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\centerline{POSITIVITY OF ENTROPY PRODUCTION}
\bigskip
\centerline{IN THE PRESENCE OF A RANDOM THERMOSTAT.}
\bigskip
\bigskip
\centerline{by David Ruelle\footnote{*}{IHES (91440 Bures sur Yvette, France),
and Math. Dept., Rutgers University (New Brunswick, NJ 08903, USA).}}
\bigskip
\bigskip
\indent
          {\it Abstract.}  We study nonequilibrium statistical mechanics
in the presence of a thermostat acting by random forces, and propose a
formula for the rate of entropy production $e(\mu)$ in a state $\mu$. 
When $\mu$ is a natural nonequilibrium steady state we show that
$e(\mu)\ge0$, and sometimes we can prove $e(\mu)>0$.
\bigskip
\bigskip
\indent
	{\it Key words and phrases:} entropy production,
nonequilibrium stationary state, nonequilibrium statistical mechanics,
random dynamics, SRB state, thermostat.
\vfill\eject           	
\centerline{0. Physical introduction.}
\bigskip
\bigskip
\indent
	The production of entropy in nonequilibrium statistical
mechanics was analyzed in various settings in [19].  Here we continue this
analysis by studying the role of a heat bath, and its idealization by
random external forces.
\medskip\indent
	As discussed in [19], we maintain a system outside of equilibrium
by external forces which, in general, do not keep the energy constant. 
If no precaution is taken, the time evolution is then represented by an
unbounded orbit in the noncompact phase space ${\cal S}$ of the system,
{\it i.e.}, the energy grows indefinitely.  Physically, this heating up is
prevented by coupling the system to a thermostat, or heat bath.  The
energy changes are diluted in the large heat bath, so that the system
keeps a bounded energy.
\medskip\indent
	If $\Omega$ is the phase space of the heat bath, we assume thus
a deterministic time evolution in $\Omega\times{\cal S}$, and we are
interested in its projection in ${\cal S}$.  A priori, however, the
projected time evolution in ${\cal S}$ has no simple description because
the mutual interactions of the system and the heat bath cannot be
disentangled.  (See however Jak\v si\'c and Pillet [7]).  One is thus led to
considering a simplified model where the heat bath acts on the system, but
the system does not act back on the heat bath.  We shall return in a moment
to the physical meaning of this simplification.
\medskip\indent
	The previous paper [19] and the present one are in the line of a
renewed attack on the problems of nonequilibrium mechanics, by using the
ideas and methods of differentiable mechanics.  This involves in
particular the important work of Gallavotti and Cohen [5], Chernov,
Eyinck, Lebowitz and Sinai [4], and a number of other papers referred to
in [19]. 
\medskip\indent
{\it Random dynamics}
\medskip\indent
	Our simplified model of a system coupled with a heat bath will
be described by a {\it random dynamical system} (with discrete time for
simplicity).  The heat bath itself is described by a probability space
$(\Omega, {\bf P})$, with an invertible measurable ${\bf P}$-preserving
map $\tau$ describing time evolution; ${\bf P}$ is assumed to be
$\tau$-ergodic.  (Further technical assumptions will be made later).
\medskip\indent
	For each $\omega\in\Omega$, a diffeomorphism $f_\omega$ of the
smooth (noncompact) manifold ${\cal S}$ is given, and the time evolution
on $\Omega\times{\cal S}$ is the skew-product transformation $f$ on
$\Omega\times{\cal S}$ defined by
$$	f(\omega,x)=(\tau\omega,f_\omega x)	$$
(Assumptions of smoothness of $f_\omega$ and measurability of
$\omega\mapsto f_\omega$ are to be discussed later).
\medskip\indent
{\it Thermostatic action of random forces.}
\medskip\indent
	It is easy to understand qualitatively how a heat bath prevents
the heating up of a system.  Suppose for example that the system is a
gas enclosed in a container, and that the thermostatic action of the
heat bath takes place when the particles of the gas hit the walls of the
container .  Shocks with the wall are not elastic .  When a particle
hits the wall at very high speed, it is released in the average with a
lower speed.  In this manner the energy of the gas is prevented from
increasing indefinitely (even though there are forces acting on the gas,
that maintain it outside of equilibrium, and usually transfer energy to
it).   
\medskip\indent
	The above thermostatic mechanism may be translated into the
language of random dynamical systems.  Suppose for example that the
$f_\omega$ are independent, identically distributed, and that there is
an energy function $x\mapsto E(x)\ge 0$ and constants $\bar E>0$, $A>0$,
$\alpha>0$, and $\beta\in (0,1)$ such that
$$	\langle e^{\alpha E(f_\omega x)}\rangle_\omega \le A\qquad{\rm if}
\qquad E(x)\le \bar E	$$
$$	\langle e^{\alpha E(f_\omega x)}\rangle_\omega \le\beta
e^{\alpha E(x)}\qquad{\rm if}\qquad E(x)>\bar E	$$
where $\langle\cdots\rangle_\omega$ is the average over $\omega$.  Let $\rho_0$
be a probability measure on ${\cal S}$, and $\rho_n$ the measure
obtained from it at time $n$ ({\it i.e.}, $\rho_{n+1}=
\langle f_\omega \rho_n\rangle_\omega$ where $f_\omega \rho$ is the
direct image of $\rho$ by $f_\omega$).  Write
$$	c_n=\int\rho_n (dx) e^{\alpha E(x)}	$$
and suppose $c_0<+\infty$; we have then
$$	c_{n+1}
=\int\langle(f_\omega \rho_n)(dx)\rangle_\omega e^{\alpha E(x)}     $$
$$	=\langle\int(f_\omega \rho_n)(dx) e^{\alpha E(x)}\rangle_\omega
=\langle\int\rho_n (dx) e^{\alpha E(f_\omega x)}\rangle_\omega				
=\int\rho_n (dx)\langle e^{\alpha E(f_\omega x)}\rangle_\omega      $$
$$	\le A\rho_n\{x:E(x)\le\bar E\}
+\beta\int_{E(x)>\bar E}\rho_n(dx)e^{\alpha E(x)}\le A+\beta c_n      $$
Therefore by induction
$$	c_n\le\beta^n c_0+{A\over{1-\beta}}	$$
which shows that
$$	\int\rho_n(dx)e^{\alpha E(x)}      $$
is bounded independently of $n$.
