\t(x)=\d_{-\t}(ix)\Eq(4.5)$$
%
which are identities (see [GC2]); for the definitions of
$\L_s,\lis\L_{s,\t}$, see the lines following \equ(4.2). Furthermore the
volume measure and the expansion and contraction rates are related by:
%
$$\lis\L_{u,\t}(x)\lis\L_{s,\t}(x)\fra{\d_\t(x)}{\d_{-\t}(x)}\=e^{-\t
t_0\lis\s_{\t}(x)}\Eq(4.6)
$$
%
where $t_0$ is the average time interval between successive timing
events and the phase space volume contraction for a single
transformation is written $e^{-t_0\s(x)}$, thus {\it defining}
$\s(x)$ and $\lis\s_{\t}(x)$:
%
$$\lis\s_{\t}(x)\,{\buildrel def \over
=}\,\fra1\t\sum_{r=-\t/2}^{\t/2}\s(S^rx)\Eq(4.7)$$
%
The \equ(4.6) is is obtained from the relation
$\L_u(x)\L_s(x)\fra{\d(Sx)}{\d(x)}=e^{-t_0\s(x)}$ by evaluating it on
the points $S^kx$, $k=-\fra{\t}2,\ldots,\fra{\t}2$ and multiplying the
results.
If the time interval $t(x)$ between the timing event $x\in\CC$ and the
successive one is very small and if its fluctuations can be neglected
toghether with those of $\g(x)$ see \equ(2.3), \equ(3.5) (within the
same time interval) then one simply has $\s(x)=\g(x)$. Note that {\it
in all cases} with any reasonable definition of timing events the time
$t(x)$ will tend to zero in the thermodynamic limit (as $O(N^{-1})$),
but also $\g$ will tend to infinity as $O(N)$.
More generally there is a simple relation between the function $\s(x)$
above and the function $\g(x)$ which describes the phase space
contraction rate in the differential equations giving rise to the map
$S$ (see \equ(2.3), \equ(3.5)); namely:
%
$$\s(x)=\fra1{t_0}\ig_0^{t(x)} \g(S_t x)\,dt\Eq(4.8)$$
%
But the use of \equ(4.8) is quite clumsy and one can always think that
the timing events are chosen, artificially, much closer than the natural
$t_0=O(1/N)$ and observed at constant time intervals so that no
difference really exixts between $\g(x)$ and $\s(x)$. If necessity
arises one can always use the precise relation
\equ(4.8), at the expense of some formal algebraic complications in the
intermediate step of our coming deductions.
%\ifnum\mgnf=0\pagina\fi
\*
{\bf\S5: Applications to the models. Onsager reciprocity and
fluctuation--dissipation theorem.}
\numsec=5\numfor=1\pgn=1
\*
In this section we neglect for simplicity of exposition the difference
between $\s(x)$ and $\g(x)$, \ie we suppose that $t(x)=t_0$ is constant
and that $\g$ is constant on the path traveled in the time interval
$t(x)$: this simplifies considerably the algebra and the reader should
have no trouble checking that the proper relation
\equ(4.8) could be consistently used leading to no corrections to the
final results below (because in the end we shall set $\V E=\V0$).
Relation \equ(4.3) has the form of a statistical average and we shall
try to use it in the "same" way as in equilibrium statistical
mechanics. We shall first study here the two currents $J_h, \,h=1,2$,
generated by the pair of fields $E_h$ in the diffusion model of \S3.
The two currents are, if $\r_h, v_h$ are the density and average
velocity of the species $h$ :
%
$$J_h=\r_h v_h=\fra{N_h}{L^2}\fra{\sum_{j\in\{h\}}\pp_j\cdot\V i}{N_h
m}= \fra{\sum_j\pp_j^2/m}{L^2(2N-1)}\,\dpr_{E_h}\s\Eq(5.1)$$
%
where $j\in\{h\}$ means that $j$ is a species $h$ particle, and
quantities of $O(N^{-1})$ have been neglected (see \equ(2.3)) and $\s$
is $\fra1{k_B}$ times the entropy production rate, see
\equ(4.6),\equ(4.7) and \equ(4.8). Hence if we define $T(x)$, for each
configuration $x$, by $\sum_j\fra{\pp_j^2}{2m}= N kT(x)$ and:
%
$$j_h\,{\buildrel def \over =}\,\fra{2N}{2N-1}\media{\fra{J_h}{kT}}\,=
\lim_{\t\to\io}\fra1{L^2}
\fra{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)\d^{-1}_\t(x_j)\,\dpr_{E_h
}\lis\s_{\t}(x_j)}{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)
\d^{-1}_\t(x_j)}\Eq(5.2)$$
%
where $\lis\s_{\t}(x)=\fra1\t\sum_{r=-\t/2}^{\t/2-1}\s(S^rx)$, see
\equ(4.7). If we recall \equ(4.3) with $\fra12\t=M$ we see that $j_h$
can be regarded as the SRB average of $\fra{J_h}{kT},\,h=1,2$.
