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\centerline{NONEQUILIBRIUM STATISTICAL MECHANICS NEAR EQUILIBRIUM:}
\bigskip
\centerline{COMPUTING HIGHER ORDER TERMS.}
\bigskip
\bigskip
\centerline{by David Ruelle\footnote{*}{IHES (91440 Bures sur Yvette, France)
$<$ruelle@ihes.fr$>$}}.
\bigskip
\bigskip
\indent
{\it Abstract.} Using SRB measures to describe nonequilibrium
steady states, one can in principle compute the coefficients of
expansions around equilibrium. We discuss here how this can be done in
practice, and how the results correspond to the zero noise limit when
there is a stochastic perturbation. The approach used is formal rather
than rigorous.
\bigskip
\bigskip
\indent
{\it Key words and phrases:} chaotic principle, statistical
mechanics near equilibrium, nonequilibrium steady state, nonequilibrium
statistical mechanics, random dynamics, SRB state, thermostat.
\vfill\eject
{\bf 0. Generalities.}
\bigskip
This article follows the lines of [12] in discussing
nonequilibrium steady states near equilibrium. Let $(f^t)$ describe the
microscopic time evolution of a system submitted to external forces and
the action of a thermostat. We assume that $(f^t)$ acts on a compact phase
space $M$ (this is the situation obtained with a so-called {\it Gaussian
thermostat} [13]). In general ({\it i.e.}, outside of equilibrium), $(f^t)$
has no invariant measure absolutely continuous with respect to a
Riemannian volume element $dx$ on $M$. Let us assume that the time
evolution $(f^t)$ is strongly chaotic (for instance {\it uniformly
hyperbolic}), and let $\rho$ be an {\it SRB measure} ({\it i.e.}, a limit
when $t\to\infty$ of the image by $f^t$ of a measure absolutely
continuous with respect to $dx$). We assume that a nonequilibrium
steady state is described by the SRB measure $\rho$, and study the
dependence of $\rho$ on $(f^t)$ near equilibrium.
\medskip
We call {\it chaotic principle} (after Gallavotti and Cohen [11])
the above assumption that physical time evolution satisfies strong
chaoticity (or hyperbolicity) conditions, and that the nonequilibrium
steady state is an SRB measure. The present paper is part of
the rediscussion of fundamental issues of nonequilibrium statistical
mechanics currently taking place on the basis of such ideas. The idea
of using SRB measures to describe nonequilibrium states is not new
(see in particular [20]) but has only recently become productive of specific
results when Gallavotti and Cohen [11] proved their fluctuation
theorem for entropy production. There is some question as to what
chaoticity assumption one wants to make\footnote{*}{In his {\it chaotic
hypothesis} Gallavotti usually also assumes (microscopic) reversibility,
which will not be used here.}: the maximal requirement that $(f^t)$ is
an exponentially mixing Anosov flow is ideal for proving theorems, but
physically unrealistic. At the other extreme one may make only the
minimal request that the SRB measures considered have nonzero Lyapunov
exponents, and that certain time correlation functions tend to zero
sufficiently fast at infinity to give convergent integrals. These
assumptions reflect the idea that the microscopic time evolution is very
chaotic. Note that in physical applications a thermodynamic limit (many
degrees of freedom) is often implied, and this may help in obtaining
time correlation functions which tend to zero fast at infinity. Here
we shall leave the explicit assumptions at a minimum, and lay no
claim to mathematical rigor. Even with this mathematical haziness,
the chaotic principle yields more definite results than the usual
physical approaches. Specifically, we shall see that it provides
explicit expressions for higher order terms in the expansion of
nonequilibrium quantities around equilibrium. These results can probably
be made rigorous for uniformly hyperbolic systems, but should remain
exact in more general (and physically interesting) circumstances.
\medskip
For the purposes of the present paper the chaotic principle will
mostly be invoqued at equilibrium. We assume that the {\it equilibrium time
evolution} $(f_0^t)$ preserves the volume element $dx$ associated with some
Riemann metric. (In coordinates $dx=\sqrt{g}\prod dx^i$, where $g$ is
the determinant of the metric tensor $(g_{ij})$). Therefore, at equilibrium,
$dx$ is an SRB measure, and we write $\rho_0(dx)=dx$. Note that a measure
$\phi(x)dx$ with smooth density $\phi>0$ may be rewritten as the Riemann
volume for a modified Riemann metric. To justify our choice of equilibrium
SRB measure, notice that for a Hamiltonian time evolution $(f^t)$ the {\it
microcanonical ensemble}, {\it i.e.}, the Liouville volume element restricted
to an energy surface $M$, can be written in the form $\rho_0(dx)=dx$.