\medskip\indent
	In the above example the energy $E(x)$ is unbounded, but states
of high energy are rarely visited: the system does not heat up.  Notice
that we are not here separating the deterministic forces driving the
system outside of equilibrium from the random forces associated with the
heat bath.
\medskip\indent
{\it Using a compact phase space.}			
\medskip\indent
	We have just seen that in the presence of suitable random
forces, a system rarely comes close to infinity on ${\cal S}$. 
Physically, this is in agreement with the remarkable metastability of
systems that are, strictly speaking, unstable (like a mixture of oxygen
and hydrogen at room temperature).  In the presence of random forces
which prevent heating up of the system we might as well, physically,
assume that the phase space ${\cal S}$ is compact.  In other words we
want to modify ${\cal S}$ and the $f_\omega$ near infinity and replace
them by a compact manifold $M$ and diffeomorphisms (again denoted by
$f_\omega$) of $M$.  Note that this is a physical approximation, not a
mathematical compactification of ${\cal S}$.
\medskip\indent
	The reason for this modification of our setup is one of
mathematical convenience.  Namely that the theorems on random dynamical
systems which we shall use have been proved for compact manifolds. 
Extensions to noncompact manifolds presumably exist, but it does not
appear justified to spend much effort in proving them at this time.
\medskip\indent
{\it Loss of correlations in the heat bath.}
\medskip\indent
	Let us briefly return to the difference between a real heat bath
and random external forces.  When a real heat bath interacts with a
system, it is acted upon by the system, which transfers to it energy and
information.  The transfer of information means that correlations are
created between the state of the heat bath and that of the system.  Such
correlations do not exist in the case of a random dynamical system,
where the time evolution in $\Omega$ is independent of the factor
${\cal S}$.  A good heat bath has a short relaxation time: correlations
diffuse in it quickly, so that the action on the system appears random
(with a short correlation time) independently of the behavior of the
system.  A random dynamical system is thus an idealization of a
dynamical system in contact with a good heat bath.  Note that the
diffusion and loss of correlations in the heat bath corresponds to an
increase of the global entropy.  More generally, loss of correlations
may be viewed as the basic physical mechanism leading to increase of
entropy and irreversibility (this fits with the discussion by Lebowitz
[10]).  Admittedly, we are lacking a detailed physical understanding of how
correlations diffuse and are lost in a heat bath.  In the present paper
we bypass this fundamental question by using a random dynamical system
instead of a realistic heat bath.  It is physically reasonable to assume
that the $f_{{\tau^n}\omega}$ are independent and identically
distributed (i.i.d.) but since this assumption is unnecessary and tends
to confuse the issues, we shall first study the general case.
\medskip\indent
{\it Scope of the paper.}
\medskip\indent
	In Section 1 we review some properties of random differentiable
dynamical systems (without i.i.d. assumption).  In Section 2 we obtain
the formula for the entropy production, study its positivity.  In
Section 3 we discuss the special i.i.d. case.      
\medskip\indent
{\it Acknowledgements.}
\medskip\indent
	I am indebted to Viviane Baladi and Lai-Sang Young for useful
discussions on the issues involved in the present paper.
\vfill\eject
\centerline{1. Ergodic theory of random dynamical systems.}
\bigskip
\bigskip
\indent
	{\it Assumptions.}
\medskip\indent
	Let us fix our mathematical framework\footnote{*}{We follow here
the presentation by Liu [14] which can be consulted for some more details
and references.  The special i.i.d. case, which was studied earlier (see
Kiefer [8], Ledrappier and Young [13]), will be discussed in Section 3.  For
further background material, see Kiefer [9],Liu and Qian [15].}.  We
consider a random system consisting of a probability space $(\Omega,{\bf P})$,
a map $\tau:\Omega\to\Omega$ such that ${\bf P}$ is $\tau$-ergodic, a compact
manifold $M$, and a family $(f_\omega)_{\omega\in\Omega}$ of diffeomorphisms
of $M$.  To be specific, we make the following standing technical
assumptions:
\medskip\indent
	$\bullet$\quad$\Omega$ is a Polish space\footnote{**}{A
topological space $\Omega$ is called a Polish space if (a) $\Omega$ is
separable,{\it i.e.}, it contains a countable dense set, (b) there is a
metric on $\Omega$ which defines the topology, and for which $\Omega$ is
complete.  Part of the results below would hold without supposing
$\Omega$ Polish, but this assumption (made by Liu [14]) is quite
acceptable for our purposes (we could in fact make the stronger
assumption that $\Omega$ is metrizable compact).  }, ${\bf P}$
is a Borel probability measure on $\Omega$, $\tau:\Omega\to\Omega$ is
invertible, $\tau$ and $\tau^{-1}$ are Borel, ${\bf P}$ is
$\tau$-invariant and ergodic.