This expression is similar to the formula derived from the generating
function of the Helfand moments in [GD2], [GD1]: but it is not the same
because in [GD2] the SRB is represented by using the notion of
$(\e,\t)$--separated sets (which are a somewhat more primitive or less
concrete version of the parallelograms of the Markov partitions).
We shall also define $l_{u,\t},l_{s,\t}$ as:
%
$$\lis\L_{u,\t}^{\,-1}(x)\d_\t(x)^{\,-1}=e^{\t l_{u,\t}(x)},\qquad
\lis\L_{s,\t}(x)\d_{-\t}^{\,-1}(x)=e^{\t l_{s,\t}(x)}\Eq(5.3)$$
%
so that \equ(4.5),\equ(4.6) imply $l_{u\t}(x)-l_{s\t}(x)=\t t_0
\lis\s_{\t}(x)$. Hence we see that, if $\dpr_k\=\dpr_{E_k}$:
%
$$\eqalignno{
\dpr_k j_h=&\fra1{L^2}\fra{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)
\d_\t^{\,-1}(x_j)\big(\dpr_{hk}\lis\s_{\t}(x_j)+\t\dpr_k
l_{u,\t}(x_j)\dpr_h\lis\s_{\t}(x_j)\big)}{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)
\d_\t^{\,-1}(x_j)}+\cr
&-\fra1{L^2}
\fra{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)\d_\t^{\,-1}(x_j)
\t \dpr_k l_{u\t}(x_j)}
{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)\d^{-1}_\t(x_j)}\cdot
\fra{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)\d_\t^{\,-1}(x_j)
\dpr_h\lis\s_{\t}(x_j)}
{\sum_j\lis\L_{u,\t}^{\,-1}(x_j)\d^{-1}_\t(x_j)}\cr
=&\fra1{
L^2}\media{\dpr_{hk}\lis\s_{\t}}-\fra\t{L^2}\big(\media{\dpr_k l_{u,\t}
\dpr_h\lis\s_{\t}}+
\media{\dpr_k l_{u,\t}}\media{\dpr_h\lis\s_{\t}}\big)&\eq(5.4)\cr
\cr}$$
%
Here we have interpreted the derivatives with respect to $E_h$ of
$\lis\s_{\t}(x)$ and $l_{u\t}(x)$ by regarding $x$ as $\V E$
independent. However this is {\it not} quite correct: in fact it is
clear that we must consider such functions as defined on the attractor,
not on the full phase space. The attractor depends on $\V E$: it can in
fact be identified with the unstable manifold $W^u_O$ of a fixed point
$O$ or of a periodic orbit $O$ (see [GC2], \S4): hence the point $x_j$,
which has to be thought as a point on the attractor, will change with
$\V E$ even though it keeps the same symbolic representation (note that
the Markov partition changes with $\V E$ although the compatibility
matrix does not, by the structural $\V E$ stability theorem of Anosov,
[AA], at least if $\V E$ is small).
In taking the derivatives with respect to $\V E$ of $l_{u\t}(x_j)$ and
in defining the current as $\dpr_{E_h}\lis\s_{\t}(x_j)$ there are therefore
additional contributions proportional to $\dpr_{E_h} x$. The latter
quantity can be considered as a function of the symbolic history of
$x$, \ie as the function $\dpr_{E_h}x(\V j)$ and in [G4] it is
conjectured that:
\*
\0{\it Conjecture: The function $\dpr_{E_h}x(\V j)$ is a {\it local}
function in the sense of the second theorem in \S5 for all Anosov
systems, or axiom A systems, depending smoothly on parameters $\V E$.}
\*
Assuming the validity of the conjecture and using it to perform
rigorously an interchange of limits one can check, see [G4] for details,
that the extra terms in the $\V E$--derivatives of $\lis\s_{\t}(x(\V
j)),l_{u,\t}(x(\V j))$ at fixed history $\V j$ just discussed {\it give
no contribution} to the end result, \ie they do not alter the validity
of Onsager's reciprocity or of the fluctuation dissipation relation,
derived below. Therefore, to avoid formal intricacies, we
shall not take into account the extra terms and we proceed by ignoring
them in \equ(5.4) as well as in the following. The above conjecture has
a mathematical nature and I do not discuss its proof here: I have not
attempted to prove it (it seems closely related to Anosov's structural
stability theorem, see [AA]).