\medskip
We refer the reader to the literature for the theory of SRB
measures\footnote{*}{See Sinai [24], Ruelle [19], Bowen and Ruelle [4] for
the uniformly hyperbolic case, Ledrappier and Strelcyn [17], and
Ledrappier and Young [18] for the general theory, and Eckmann and Ruelle
[7] for a physical introduction.}. Here we recall a couple of basic
facts. For any ergodic measure $\rho$ one can define Lyapunov exponents
$\lambda_i$ (as many as the dimension ${\rm dim}M$ of $M$) and, at
$\rho$-almost every $x\in M$, stable and unstable subspaces $V_x^s$,
$V_x^u\subset T_xM$ (the tangent space to $M$ at $x$). If $\xi\in V_x^u$
we have $(1/t)\log\|Tf^t\xi\|\rightarrow{\rm some}\enspace\lambda_i>0$
for $t\rightarrow -\infty$, and if $\xi\in V_x^s$ we have
$(1/t)\log\|Tf^t\xi\|\rightarrow{\rm some}\enspace\lambda_i<0$
for $t\rightarrow +\infty$. We assume that only one of the $\lambda_i$
vanishes (corresponding to the direction $F(x)=df^t(x)/dt|_{t=0}$ of the flow,
so that $T_xM={\bf R}.F(x)\oplus V_x^s\oplus V_x^u$. Through $\rho$-almost
all points, there is an {\it unstable manifold} ${\cal V}^u\subset M$.
This is a smooth manifold, with dimension equal to the number of positive
Lyapunov exponents, tangent to $V_x^u$ at each $x\in{\cal V}^u$. There
is a natural volume element $\sigma^u(dx)$ on each ${\cal V}^u$ (defined
up to a multiplicative constant) such that $(f^t)^*(\sigma^u|{\cal V})$
is (up to multiplication by a constant) $\sigma^u|f^t{\cal V}^u$. One
can always write a {\it disintegration} $\rho=\int d\alpha\,\rho_\alpha$
where each $\rho_\alpha$ is carrried by an unstable manifold. The SRB
measures $\rho$ are those for which each $\rho_\alpha$ is absolutely continuous
with respect to the Riemann volume on the unstable manifold carrying it,
{\it i.e.}, $\rho_\alpha$ is proportional to $\sigma^u$ on a piece of
unstable manifold.
\medskip
We shall use below the {\it divergence} ${\rm div}X$ of a vector
field $X$ on $M$ defined with respect to the volume element $dx$.
(In coordinates, ${\rm div}X=(1/\sqrt{g})\sum_i\partial(\sqrt{g}\,X^i)/
\partial x^i$, where $g$ is the determinant of the metric tensor
$(g_{ij})$). In particular, if $\rho$ is an SRB measure for the flow
$(f^t)$ generated by the vector field $F$, the corresponding rate of
{\it entropy production}\footnote{**}{This formula appears for instance
in Andrey [1]; for a general discussion of entropy production, see Ruelle
[21].} is
$$ e=\rho(-{\rm div} F) $$
We shall also need the {\it unstable divergence} ${\rm div}^uX
={\rm div}^uX^u$. To define this we first take the component
$X_x^u$ of $X$ along the unstable subspace $V_x^u$ at $x$ by projecting
along ${\bf R}.F(x)\oplus V_x^s$ (where $V_x^s$ is the stable
subspace, and $F(x)=df^t(x)/dt|_{t=0}$ as above). By definition, ${\rm
div}^uX^u$ is the divergence of $X^u$ along an unstable manifold ${\cal
V}^u$ , taken with respect to the volume element $\sigma^u(dx)$. In the
uniformly hyperbolic case, ${\rm div}^uX^u$ is a honest H\"older
continuous function on the support of $\rho$, otherwise little can be
said in general.
\medskip
The dependence of the SRB measure $\rho$ on $(f^t)$ was studied
in [23]; in the linear approximation near equilibrium this yields a
justification of the Onsager relations (see [12], following the earlier [9],
[10]). In the present paper we want to go beyond the linear approximation,
and obtain power series expansions for the state $\rho$, the rate of
entropy production $e$, and the thermodynamic fluxes ${\cal J}$ around
equilibrium. In Section 1, we discuss the simplest case of a
continuous time deterministic evolution or {\it flow} $(f^t)$. In order
to obtain more explicit formulae, we deal in Section 2 with the
special example of the {\it isokinetic model}. In Section 3, we examine
the complications that arise when we have {\it random forces}, and look
in particular at the zero noise limit for a stochastic perturbation of a
deterministic flow. Finally, in Section 4 we consider {\it discrete time}
dynamics.