\medskip\indent
	$\bullet$\quad$M$ is a compact $C^\infty$ manifold.
\medskip\indent	
	$\bullet$\quad$\omega\mapsto f_{\omega}$ is a Borel map
$\Omega\to{\hbox{Diff}}^r (M)$.
\par\noindent
(allowed values of $r\ge 1$ will be specified as needed).
\medskip\indent
	$\bullet$\quad If $J_\omega$ is the Jacobian of $f_\omega$ with
respect to some Riemann metric, and $\ell(\omega)=\sup_x|\log J_\omega(x)|$,
then $\ell\in L^1({\bf P})$
\medskip\indent
	{\it Time entropy.}
\medskip\indent
	A Borel map $f:\Omega\times M\to\Omega\times M$ is defined
by $f(\omega,x)=(\tau\omega,f_\omega x)$ and we denote by $\pi:\Omega\times
M\to\Omega$ the canonical projection.  We assume that $\mu$ is
an $f$-invariant probability measure on $\Omega\times M$ with projection
$\pi\mu={\bf P}$.  A disintegration ${(\mu_\omega )}_{\omega\in\Omega}$
of $\mu$ then exists (${\bf P}$-a.e. unique) such that the $\mu_\omega$
are probability measures on $M$ and
$$	\mu(d\omega dx)= {\bf P}(d\omega)\mu_\omega (dx)	$$
It is convenient to write $f^k (\omega ,x)=(f^k)_\omega (x)$ (so that
for instance $(f^{-1})_\omega =(f_{{\tau}^{-1} \omega})^{-1})$.  Let then
$\beta$ be a finite Borel partition of $M$, and
$$	\beta_\omega^{(n)}=\beta\vee f_\omega^{-1} \beta\vee\cdots\vee
((f^{n-1})_\omega)^{-1} \beta	$$
If we write
$$	H(\mu_\omega,\beta_\omega^{(n)})=-\sum_{B\in\beta_\omega^{(n)}}
\mu_\omega (B)\log \mu_\omega (B)	$$
the limit
$$	h(\mu,\beta)
=\lim_{n\to\infty} {1\over n} H(\mu_\omega,\beta_\omega^{(n)})	$$
exists and is constant for {\bf P}-almost all $\omega$.  The
{\it fiber entropy} is defined by
$$	h(\mu)=\sup_\beta h(\mu,\beta)	$$
and turns out to coincide with the relative (or conditional) entropy of
$(\mu,f)$ with respect to $\pi:\Omega\times M \to \Omega$.  (For these
results see Bogensch\"utz [3], and for background the book of Kifer [8]). 
Note that $h$ is a time entropy, different from the statistical
mechanical entropy $S$ to be discussed later.  
\medskip\indent
	{\it Lyapunov exponents and unstable manifolds.}
\medskip\indent
	Choose a Riemann metric on the tangent bundle $TM$ (with
associated distance $d$ on $M$) and assume that
$$	\int[\log^+\parallel T_x f_\omega\parallel
+\log^+\parallel T_x f_\omega^{-1}\parallel]\mu(d\omega dx) <\infty	$$
One may then write
$$	-\infty<\lambda^{(1)}(\omega,x)<\lambda^{(2)}(\omega,x)<\cdots
<\lambda^{(r(\omega,x))}(\omega,x)<\infty	$$
and
$$	T_xM=E^{(1)}(\omega,x)\oplus\cdots\oplus E^{(r(\omega,x))}(\omega,x)$$	
so that $r$, the $\lambda^{(i)}$ and $E^{(i)}$ are Borel, and for
$\mu$-almost all $(\omega,x)$
$$	\lim_{n\to\pm\infty}{1\over n}\log\parallel T_x(f^n)_\omega\xi\parallel
=\lambda^{(i)}(\omega,x)	$$
if $0\ne\xi\in E^{(i)}(\omega,x)$, $1\le i\le r(\omega,x)$.  This is a
form of the multiplicative ergodic theorem of Oseledec; the
$\lambda^{(i)}$ are called {\it Lyapunov exponents}, and the $m^{(i)}
=\dim E^{(i)}$ are their multiplicities.  We define $E^u(\omega,x)$ to
be the sum $\oplus E^{(i)}(\omega,x)$ extended over those $i$ for which
$\lambda^{(i)}(\omega,x)\ge0$ (unstable space).
\medskip\indent
	Suppose now that the $f_\omega$ are $C^2$ ({\it i.e.},
$r\ge2$),and that
$$	\int[\log^+\parallel f_\omega\parallel_{C^2}+
\log^+\parallel f_\omega^{-1}\parallel_{C^2}]{\bf P}(d\omega)<\infty	$$
Given $\omega\in\Omega$, we partition $M$ into {\it unstable manifolds}
$W_\omega^u$ such that the $W_\omega^u$ containing $x$ is
$$	W_\omega^u(x)=\{y\in M:\limsup_{n\to+\infty}{1\over n}\log
d((f^{-n})_\omega x,(f^{-n})_\omega y)<0\}	$$
One can construct an $f$-invariant Borel set $\Delta\subset\Omega\times
M$, with $\mu(\Delta)=1$ such that, if $(\omega,x)\in\Delta$,
$W_\omega^u(x)$ is the image of $E^u(\omega,x)$ by a $C^{1,1}$ injective
immersion\footnote{*}{The class $C^{1,1}$ consists of functions with
Lipschitz continuous first order derivatives.  }.  The proof of the
above results can be obtained by the methods of Ruelle [17] as noted by
Liu [14].