By using the time reversal invariance we see that:
%
$$\eqalignno{
&\media{\dpr_k l_{u,\t}\dpr_h\lis\s_{\t}}=
\fra{\sum \lis\L_{u,\t}^{\,-1}\d_\t^{-1}
\dpr_k l_{u,\t}\,\dpr_h\lis\s_{\t}}Z=&\eq(5.5)\cr
&=\fra{\sum_j\big(
\lis\L_{u,\t}^{\,-1}(x_j)
\d_\t^{\,-1}(x_j)\,\dpr_k l_{u,\t}(x_j)\,\dpr_h\lis\s_{\t}(x_j)+
\lis\L_{u,\t}^{\,-1}(i x_j)\d_\t^{\,-1}(ix_j)\dpr_k l_{u,\t}(i
x_j)\,\dpr_h\lis\s_{\t}(ix_j) \big)}{2Z}\cr}$$
%
where $Z$ denotes the ``partition sum'', \ie the sum in the denominator
of \equ(5.4), and the averages are with respect to the distribution
$\m_{\t/2,\t}$.
Recalling that (see \equ(4.5), \equ(4.6))
$l_{u,\t}(ix)=l_{s,\t}(x),\,\lis\s_{\t}(ix)=-\lis\s_{\t}(x)$ this becomes:
%
$$\fra{\sum_j\big(
\lis\L_{u,\t}^{\,-1}(x_j)\d_\t^{\,-1}(x_j)\dpr_k l_{u,\t}(x_j)
-\lis\L_{s,\t}(x_j)\d_{-\t}^{\,-1}(x_j)\dpr_k
l_{s,\t}(x_j)\big)\,\,\dpr_h\lis\s_{\t}(x_j) }{2Z}\Eq(5.6)$$
%
The derivatives at $E_1=E_2=0$ can be computed immediately by noting
that {\it in such case}, $\lis\L_{u,\t}(x)\lis\L_{s,\t}(x)
\fra{\d_\t(x)}{\d_{-\t}(x)}\=1$ (see \equ(4.6)). If we use that
\equ(4.6) implies $l_{u,\t}-l_{s,\t}=t_0\t \lis\s_{\t}$
then it follows, from \equ(5.6), that:
%
$$\Big(\media{\dpr_k l_{u,\t}\,\dpr_h\lis\s_{\t}}-
\media{\dpr_k l_{u,\t}}\media{\dpr_h\lis\s_{\t}}\Big)
\Big|_{\V E=\V0}
=\fra{\t t_0}2\Big(\media{\dpr_k\lis\s_{\t}\,\dpr_h\lis\s_{\t}}-
\media{\dpr_k\lis\s_{\t}}\media{\dpr_h\lis\s_{\t}}\Big)
\Big|_{\V E=\V0}\Eq(5.7)$$
%
We also see that (since $\media{\dpr_{hk}\lis\s_{\t}}$ vanishes in the
present case):
%
$$\dpr_k j_h=\lim_{\t\to\io}\fra{t_0}{2\t L^2}\sum_{m=-\t/2}^{\t/2-1}
\sum_{n=-\t/2}^{\t/2-1}\big(\media{\dpr_k\s(S^m\cdot)\dpr_h\s(S^m\cdot)
}-\media{\dpr_k\s(S^m\cdot)}\media{\dpr_h\s(S^m\cdot)}\big)\Eq(5.8)$$
%
where the averages in the r.h.s. are with respect to $\m_{\t/2,\t}$.