\medskip
I am indebted to Giovanni Gallavotti and Joel Lebowitz for
discussions related to the present paper.
\vfill\eject
{\bf 1. Continuous time deterministic evolution.}
\bigskip
Let $(f_\lambda^t)$ be the flow associated with the vector field
$F+\lambda X$ on $M$. (The assumed affine dependence of the vector field
on the parameter $\lambda$ simplifies formulae; the isokinetic example
discussed below satisfies this condition). We define operators
$P$, $Q^t$ acting on smooth functions $M\to{\bf R}$ by
$$ P\Psi=\sum_iX^i\partial_i\Psi $$
$$ Q^t\Psi=\Psi\circ f_0^t $$
We have used local coordinates $x^i$ to define
$\partial_i=\partial/\partial x^i$ and the components $X^i$ of $X$.
\medskip
{\bf 1.1. SRB states.}
\medskip
If $\Phi:M\to{\bf R}$ is smooth and independent of $\lambda$ we
have\footnote{*}{$\delta$ is used to denote differentiation with respect
to $\lambda$.}
$${\delta^r\over\delta\lambda^r}(\Phi\circ f_\lambda^T)\big\vert_{\lambda=0}$$
$$ =r!{\int\cdots\int}_{0From this we can compute formally the derivatives with respect to
$\lambda$ of the SRB measure $\rho_\lambda$ for the flow $(f_\lambda^t)$.
We assume that $(f_\lambda^T)^*\mu\to\rho_\lambda$ when $T\to\infty$,
for a probability measure $\mu$ with smooth density\footnote{**}
{In the uniformly hyperbolic Axiom A case, one would assume that $\rho_f$
has its support on an attractor, and that $\mu$ has support close to this
attractor.}. Thus (formally) at $\lambda=0$
$$ ({\delta^r\over\delta\lambda^r}\rho_\lambda|_{\lambda=0})(\Phi)
=\lim_{T\to\infty}\mu({\delta^r\over\delta\lambda^r}(\Phi\circ f^T)) $$
$$ =r!\int_0^\infty d\tau_1\cdots\int_0^\infty d\tau_r\,
\rho_0(PQ^{\tau_r}P\ldots PQ^{\tau_1}\Phi) $$
so that
$$ \rho_\lambda(\Phi)=\sum_{r=0}^\infty{\lambda^r\over r!}
({\delta^r\over\delta\lambda^r}\rho|_{\lambda=0})(\Phi) $$
$$=\sum_{r=0}^\infty\lambda^r\int_0^\infty d\tau_1\cdots\int_0^\infty d\tau_r\,
\rho_0(PQ^{\tau_r}P\ldots PQ^{\tau_1}\Phi)\eqno{(1.1)} $$
This formula deserves a couple of comments.
\medskip
(a) We may write (1.1) formally as
$$\rho_\lambda(\Phi)=\rho_0((1-\lambda\int_0^\infty d\tau PQ^\tau)^{-1}\Phi)$$
Since the orthogonal projection $\pi_0$ on constants in $L^2(\rho_0)$ is
given by
$$ \pi_0\Phi=\int\rho_0(dx)\Phi(x) $$
we have
$$ P\pi_0=0\qquad ,\qquad\pi_0Q^\tau=Q^\tau\pi_0 $$
hence
$$ PQ^\tau=P(1-\pi_0)Q^\tau=PQ^\tau(1-\pi_0) $$
Therefore (1.1) may also be rewritten as
$$ \rho_\lambda(\Phi)=
\rho_0((1-\lambda\int_0^\infty d\tau PQ^\tau(1-\pi_0))^{-1}\Phi) $$
\par
(b) The simplicity of (1.1) is deceptive: it is not clear why
the integrals should converge. The following calculation will clarify
this point partially. We may take $X=\phi F+X^s+X^u$, where
$F(x)=df_0^t(x)/dt|_{t=0}$, and $X^s$, $X^u$ are the components of $X$
in the (strong) stable and unstable directions with respect to $(f_0^t)$.