\medskip\indent
	The SRB condition for the measure $\mu$ is, roughly speaking,
that the conditional measures on the unstable manifolds $(\omega, W_\omega^u)
\subset\Omega\times M$ be absolutely continuous with respect to the Riemann
measure.  Technically, however, one cannot directly define conditional
measures\footnote{**}{For the theory of conditional measures associated
with measurable partitions see Rohlin [16]} on the $(\omega, W_\omega^u)$.
 This is because they usually do not form a measurable partition of
$\Omega\times M$ (each $W_\omega^u$ may be folded over upon itself so
that its closure is $M$).  One defines a {\it local unstable manifold}
$W_\omega^u(x,\hbox{local}$) to be the graph of a smooth map from an open
neighborhood of $x$ in $E^u(\omega,x)$ to $E^u(\omega,x)^\perp$.
It is then possible to define measurable partitions of $\Omega\times M$
into sets $(\omega,S)$ where $S$ is on open subset of a local unstable
manifold (for its induced topology).  If the conditional measures of
$\mu$ on the $(\omega,S)$ are absolutely continuous with respect to the
Riemann measure of the unstable manifolds, then $\mu$ is said to satisfy
the SRB condition.
\medskip\indent
	We can now state two results of the ergodic theory of random
differentiable dynamical systems, which we shall use later in the
discussion of entropy production.  (For the proofs we refer to the
original papers).  
\medskip\indent
	{\it 1.1 Theorem.  If $r\ge1$ and
$$	\int {\bf P}(d\omega)\log^+\parallel f_\omega\parallel_{C^1}
\le+\infty	$$
then the fiber entropy (=conditional entropy of $(\mu,f)$ with respect to
the projection $\pi:\Omega\times M\to\Omega$) satisfies}
$$	h(\mu)\le\int\mu(d\omega dx)
\sum_{i:\lambda^{(i)}>0}\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)	$$
\indent
	This was first proved for a single $C^1$ map ({\it i.e.}, $\Omega$
reduced to one point), see Ruelle [18].  For the extension to the present
situation see Bahnm\"uller and Bogensch\"utz [2]\footnote{***}{A gap in
[2] is fixed in [1].  Note that, for Theorem 1.1, $\Omega$ need not
be Polish, and the $f_\omega$ are not required to be diffeomorphisms (it
suffices, as done in [18], to assume that they are $C^1$ maps).  } (their
paper gives a history of related results).
\medskip\indent
	{\it 1.2 Theorem.  If $r\ge2$ and
$$	\int {\bf P}(d\omega)[\log^+\parallel f_\omega\parallel_{C^2}
+\log^+\parallel f_\omega^{-1}\parallel_{C^2}]\le+\infty	$$
and if $\mu$ satisfies the SRB condition, then}
$$	h(\mu)=\int\mu(d\omega dx)
\sum_{i:\lambda^{(i)}>0}\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)	$$
\indent
	In the case of a single map, the above equality is known as
Pesin's formula, and it was shown to follow from the SRB condition by
Ledrappier and Strelcyn [11].  (In fact Pesin's formula is equivalent to
the SRB condition, as proved by Ledrappier and Young [12]).  The
generalization to random dynamical systems of the result of Ledrappier
and Strelcyn is due to Liu [14].  Liu's assumption that $\Omega$ is Polish
allows him to apply Lusin's Theorem\footnote{*}{To the effect that a
measurable function is in fact continuous outside of a set of small
measure.  More precisely, let $E$ be a Polish space, $F$ a topological
space with countable base, $\mu$ a (Borel, bounded) positive measure on
$E$, and $f:E\to F$ a Borel map.  Then there is $\tilde f$ equal
$\mu$-a.e. to $f$, and for each $\epsilon>0$ there is a compact set
$K\subset E$ such that $\mu(E\backslash K)\le\epsilon$ and $\tilde f|K$
is continuous.  } to connect arguments of abstract measure theory (measurable
partitions of Lebesgue spaces) and the topology of the problem.
\medskip\indent
	{\it Time reversal.}
\medskip\indent
	The measure $\mu$ is invariant under $f^{-1}:(\omega,x)\mapsto
(\tau^{-1}\omega,f_{\tau^{-1}}^{-1}x)$.  The fiber entropy computed with
respect to $f^{-1}$ is again $h(\mu)$.  (To see this one can for
instance use the fact that the relative entropies of $(\mu,f)$ and
$(\mu,f^{-1})$ with respect to the projection $\Omega\times M\to\Omega$
are the same).  It follows from the definition that the Lyapunov
exponents associated with $f^{-1}$ are the $-\lambda^{(i)}(\omega,x)$,
with multiplicity $m^{(i)}(\omega,x)$.  The unstable manifolds
associated with $f^{-1}$ are the stable manifolds associated with $f$
(we shall not use them).