Hence we see that, apart from a further problem of interchange of
limits (see below), \equ(5.8) becomes:
%
$$\dpr_k j_h=\fra{t_0}{2L^2}
\sum_{m=-\io}^\io \big(\,\media{\dpr_k\s(S^m\cdot)
\dpr_h\s(\cdot)}-\media{\dpr_k\s(\cdot)}\media{\dpr_h\s(\cdot)}\,
\big)\Eq(5.9)$$
%
where the averages are with respect to the SRB distribution (\ie to the
limit of $\m_{\t/2,\t}$).
The problem of interchange of limits is easily solved: under our
assumption that the system is a transitive Anosov system the
correlations of smooth observables decay exponentially (because they
become local observables in the symbolic dynamics interpretation of the
evolution, provided by the Markov partitions), not only for $\m$ but
also for $\m_{\t/2,\t}$ (in the natural sense in which this may be
interpreted in a finite $\t$ case; \eg by regarding the interval
$[-\fra\t2,\fra\t2]$ as a circle), and uniformly in $\t$.
The relation \equ(5.9) implies that setting $L_{hk}=\media{\dpr_h
j_k}|_{\V E=\V0}$ then:
%
$$L_{hk}=L_{kh}\Eq(5.10)$$
%
follows. Note that \equ(5.9) expresses the {\it fluctuation dissipation}
relation between the transport matrix $L$ and the current--current
equilibrium correlation.
In the case of the model in \S4 the situation is similar. If $\sum_j
\fra{\pp_j^2}m=2Nk_B T(x)$ and $J_q$ denotes the heat $-q_+=-Q_+/L^2$
received by the gas from the thermostat $\RR_+$ per unit time and
volume:
%
$$\eqalign{\fra{J}{k_BT}=&\fra{2N-1)}{L^2}\fra{\sum_j
\pp_j\cdot\V i/m}{\sum\pp_j^2/m}=\fra1{L^2}\dpr_E\s\cr
\fra1{T_+}\fra{J_q}{T_+}=&\fra{2N_+-1}{L^2}\fra{-Q_+}{2N_+k_B
T_+^2}=\fra1{L^2}\dpr_{\d T}\s\cr}\Eq(5.11)$$
%
Hence the above argument yields, for the model in \S4:
%
$$\dpr_{\d T}\media{\fra{J}{k_BT}}\Big|_{\d
T,E=0}=\dpr_E\media{\fra1T\fra{J_q}{k_BT}}\Big|_{\d T,E=0}\Eq(5.12)$$
%
In general we can consider changing two parameters denoted $a,b$, {\it
thermodynamic forces}, which control equations of motion of the system.
Suppose that the entropy generation per unit time $\s$ has the form:
%
$$\s=\sum_r D_r\fra{Q_r}{\sum^r\pp_j^2/m}\Eq(5.13)$$
%
where $\sum^r$ denotes that the coordinates $\qq_j$ are coordinates of a
particle belonging to a group of $N_r$ particles whose phase points are
constrained by the $r$-th constraint that we impose on the system (to
fix the coordinates that would evolve with a zero Lyapunov exponent, in
the thermodynamic limit). And let $D_r$ be the number of degrees of
freedom of the $r$-th group of particles (in $2$ space dimensions
$D_r\simeq 2N_r$); then the above argument can be immediately
generalized to yield that the {\it flows} $J_a=\media{\dpr_a\s}$ and
$J_b=\media{\dpr_b\s}$ verify:
%
$$\dpr_b J_a\,\Big|_{a,b=0}{\buildrel def\over=}
\,L_{12}=L_{21}\,{\buildrel def\over=}\,\dpr_a J_b\,\Big|_{a,b=0}
\Eq(5.14)$$
%
which is a general Onsager reciprocity relation between "thermodynamic
forces" and "currents". From \equ(5.7) we also see that the matrix
$\dpr_b J_a$ is positive definite.
Note that, as mentioned above, in defining $\dpr_a\s,\dpr_b\s$ one has
really to think of $\s$ as defined on the space of the symbolic
sequences $\s=\s(x(\V j))$ (so that $\dpr_a\s=\fra{\dpr \s}{\dpr a}(x(\V
j))+\fra{\dpr \s}{\dpr x}\fra{\dpr x(\V j)}{\dpr{a}}$: this conceptually
different from the ``naive'' $\fra{\dpr\s}{\dpr a}(x(\V j))$ although
({\it in the above models} it does not affect the end result).