Then, if ${\rm div}^u$ denotes the unstable divergence defined in
Section 0, we may write (using as in [23] the fact that $\rho_0$ is SRB
to perform integrations by part)
$$ \int_0^\infty d\sigma\int_0^\infty d\tau\,
\rho_0(\Psi\ldotp Q^\sigma PQ^\tau\Phi) $$
$$ =\int_0^\infty d\sigma\int_0^\infty d\tau\,\rho_0
\big((\Psi\circ f_0^{-\sigma}).
[(\phi F+X^s+X^u)\cdot{\rm grad}(\Phi\circ f_0^\tau)]\big) $$
$$ =\int_0^\infty d\sigma\int_0^\infty d\tau\,\rho_0\big(
[(-X^u\cdotp{\rm grad})(\Psi\circ f_0^{-\sigma}].(\Phi\circ f_0^\tau)
+(\Psi\circ f_0^{-\sigma}).(-{\rm div}^uX^u).(\Phi\circ f_0^\tau)$$
$$ +(\Psi\circ f_0^{-\sigma}).\phi.
[(F\cdotp{\rm grad}(\Phi\circ f_0^\tau)]
+(\Psi\circ f_0^{-\sigma}).
[(X^s\cdotp{\rm grad})(\Phi\circ f_0^\tau)]\big) $$
$$ =\int_0^\infty d\sigma\int_0^\infty d\tau\,\rho_0\big(
-[(Tf_0^{-\sigma})X^u)\cdot(({\rm grad}\Psi)\circ f_0^{-\sigma})]
.(\Phi\circ f_0^\tau)-(\Psi\circ f_0^{-\sigma}).
({\rm div}^uX^u).(\Phi\circ f_0^\tau) $$
$$ +(\Psi\circ f_0^{-\sigma}).\phi.
((F\cdot{\rm grad}\Phi)\circ f_0^\tau)
+(\Psi\circ f_0^{-\sigma}).
[((Tf_0^\tau)X^s)\cdot(({\rm grad}\Phi)\circ f_0^\tau)]\big)\eqno{(1.2)} $$
If we assume that $\Phi$, $\Psi$ are smooth and satisfy
$\rho_0(\Phi)=\rho_0(\Psi)=0$, the chaotic principle implies that the
integrand in the right-hand side tends to $0$ rapidly when $\sigma$,
$\tau\to\infty$, so that the double integral is convergent and the above
expressions are well defined.
\medskip
Note that the above formulae do not assume that we have
equilibrium at $\lambda=0$.
\medskip
{\bf 1.2. Entropy production and thermodynamic fluxes.}
\medskip
Let us replace the vector field $F+\lambda X$ by $F+\sum_\alpha
\lambda_\alpha X_\alpha$, and define $P_\alpha$ by
$$ P_\alpha\Psi=\sum_iX_\alpha^i\partial_i\Psi $$
We write $\lambda=(\lambda_\alpha)$ and denote by $\rho_\lambda$ the
SRB measure corresponding to $F+\sum_\alpha\lambda_\alpha X_\alpha$,
obtaining formally
$$ \rho_\lambda(\Phi)=\rho_0((1-
\sum_\alpha\lambda_\alpha\int_0^\infty d\tau P_\alpha Q^\tau)^{-1}\Phi) $$
Let us now assume that we have equilibrium at $\lambda=0$, {\it i.e.},
$\rho_0(dx)=dx$, hence ${\rm div}F=0$. The entropy production is then
$$ e_\lambda=\rho_0((1-\sum_\alpha\lambda_\alpha
\int_0^\infty d\tau P_\alpha Q^\tau)^{-1}(-\sum_\alpha\lambda_\alpha
{\rm div}X_\alpha)) $$
and can be expanded in powers of the $\lambda_\alpha$. Omitting the
index $\lambda$, we define the {\it fluxes} (see [9], [10], [12])
$$ {\cal J}_\alpha=\rho_\lambda({\partial\over\partial\lambda_\alpha}
(-{\rm div}(F+\sum_\alpha\lambda_\alpha X_\alpha)))
=-\rho_\lambda({\rm div}X_\alpha) $$
so that
$$ e_\lambda=\sum_\alpha\lambda_\alpha{\cal J}_\alpha $$
The ${\cal J}_\alpha$ are nonlinear functions of $\lambda$ vanishing
for $\lambda=0$ (because $\rho_0({\rm divergence})=0$). The positivity of
the entropy production (see [21]) implies that
$$ \sum_\alpha\lambda_\alpha{\cal J}_\alpha\ge0 $$
identically. We may write
$$ {\cal J}_\alpha=\sum_\beta L_{\alpha\beta}^{(2)}\lambda_\beta
+\sum_{\beta\gamma} L_{\alpha\beta\gamma}^{(3)}\lambda_\beta\lambda_\gamma
+\ldots\eqno{(1.3)} $$
where the matrix $(L_{\alpha\beta}^{(2)})$ of {\it transport coefficients}
is the object of the Onsager reciprocity relations $L_{\alpha\beta}^{(2)}
=\pm L_{\beta\alpha}^{(2)}$ as discussed in [9], [10], [12] and an expression
for $L_{\alpha\beta}^{(2)}$, $L_{\alpha\beta\gamma}^{(3)}$, is given below.