\vfill\eject
\centerline{2. Entropy production.  }
\bigskip
\bigskip
\indent
	Keeping the assumptions of Sections 1, we let $J_\omega$ denote
the absolute value of the Jacobian of $f_\omega$ with respect to some
Riemann metric.  If $\rho$ is any (Borel) probability measure on
$\Omega\times M$ we may define the entropy production $e_f(\rho)$ by
$$	e_f(\rho)=-\int\rho(d\omega dx)\log J_\omega(x)	  \eqno(2.1)  $$
We shall use this definition only when $\pi\rho={\bf P}$ (where $\pi$ is
the projection $\Omega\times M\to\Omega$).  We have then
$$	|e_f(\rho)|\le\int {\bf P}(d\omega)\sup_x|\log
J_\omega(x)|\le\infty    $$
\medskip\indent
	{\it 2.1 Lemma.  The entropy production $e_f(\mu)$ for the
$f$-invariant probability measure $\mu$ is independent of the choice of
Riemann metric on $M$, and
$$	e_f(\mu)=-\int\mu(d\omega dx)\sum_i
\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)	$$ }
\indent
	This expression of $e_f(\mu)$ in terms of the Lyapunov exponents
follows from the multiplicative ergodic theorem, and implies independence
of the choice of metric.\qed
\medskip\indent
	{\it 2.2 Theorem.  If the f-invariant probability measure $\mu$
satisfies the SRB condition, then $e_f(\mu)\ge0$.  }
\medskip\indent
	By Lemma 2.1, we have
$$e_f(\mu)=-\int\mu(d\omega dx)\sum_i\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)$$	
$$	=h(\mu)-\int\mu(d\omega dx)\sum_{i:\lambda^{(i)}>0}
	\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)     $$
$$	-[h(\mu)+\int\mu(d\omega dx)\sum_{i:\lambda^{(i)}<0}
	\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)]	$$
so that, by Theorem 1.2,
$$	e_f(\mu)=-[h(\mu)-\int\mu(d\omega dx)\sum_{i:\lambda^{(i)}<0}
	|\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)|]	$$
If $\bar h$, $\bar\lambda$ denote the fiber entropy and Lyapunov
exponents for the time reversed system ({\it i.e.}, when $f$ is replaced
by $f^{-1}$, see end of Section 1) we have thus
$$	e_f(\mu)=-[\bar h(\mu)-\int\mu(d\omega dx)\sum_{i:\bar\lambda^{(i)}>0}	
	\bar\lambda^{(i)}(\omega,x)m^{(i)}(\omega,x)]	$$
Finally, Theorem 1.1 applied to the time reversed system gives
$e_f(\mu)\ge0$.\qed
\medskip\indent
	{\it Comment.  }
\medskip\indent
	If the probability measure $\rho$ is absolutely continuous with
respect to the Riemann volume on $M$, one expects a limit $\mu$ of
$f^k\rho$ when $k\to+\infty$ to be smooth along unstable directions, {\it
i.e.}, to be an SRB measure.  If we accept that (2.1) represents the
physical entropy production (see below) then Theorem 2.2 means that the
physical entropy production is positive. 
\medskip\indent
	{\it Relation of $e_f(\rho)$ with statistical mecanical entropy.  }
\medskip\indent
	For any (Borel) probability measure $\rho$ on $\Omega\times M$, such
that $\pi\rho={\bf P}$, we have the ({\bf P}-a.e. unique) disintegration
$(\rho_\omega)_{\omega\in\Omega}$ where the $\rho_\omega$ are
probability measures on $M$, and
$$	\rho(d\omega dx)={\bf P}(d\omega)\rho_\omega(dx)	$$
We shall use in a moment the fact that\footnote{*}{We have indeed
$$	\int (f\rho)(d\omega dx)\Phi(\omega,x)=\int\rho(d\omega dx)
\Phi(\tau\omega,f_\omega x)	
	=\int{\bf P}(d\omega)\rho_\omega(dx)\Phi(\tau\omega,f_\omega x)  $$
$$	=\int{\bf P}(d\omega)(f_\omega\rho_\omega)(dx)\Phi(\tau\omega,x)  
	=\int{\bf P}(d\omega)(f_{\tau^{-1}\omega}\rho_{\tau^{-1}\omega})(dx)
	 \Phi(\omega,x)	  $$}
$$	(f\rho)(d\omega dx)
={\bf P}(d\omega)\times(f_{\tau^{-1}\omega}\rho_{\tau^{-1}\omega})(dx) 
\eqno(2.2)     $$
Let us assume that the $\rho_\omega$ are absolutely continuous with
respect to the Riemann volume on $M$.  We write then $\rho_\omega(dx)
=\underline\rho_\omega(x)dx$, where $\underline\rho_\omega$ is the density of
$\rho_\omega$ with respect to the Riemann volume element $dx$.  Interpreting
$dx$ as phase space volume element we define the entropy
$$	S(\underline\rho_\omega)=-\int dx
\underline\rho_\omega(x)\log\underline\rho_\omega(x)       $$
(which is $\le$log vol$M$).  The entropy corresponding to $\rho$ is then
the average
$$	S_\rho=\int{\bf P}(d\omega)S(\underline\rho_\omega)
=-\int\rho(d\omega dx)\log\underline\rho_\omega(x)     $$
and $-\infty\le S_\rho\le$log vol$M$.