\*
\0{\bf\S6. Remarks.}
\numsec=6\numfor=1\pgn=1
\*
(1) The models in \S3,\S4 have been considered as undergoing
transformations at constant energy $U$. This is not very satisfactory as
one also, and mainly, wants to understand also transformations in which
the internal energy is allowed to change. According to the ideas in \S2
this case can be treated by imposing that $U$ is constant; but the
constant value is fixed on the basis of the {\it equation of state} of
the system. The latter is the relation linking $U$ to the other system
parameters:
%
$$U=f(E^1,E^2),\qquad U=f(E,T_-,T_+)\Eq(6.1)$$
%
in the cases of the models of \S3,\S4. Here $f$ is determined by the
dynamics itself, but its computation requires mathematical difficulties
that we cannot expect to be able to solve (in general).
Nevertheless we can proceed as in the previous section: in taking the
derivatives with respect to the parameters there will be extra terms
that arise from the fact that the energy surface changes as described by
\equ(6.1), but the basic symmetry $L_{hk}=L_{kh}$ remains, as one can
check (by taking also into account that in the equilibrium state
obtained when the thermodynamic forces, \ie the variations from the
equilibrium values of the parameters, are set to $0$ then the currents
vanish).
\*
(2) It is quite clear that the discussion of the previous sections can
be extended to many other models. But it is not clear how far one can
really extend the considerations. For instance it would be desirable to
extend, if possible, the considerations to a microscopic model of a
macroscopic continuum obeying the macroscopic equations for a fluid or a
mixture of fluids (possibly with chemical reactions allowed), as defined
in [DGM].
\*
(3) Onsager relations are often regarded to be consequences solely of
the reversibility of the equilibrium dynamics, (see, however, [DJL]).
Therefore they {\it must} hold also for our models simply because they
could be derived "as usual", see [DGM], p. 100-101, for a "usual
derivation".
Hence it is worth pointing out that the "usual derivation" rests on
several assumptions, none of which is needed {\it if} the chaotic
hypothesis is retained, at least in a dynamics of the type considered
in the above models.
\*
(4) The "usual derivation" assumptions are the following;
\*
\0(a) {\it linear regression}, \ie the "time behaviour of the state
parameters can be described by linear equations", see [DGM], p. 36 and
p.100.
\\
(b) The "Boltzmann postulate": \ie the equilibrium distribution is such
that the "state parameters" $\V a$ obey a gaussian {\it large deviation
law}, see [DGM], p. 91 eq. (46). This means that they have an
equilibrium probability distribution $p(\V a)$ proportional to
$e^{-\fra1{2k_B} G\V a\cdot\V a}= e^{S(\V a)/k_B}$ where $S$ is the
"entropy" of the state with state parameters $\V a$. The entropy is
{\it defined} by the above formula but it is treated as if it had the
properties we may expect from a genuine equilibrium entropy function
(note that this is an assumption). By "large" one means an amount much
larger than the root mean square values at equilibrium (see [DGM],
p. 100), \ie macroscopic (but still very small). One should remark that
the gaussian nature {\it is not} a consequence of a normal distribution
assumption as the latter usually concerns {\it small deviation} of the
order of the root mean square values. On the other hand a careful
examination of the proof shows that all is really needed is that $-S(\V
a)$ is a convex smooth function near $\V a=\V0$. Therefore the really
strong part of the assumption above is that $S(\V a)$ {\it behaves as an
ordinary entropy function} (a concept that would require some more
precise definition).
\\
(c) The equilibrium evolution is {\it reversible}.
\*
Then it follows that the time evolution of a fluctuation is such that
the "state parameters" $\V a$ verify $\dot{\V a}= L \V X$ with $\V
X=-G\V a$ and the symmetry $L=L^T$, see [DGM], p. 101$\div$102.
An initial (distribution of) microscopic configurations, close to the
equilibrium state, generates a macroscopic state in which fluctuations
are possible: so one can consider the free evolution of a state in which
the initial "state parameters" have an average value off by $\V a$ from
the equilibrium value $\V0$. The evolution will then verify the above
"symmetry" relation.
For instance $\V a=(a_1,a_2)$ could be the horizontal center of mass
coordinates of the two species of particles in model 1 above.
\*
(5) The connection with "reality" requires further assumptions.