\medskip
{\bf 1.3. Coefficients for the expansion of $e_\lambda$,
${\cal J}_\alpha$.}
\medskip
In the simple case of a vector field $F+\lambda X$ (with
$\rho_0(dx)=dx$) we may compute the entropy production as
$$ e_\lambda=-\lambda\rho_\lambda({\rm div}X)
=-\sum_{r=1}^\infty\lambda^{r+1}\int_0^\infty d\tau_1\cdots
\int_0^\infty d\tau_r\,\rho_0(PQ^{\tau_r}P\ldots PQ^{\tau_1}{\rm div}X) $$
$$ =\sum_{r=1}^\infty\lambda^{r+1}\int_0^\infty d\tau_1\cdots
\int_0^\infty d\tau_r\,\rho_0({\rm div}X\cdotp Q^{\tau_r}P\ldots PQ^{\tau_1}
{\rm div}X)=\sum_{r=1}^\infty L^{(r+1)}\lambda^{r+1} $$
where
$$ L^{(2)}=\int_0^\infty d\tau_1\rho_0({\rm div}X.Q^{\tau_1}{\rm div}X)
={1\over2}\int_{-\infty}^\infty d\tau_1\,
\rho_0({\rm div}X\ldotp({\rm div}X)\circ f^{\tau_1}) $$
$$ L^{(3)}=\int_0^\infty d\tau_1\int_0^\infty d\tau_2\,
\rho_0({\rm div}X.Q^{\tau_2}PQ^{\tau_1}{\rm div}X) $$
$$ =\int_0^\infty d\tau_1\int_0^\infty d\tau_2\,
\rho_0\big((({\rm div}X)\circ f^{-\tau_2})
.P(({\rm div}X)\circ f^{\tau_1})\big) $$
Restoring the vector field $F+\sum_\alpha\lambda_\alpha X_\alpha$ we may
similarly write ${\cal J}_\alpha$ in the form (1.3), hence
$$ e=\sum_{\alpha\beta} L_{\alpha\beta}^{(2)}\lambda_\alpha\lambda_\beta
+\sum_{\alpha\beta\gamma} L_{\alpha\beta\gamma}^{(3)}
\lambda_\alpha\lambda_\beta\lambda_\gamma+\ldots $$
where
$$ L_{\alpha\beta}^{(2)}=\int_0^\infty d\tau_1\,\rho_0
({\rm div}X_\beta.({\rm div}X_\alpha)\circ f^{\tau_1})\eqno{(1.4)} $$
$$ L_{\alpha\beta\gamma}^{(3)}=\int_0^\infty d\tau_1\int_0^\infty d\tau_2
\,\rho_0\big((({\rm div}X_\beta)\circ f^{-\tau_2}).
P_\gamma(({\rm div}X_\alpha)\circ f^{\tau_1})\big)\eqno{(1.5)} $$
Similar formulae are easily obtained for higher orders. Note that the
above expressions for $L^{(3)}$ and $L_{\alpha\beta\gamma}^{(3)}$ can be
rewritten following (1.2) yielding, hopefully, convergent integrals. In
practice, however, this rewriting cannot be done explicitly, because of
the difficulty of determining the stable and unstable directions. The
point of invoquing (1.2) is to give evidence that the integrals for
$L^{(3)}$, $L_{\alpha\beta\gamma}^{(3)}$ do in fact converge.