\medskip\indent
	Note that
$$	(f_\omega\rho_\omega)(dx)={\underline\rho_\omega(f_\omega^{-1}x)
\over J_\omega(f_\omega^{-1}x)}dx   $$
Using (2.2) thus yields
$$	(f\rho)(d\omega dx)={\bf P}(d\omega)\times
\underline\rho_{\tau^{-1}\omega}((f^{-1})_\omega x)
\bar J_{\tau^{-1}\omega}((f^{-1})_\omega x)dx   $$
where we have written $\bar J=1/J$.  Therefore
$$	S_{f\rho}=-\int(f\rho)(d\omega dx)
[\log\underline\rho_{\tau^{-1}\omega}(f^{-1}_\omega x)
+\log\bar J_{\tau^{-1}\omega}(f^{-1}_\omega x)]	$$
$$	=-\int\rho(d\omega dx)
[\log\underline\rho_\omega((f^{-1})_{\tau\omega}f_\omega x)
+\log J_\omega((f^{-1})_{\tau\omega}f_\omega x)]	$$
$$	=-\int\rho(d\omega dx)[\log\underline\rho_\omega(x)
+\log\bar J_\omega(x)]	   $$
so that
$$	-[S_{f\rho}-S_\rho]=\int\rho(d\omega dx)\log\bar J_\omega(x)
	\eqno(2.3)	$$
This expression is minus the entropy put into the system in one time
step, which is equal to the entropy pumped out of the system, or
produced by the system.  
\medskip\indent
	The expression (2.3) agrees with the expression (2.1) postulated
for the entropy production.  This justifies our definition in a special
case, and in more general cases obtained by suitable limits.  We are
thus led to investigating the topology of probability measures $\mu$
such that $\pi\mu={\bf P}$.
\medskip\indent
	{\bf P}{\it -vague topology.  }
\medskip\indent
	Let us define
$$	E=\{\mu:\mu\hbox{ is a Borel probability measure on }\Omega\times M
\hbox{ and }\pi\mu={\bf P}\}      $$
and
$$	\Psi_{A\phi}(\mu)=\int_{A\times M}\mu(d\omega dx)\phi(x)   $$
where $A$ is a {\bf P}-measurable subset of $\Omega$, and $\phi$ a
continuous function $M\to{\bf C}$.  We call {\bf P}-vague topology the
coarsest topology on $E$ for which all the functions $\Psi_{A\phi}$ are
continuous\footnote{*}{Given a finite number of pairs $(A_1,\phi_1)$,\dots,
$(A_n,\phi_n)$, and $\epsilon>0$, let $N=\{\mu\in E:|\Psi_{A_i\phi_i}(\mu)
-\Psi_{A_i\phi_i}(\mu_0)|<\epsilon\hbox{ for }i=1,\ldots,n\}$.  Such sets
form a basis of neighborhoods of $\mu_0\in E$ for the {\bf P}-vague topology.}.
\medskip\indent
	{\it 2.3 Proposition.  With respect to the }{\bf P}{\it -vague topology,
 $E$ is metrizable compact.  }
\medskip\indent
	In order to prove this we shall first replace $\Omega$ by a
metrizable compact space.  This is possible because on a Polish space
$\Omega$ there is a metrizable compact topology which gives the same
Borel sets [6].  More simply we can use the fact that a Polish space is
homeomorphic to a countable intersection of open sets in a metrizable
compact space.
\medskip\indent
	Now that $\Omega\times M$ is metrizable compact, the Borel
measures coincide with the Radon measures, and the set of all
probability measures on $\Omega\times M$, with the vague
topology\footnote{*}{The vague topology on measures on the compact set
$\Omega\times M$ is the topology of pointwise convergence on the space
${\cal C}(\Omega\times M)$, of continuous functions.  In other words the
vague topology is the $w^*$-topology of the dual ${\cal C}(\Omega\times M)^*$.}
 is metrizable compact.  The subset $E$ of probability measures 
such that $\pi\mu={\bf P}$ is vaguely closed, and $E$ is thus metrizable
compact for the vague topology.  We are going to see that this vague
topology on $E$ coincides with the {\bf P}-vague topology defined above.
\medskip\indent
	Let us show that the {\bf P}-vague topology on $E$ is coarser
than the vague topology.  It suffices to prove that
$$	N=\{\mu\in E:|\Psi_{A\phi}(\mu)-\Psi_{A\phi}(\mu_0)|<\epsilon\}  $$
contains a vague neighborhood of $\mu_0$ in $E$.  We may assume $\phi\ne0$
and choose a continuous function $\psi:\Omega\to{\bf R}$ so that it is
close to the characteristic function of $A$ in the $L^1({\bf P})$ norm:
$$	\parallel\psi-\chi_A\parallel_{L^1}\le
{\epsilon\over{3\parallel\phi\parallel}}	$$
where $\parallel\phi\parallel$ is the uniform (=sup) norm of $\phi$.Then
$$	|\mu(\psi\otimes\phi)-\Psi_{A\phi}(\mu)|
	=|\int\mu(d\omega dx)[\psi(\omega)-\chi_A(\omega)]\phi(x)]|    $$
$$	\le\parallel\phi\parallel\cdot\parallel\psi-\chi_A\parallel_{L^1}
	\le{\epsilon/3}	  $$
The vague neighborhood of $\mu_0$ defined by
$$	\{\mu\in E:|\mu(\psi\otimes\phi)-\mu_0(\psi\otimes\phi)|
\le{\epsilon/3}\}     $$
is then contained in $N$ as announced.