Considering our model 1 for definiteness, suppose that we act on the
system with small forces thus driving an evolution of the average values
of the "state parameters" $t\to\V a(t)$, and creating an entropy per
unit time $\fra{(\dot a_1 N_1E^1 +\dot a_2 N_2 E^2)}{T}$ (see \S2: this makes
sense for small fields when the temperature $T$ can be identified with
the average kinetic energy per particle). Note also that, by (b) above,
the entropy creation rate is in general $\dot S=-G\V a \cdot\V a\=\V
X\cdot\dot{\V a}$ with $\V X=-G\V a$.
\\
(d) Then Onsager supposes (see [O]: ``As before we shall assume that the
average regression of fluctuations obey the same laws as the
corresponding macroscopic irreversible processes'') that a fluctuation
forced by external forces evolves as if it had occurred spontaneouly;
\ie if $\V X$ is given, $\V X=\fra{(N_1E^1,N_2E^2)}T$,
recalling that $N_1=N_2=\fra{N}2$, it is $\dot{\V a}=\fra{N}2 L
\fra{\V E}T$ (or $\V j=\fra{N}2 L \fra{\V E}T$, in the notations of the
present paper), with $L_{12}=L_{21}$. Note that this is done (and can
only be correct) up to corrections $O(\V E^2)$.
Other derivations are more fundamental and are based on kinetic theory
(see [C], [DGM]) or on the pure microscopic dynamics ([DGM], [D]), but
still they rely on various assumptions besides time reversibility of the
equilibrium dynamics.
\*
\ifnum\mgnf=0\pagina\fi
(6) With our chaotic hypothesis all the above assumptions (a)$\div$(d)
are {\it not necessary}, if one considers the models in \S2,\S3, because
the final result (\ie $\V j$ proportional to $L \V E$ with $L$ symmetric
and positive definite) has been drawn without further hypotheses (other
than the mathematical conjecture in
\S5). Of course assuming the reversibility of the dynamics {\it even}
in nonequilibrium (close to equilibrium) {\it and} the chaotic
hypothesis is in some sense much stronger than the assumptions
(a)$\div$(d) (but note that the reversibility in nonequilibrium ,
close to equilibrium, is a form of the assumption (d)).
It has, {\it however}, the basic advantage of being a conceptually
simple general assumption for nonequilibrium statistical mechanics,
which should furthermore be valid without even the restriction of being
close to equilibrium.
\*
As a final remark one should add that, although reversibility of the
nonequilibrium dynamics is assumed in this paper, it is quite likely
that the ideas and methods can be extended to genuinely non reversible
models. Attempts at such applications can be already found in
[GC1],[GC2],[G4] and I hope that they can be generalized to more general
physical situations quite beyond the, so far special, cases mentioned.
A much debated question is the extension of the Onsager relations to the
really nonequilibrium regime (\ie not close to equilibrium). In this
case the very notion of reciprocity becomes ill defined: recently the
proposal of interpreting the relations as the statement that the flow of
the state parameters is, in a suitable sense, a ``gradient flow'' has
emerged, [GJL]. The \CH\ in principle applies also in the nonlinear
regimes but it is unlikely that it can be as powerful a tool to cover
deterministic versions of the concrete, exactly treated, stochastic
models of [GJL].
\*\*
{\bf Appendix A1:\it The Gauss minimal constraint principle.}
\numsec=1\numfor=1\pgn=1
\*
Let $\f(\dot \xx,\xx)=0$ be a constraint and let $\V R(\dot\xx,\xx)$ be
the constraint reaction and $\V F(\dot\xx,\xx)$ the active force.
Consider all the possible accelerations $\V a$ compatible with the
constraints and a given initial state $\dot\xx,\xx$. Then $\V R$ is {\it
ideal} or {\it verifies the principle of minimal constraint} if the
actual accelerations $\V a_i=\fra1{m_i} (\V F_i+\V R_i)$ minimize the
{\it effort}:
$$\sum_{i=1}^N\fra1{m_i} (\V F_i-m_i\V a_i)^2\ \otto\
\sum_{i=1}^N (\V F_i-m_i\V a_i)\cdot\d \V a_i=0\Eqa(A1.1)$$
\0for all possible variations $\d \V a_i$ compatible with the
constraint $\f$.
Since all possible accelerations following $\dot\xx,\xx$ are such that
$\sum_{i=1}^N \dpr_{\dot\xx_i}\f(\dot\xx,\xx)\cdot\d \V a_i=0$ we can write:
$$\V F_i-m_i\V a_i-\a\,\dpr_{\dot\xx} \f(\dot\xx,\xx)=\V0\Eqa(A1.2)$$
\0with $\a$ such that $\fra{d}{dt}\f(\dot\xx,\xx)=0$, \ie:
$$\a=\fra{\sum_i\,(\dot\xx_i\cdot\dpr_{\dot\xx_i} \f+\fra1{m_i} \V
F_i\cdot\dpr_{\dot\xx_i}\f)}{\sum_i m_i^{-1}(\dpr_{\dot\xx_i}\f)^2}
\Eqa(A1.3)$$
\0which is the analytical expression of the Gauss' principle.
\*
\0{\it Acknowledgements:} I am indebted to J.L. Lebowitz, G.L.
Eyink and G. Jona for very helpful comments and in particular to
E.G.D. Cohen for many very valuable and inspiring suggestions and hints
and for his constant interest and encouragement. This work is part of
the research program of the European Network on: "Stability and
Universality in Classical Mechanics", \# ERBCHRXCT940460.
%\pagina
\*
\0{\bf References.}
\*
%\let\item=\0
\item{[AA]} Arnold, V., Avez, A.: {\it Ergodic problems of classical
mechanics}, Benjamin, 1966.
\item{[Bo]} Bowen, R.: {\it Equilibrium states and the ergodic theory of
Anosov diffeomorphisms}, Lecture notes in mathematics, vol. {\bf 470},
Springer Verlag, 1975.
\item{[BEC]} Baranyai, A., Evans, D.T., Cohen, E.G.D.: {\it
Field dependent conductivity and diffusion in a two dimensional Lorentz
gas}, Journal of Statistical Physics, {\bf70}, 1085-- 1098, 1993.
\item{[C]} Cohen, E.G.D.: {\it },
H.J.M. Hanley, ed., Marcel Dekker, 1969.
\item{[CG1]} Gallavotti, G., Cohen, E.G.D.: {\it Dynamical ensembles in
nonequilibrium statistical mechanics}, Physical Review Letters,
{\bf74}, 2694--2697, 1995.
\item{[CG2]} Gallavotti, G., Cohen, E.G.D.: {\it Dynamical ensembles in
stationary states}, in print in Journal of Statistical Physics, 1995.
\item{[D]} Dorfman, J.R.: {\it Transport coefficients}, entry for the
"Enciclopedia delle Scienze Fisiche", Enciclopedia Italiana, Roma, 1995
(in italian, preprint in english available from the author), p. 1-31.
\item{[DGM]} de Groot, S., Mazur, P.: {\it Non equilibrium thermodynamics},
Dover, 1984.
\item{[E]} Eyink, G.: {\it Turbulence noise}, in {\it mp$\_$arc@ math.
utexas. edu}, \#95-254, 1995.
\item{[ECM1]} Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Viscosity of a
simple fluid from its maximal Lyapunov exponents}, Physical Review, {\bf
42A}, 5990--\-5997, 1990.
\item{[ECM2]} Evans, D.J.,Cohen, E.G.D., Morriss, G.P.: {\it Probability
of second law violations in shearing steady flows}, Physical Review
Letters, {\bf 71}, 2401--2404, 1993.
\item{[ELS]} Eyink, G.L., Lebowitz, J.L., Spohn, H.: {\it Hydrodynamics and
Fluctuations Outside of Local Equilibrium: Driven Diffusive Systems},
in {\it mp$\_$arc@math.utexas.edu}, \#\-95\--168.
\item{[EM]} Evans, D.J., Morriss, G.P.: {\it Statistical Mechanics of
Nonequilibrium fluids}, Academic Press, New York, 1990.
\item{[EMY]} Esposito, R., Marra, R., Yau, H.T.: {\it Diffusive Limit of
Asymmetric Simple Exclusion}, In: ``The State of Matter'',
ed. M. Aizenmann, H. Araki., World Scientific, Singapore, 1994.
\item{[ES]} Evans, D.J., Searles, D.J.: {\it Equilibrium microstates which
generate second law violating steady states}, Research school of
chemistry, Canberra, ACT, 0200, preprint, 1993.
\item{[G1]} Gallavotti, G.: {\it Topics in chaotic dynamics}, Lectures
at the Granada school, ed. Garrido--Marro, Lecture Notes in Physics,
{\bf 448}, 1995.