\medskip\indent
	The {\bf P}-vague topology separates points of $E$ and is
coarser than the vague topology, for which $E$ is compact.  Therefore
the {\bf P}-vague and the vague topology coincide on $E$, and the
proposition follows.\qed 
\medskip\indent
	{\it 2.4 Proposition.  If $\rho^{(m)}$ tends to $\mu$ for the
{\bf P}-vague topology, then}
$$	\lim_{m\to\infty}\int\rho^{(m)}(d\omega dx)\log\bar J_\omega(x)
	=\int\mu(d\omega dx)\log\bar J_\omega(x)     $$
\indent
	As in the previous proof we may take $\Omega$ to be compact
metrizable.  We apply Lusin's theorem to the measure
$(1+\ell(\omega)){\bf P}(d\omega)$ where $\ell(\omega)=\sup_x|\log
J_\omega(x)|$ and to the map $\omega\to\log\bar J_\omega(\cdot)$ of
$\Omega$ to ${\cal C}(M)$ (space of continuous functions on $M$, with
the sup-norm).  We obtain thus a compact set $K\subset\Omega$ with ${\bf P}(K)
\ge1-\epsilon$, $\int_{\Omega\backslash K}{\bf P}(d\omega)\ell(\omega)
\le\epsilon$, and $(\omega,x)\mapsto\log\bar J_\omega(x)$ continuous on
$K\times M$.  The restriction of $\rho^{(m)}$ to $K\times M$ tends
vaguely to the restriction of $\mu$ to $K\times M$ (this results from the
proof of Proposition 2.3 with $\Omega$ replaced by $K$), therefore
$$	\int_{K\times M}\rho^{(m)}(d\omega dx)\log\bar J_\omega(x)\to
        \int_{K\times M}\mu(d\omega dx)\log\bar J_\omega(x)     $$
and
$$|\int_{(\Omega\backslash K)\times M}\rho^{(m)}(d\omega dx)\log\bar J_\omega(x)|
	\le\int_{\Omega\backslash K}{\bf P}(d\omega)\ell(\omega)<\epsilon $$
$$|\int_{(\Omega\backslash K)\times M}\mu(d\omega dx)\log\bar J_\omega(x)|
	\le\int_{\Omega\backslash K}{\bf P}(d\omega)\ell(\omega)<\epsilon $$
Since $\epsilon$ is arbitrarily small, the proposition follows.\qed 
\medskip\indent
	{\it 2.5 Theorem.  Let $\rho(d\omega dx)={\bf P}(d\omega)\rho_\omega(dx)$
.  Assume that the $\rho_\omega$ are absolutely continuous with respect
to the Riemann volume, and that $S_\rho>+\infty$.  If $\mu$ is any {\bf
P}-vague limit of the measures $\rho^{(m)}=(1/m)\sum_{k=0}^{m-1}f^k\rho$,
then $e_f(\mu)\ge0$.}
\medskip\indent
	By Proposition 2.4 and equation (2.3) we have
$$	e_f(\mu)=\lim_{m\to\infty}e_f(\rho^{(m)})
=\lim_{m\to\infty}{1\over m}\sum_{k=0}^{m-1}[-S_{f^{k+1}\rho}+S_{f^k\rho}]
=\lim_{m\to\infty}{1\over m}[-S_{f^m\rho}+S_\rho]     $$
Since $S_\rho$ is finite, and $S_{f^m}\rho$ bounded above by log vol$M$,
the limit in the right hand side is $\ge0$ as announced.\qed
\vfill\eject
\centerline{3. The i.i.d. case.\footnote{*}{See Ledrappier and Young [13]
and, for background, Kifer[8].}  }
\bigskip
\bigskip
\indent
	To describe the case where the $f_{\tau^k\omega}$ are
independent identically distributed, we write
$\omega=(\alpha_i)_{i\in{\bf Z}}$ with $\alpha_i\in A$, so that
$\Omega=A^{\bf Z}$.  We also write ${\bf P}(d\omega)
=\prod_{i\in{\bf Z}}p(d\alpha_i)$ where $p$ is a probability measure on
$A$.  We take $f_\omega$ to depend only on $\alpha_0$ and write $f_\omega
=f_{(\alpha_0)}$.  Our standing assumptions of Section 1 are thus replaced
by the following.
\medskip\indent
	$\bullet$\quad $A$ is a Polish space, $p$ a Borel probability
measure on $A$.  (We denote by $\tau$ the shift on $A^{\bf Z}$).
\medskip\indent
	$\bullet$\quad $M$ is a compact $C^\infty$ manifold.
\medskip\indent	
	$\bullet$\quad $\alpha\mapsto f_{(\alpha)}$ is a Borel map $A\to
{\hbox{Diff}}^r(M)$, $r\ge2$.
\medskip\indent
	$\bullet$\quad If $J_{(\alpha)}$ is the Jacobian of $f_{(\alpha)}$
with respect to some Riemann metric, and $\ell(\alpha)
=\sup_x|\log J_{(\alpha)}(x)|$, then $\ell\in L^1(p)$.  