\item{[G2]} Gallavotti, G.: {\it Reversible Anosov diffeomorphisms and
large deviations.}, Mathematical Physics Electronic Journal,
{\bf 1}, 1-12, 1995.
\item{[G3]} Gallavotti, G.: {\it Coarse graining and chaos},
in preparation.
\item{[G4]} Gallavotti, G.: {\it Chaotic principle: some applications to
developed turbulence}, in {\it mp$\_$arc @math. utexas. edu}, \#95-232, 1995.
\item{[GCDR]} Garcia--Colin, L.S., Del Rio, J.L.: {\it Green's
contributions to nonequilibrium statistical mechanics}, in
"Perspectives in statistical physics: M.S. Green memorial volume",
ed. H.J. Ravech\'e, p. 75--87, North Holland, 1981.
\item{[GD1]} Gaspard, P., Dorfman, J.R.: {\it Chaotic scattering theory,
of transport and reaction rate ccoefficients}, Physical Review, {\bf
E51}, 28--35, 1995.
\item{[GD2]} Gaspard, P., Dorfman, J.R.: {\it Chaotic scattering theory,
thermodynamic formalism and transport coefficients}, in {\it
chao-dyn@xyz.lanl.gov}, \# 9504014.
\item{[GJL]} Gabrielli, D., Jona--Lasino, G., Landim, C.: {\it
Microscopic reversibility and thermodynamic fluctuations},
in {\it mp$\_$arc@ math. utexas. edu}, \#95-248, 1995.
\item{[HHP]} Holian, B.L., Hoover, W.G., Posch. H.A.: {\it Resolution of
Loschmidt's paradox: the origin of irreversible behavior in reversible
atomistic dynamics}, Physical Review Letters, {\bf 59}, 10--13, 1987.
\item{[KV]} Kipnis, C., Olla, S., Varadhan, S.R.S., {\it }, Communications
in Pure and Applied Mathematics, {\bf XLII}, 243--, 1989.
\item{[LA]} Levi-Civita, T., Amaldi, U.: {\it Lezioni di Meccanica
Razionale}, Zanichelli, Bologna, 1927 (re\-prin\-ted 1974), volumes
$I,II_1,II_2$.
\item{[LOY]} Landim, C., Olla, S., Yau, H.T.; {\it First-order
correction for the hydrodynamic limit of asymmetric simple exclusion
processes in dimension $d\ge 3$}, preprint, Ecole Polytechnique,
R. I. No. 307, Nov. 1994.
\item{[LPR]} Livi, R., Politi, A., Ruffo, S.: {\it Distribution of
characteristic exponents in the thermodynamic limit}, Journal of
Physics, {\bf 19A}, 2033--2040, 1986.
\item{[O]} Onsager, L: {\it Reciprocal relations in irreversible
processes. II}, Physical Review, {\bf 38}, 2265--2279, 1932.
\item{[PH]} Posch, H.A., Hoover, W.G.: {\it Non equilibrium molecular
dynamics of a classical fluid}, in "Molecular Liquids: New Perspectives
in Physics and Chemistry", ed. J. Teixeira-Dias, Kluwer Academic
Publishers, p. 527--547, 1992.
\item{[R1]} Ruelle, D.: {\it Chaotic motions and strange attractors},
Lezioni Lincee, notes by S. Isola, Accademia Nazionale dei Lincei,
Cambridge University Press, 1989; see also: Ruelle, D.: {\it Measures
describing a turbulent flow}, Annals of the New York Academy of
Sciences, {\bf 357}, 1--9, 1980. For more technical expositions see
Ruelle, D.: {\it Ergodic theory of differentiable dynamical systems},
Publications Math\'emathiques de l' IHES, {\bf 50}, 275--306, 1980.
\item{[R2]} Ruelle, D.: {\it A measure associated with axiom A
attractors}, American Journal of Mathematics, {\bf98}, 619--654, 1976.
\item{[S]} Sinai, Y.G.: {\it Gibbs measures in ergodic theory}, Russian
Mathematical Surveys, {\bf 27}, 21--69, 1972. Also: {\it Lectures in
ergodic theory}, Lecture notes in Mathematics, Prin\-ce\-ton U. Press,
Princeton, 1977.
\item{[UF]} Uhlenbeck, G.E., Ford, G.W.: {\it Lectures in Statistical
Mechanics}, American Mathematical society, Providence, R.I.,
pp. 5,16,30, 1963.
\bye