\medskip\indent
	If the probability measure $\mu$ on $\Omega\times M$ has
projection $\pi\mu={\bf P}$ on $\Omega$, we have the disintegration
$\mu(d\omega dx)={\bf P}(d\omega)\mu_\omega(dx)$.  We denote by
$E^{\le}$ the set of those $\mu$ such that $\mu_\omega$ does not depend
on the {\it future}, i.e., $\mu_\omega$ depends only on $\alpha_i$ for
$i\le 0$.  Define also the maps $s:\Omega\times M\to A\times M$, $t:A\times
M\to M$, and $\theta=ts:\Omega\times M\to M$ such that
$$(\omega,x){\buildrel{\displaystyle s}\over{\hbox to 10mm{\rightarrowfill}}}
(\alpha_0,x){\buildrel{\displaystyle t}\over{\hbox to 10mm{\rightarrowfill}}}x
\qquad,\qquad(\omega,x)
{\buildrel{\displaystyle\theta}\over{\hbox to 10mm{\rightarrowfill}}}x$$
and let
$$	m=\theta\mu\qquad\qquad ,\qquad\qquad m_1=\theta f\mu	$$
\indent
It is readily seen that if $\mu\in E^{\le}$, then the image of $f\mu$ by $s$
is of the form
$$	(sf\mu)(d\alpha_1dx)=p(d\alpha_1)m_1(dx)	$$
and
$$	m_1=\int p(d\alpha)f_{(\alpha)} m	\eqno(3.1)	$$
Also $fE^{\le}\subset E^{\le}$ and the entropy production for
$\mu\in fE^{\le}$ has the expression
$$	e_f(\mu)=\int p(d\alpha)m(dx)\log \bar J_{(\alpha)}(x)   \eqno(3.2) $$	
\indent
	{\it 3.1 Proposition.  Let $\mu\in fE^{\le}$, and assume that
$m=\theta\mu$ has density $\underline m$ with respect to the Riemann volume
element $dx$, i.e., $m(dx)=\underline m(x)dx$.
\medskip\indent
	(a) The density $\underline m_{(\alpha)}$ of $f_{(\alpha)}m$ and the
density $\underline m_1$ of $\theta fm$ are given by
$$	\underline m_{(\alpha)}(x)
	=\underline m(f_{(\alpha)}^{-1}x)\bar J(f_{(\alpha)}^{-1}x)	$$
$$	\underline m_1(x)=\int p(d\alpha)\underline m_{(\alpha)}(x)	$$
\indent
	(b) If $S(\underline m)=-\int dx\,\underline m(x)\log\underline m(x)
>-\infty$, then
$$	e_f(\mu)=-\int p(d\alpha)S(\underline m_{(\alpha)})+S(\underline m)   $$
\indent
	(c) Let $\delta(\alpha,x)=\underline m_{(\alpha)}/\underline m_1(x)$
, then
$$	\int p(d\alpha)S(\underline m_{(\alpha)})-S(\underline m_1)
=\int m_1(dx)[-\int p(d\alpha)\delta(\alpha,x)\log\delta(\alpha,x)]\le0  $$
\indent
	(d) In particular if $\underline m=\underline m_1$ (i.e. if $\mu$
is $f$-invariant) we have
$$ 	e_f\ge0		$$
and $e_f>0$ unless $\underline m_{(\alpha)}(x)=\underline m(x)$ a.e.
with respect to $p(d\alpha)\underline m(x)dx$.}\footnote{*}{This has
been noted by Kifer [8] and Ledrappier and Young [13] Proposition (2.4.2).}
\medskip\indent
	(a) follows from the definitions and (3.1).  We have
$$	-\int p(d\alpha)S(\underline m_{(\alpha)})+S(\underline m)
=\int p(d\alpha)[-S(\underline m_{(\alpha)})+S(\underline m)]    $$
$$=\int p(d\alpha)\int dx\,\underline m_{(\alpha)}(x)\log\bar J_{(\alpha)}(x)$$
and (b) follows from (3.2).  Note that we have $\int p(d\alpha)
\delta(\alpha,x)=1$; therefore
$$	\int p(d\alpha)S(\underline m_{(\alpha)})-S(\underline m_1)
	=\int p(d\alpha)S(\delta(\alpha,\cdot)\underline m_1(\cdot))
	 -S(\underline m_1)     $$
$$	=\int m_1(dx)[-\int p(d\alpha)\delta(\alpha,x)\log\delta(\alpha,x)] $$
but $\int p(d\alpha)\delta(\alpha,x)=1$ also implies
$$	\int p(d\alpha)\delta(\alpha,x)\log{1\over\delta(\alpha,x)}
	\le\int p(d\alpha)\log1\le0     $$
proving (c).  From (b) and (c) we obtain $e_f\ge0$ when
$\underline m=\underline m_1$, and in fact $e_f>0$ unless $\delta(\alpha,x)$
(equal to $\underline m_{(\alpha)}(x)/\underline m(x)$) is 1 almost
everywhere.  This proves (d).\qed
\medskip\indent
	{\it 3.2 Remarks.  }
\medskip\indent
	(a) While in the general case (Section 2) $f$ and $f^{-1}$ play
symmetric roles, this symmetry is broken in the present Section because
$(f^{-1})_{(\alpha_0)}\ne (f_{(\alpha_0)})^{-1}$.
\medskip\indent
	(b) From Proposition 3.1(b),(c) it is clear that, in the stready
state, the entropy produced by the system is equal to minus the entropy
that it extracts from the heat bath.  
\vfill\eject	
  	
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\vfill\eject
\end