\vfill\eject
{\bf 2. Application to the isokinetic model.}
\bigskip
Consider the $2N-1$-dimensional manifold $S\times M$, where
$S=\{{\bf p}\in{\bf R}^N:{\bf p}\cdot{\bf p}/2m=K\}$ and $M$ is a
bounded open subset of ${\bf R}^N$ or of the torus ${\bf T}^N$, with
piecewise smooth boundary $\partial M$. A time evolution $(f^t)$ is
defined on (a large subset of) $S\times\overline{\!M}$ by
$$ {d\over dt}\pmatrix{{\bf p}\cr{\bf q}\cr}
=\pmatrix{\lambda({\bf\xi}-\alpha{\bf p})\cr{\bf p}/m\cr}\eqno{(2.1)}$$
when $({\bf p},{\bf q})$ is in $S\times M$, and by elastic reflection
at $S\times\partial M$. In (2.1), ${\bf\xi}$ is a smooth vector field
and $\alpha={\bf p}\cdot{\bf\xi}/{\bf p}\cdot{\bf p}
={\bf p}\cdot{\bf\xi}/2mK$. This {\it isokinetic} time evolution is of
particular interest when ${\bf\xi}$ is locally a gradient, in view of
the {\it pairing property} which then holds for Lyapunov
exponents\footnote{*}{The pairing, observed empirically by Evans, Cohen
and Morriss [8], was proved by Dettmann and Morriss [6], and also
Wojtkowski and Liverani [25].}. When $\lambda=0$, the time evolution
is that of a billiard (or of a hard sphere system). For $\lambda\ne0$
it includes the model studied in [5].
\medskip
We have here
$$ X=\pmatrix{{\bf\xi}-\alpha{\bf p}\cr0} $$
so that (without the assumption that ${\bf\xi}$ is locally a gradient):
$$ {\rm div}X=-(N-1)\alpha\eqno{(2.2)} $$
Define $N\times N$ matrices $L_{pp}(t)$, $L_{pq}(t)$, $L_{pq}(t)$,
$L_{qq}(t)$, depending on ${\bf p}={\bf p}(0)$, ${\bf q}={\bf q}(0)$
and $t$, so that
$$ \pmatrix{d{\bf p}(t)\cr d{\bf q}(t)\cr}
=\pmatrix{L_{pp}(t)&L_{pq}(t)\cr L_{qp}(t)&L_{qq}(t)\cr}
\pmatrix{d{\bf p}(0)\cr d{\bf q}(0)\cr} $$
[This is obtained by solving the "linearized equation" corresponding to
(2.1) with the initial conditions $L_{pp}(0)=1$, $L_{pq}(0)=0$,
$L_{pq}(0)=0$, $L_{qq}(0)=1$]. The gradient of $\xi$ is a $N\times N$
matrix $M({\bf q})$ such that
$$M({\bf q}){\bf u}={\rm grad}_{\bf q}({\bf\xi}({\bf q})\cdot{\bf u})$$
With this notation we have
$$ P(({\rm div}X)\circ f^t)
=-{(N-1)\over 2mK}P({\bf p}(t)\cdot{\bf\xi}({\bf q}(t))) $$
$$ =-{(N-1)\over 2mK}({\bf\xi}({\bf q})-\alpha{\bf p})\cdot
{\rm grad}_{\bf p}({\bf p}(t)\cdot{\bf\xi}({\bf q}(t))) $$
$$ =-{(N-1)\over 2mK}[{\bf\xi}({\bf q}(t))\cdot
L_{pp}(t)({\bf\xi}({\bf q})-\alpha{\bf p})
+(M({\bf q}(t)){\bf p}(t))\cdot
L_{qp}(t)({\bf\xi}({\bf q})-\alpha{\bf p})]\eqno{(2.3)} $$
Using (2.2) and (2.3) one can compute the coefficients $L^{(2)}$ and
$L^{(3)}$ in the $\lambda$-expansion of the entropy production
$e_\lambda$, and similarly for $L^{(2)}_{\alpha\beta}$,
$L^{(3)}_{\alpha\beta\gamma}$.
\vfill\eject
{\bf 3. Continuous time random dynamical systems.}
\bigskip
Here we start with a probability space $(\Omega,{\bf P})$ and a
continuous one-parameter group $(\theta^t)$ of transformations of
$\Omega$ with respect to which ${\bf P}$ is ergodic. A one-parameter
group $({\bf f}^t)$ of transformations of $\Omega\times M$ is given such
that ${\bf f}^t(\omega,x)=(\theta^t\omega,f_\omega^tx)$. One can
again define SRB states under suitable hyperbolicity conditions (the
discrete time case is discussed in [3]).
\medskip
{\bf 3.1. $\lambda$-expansions.}
\medskip
In what follows we shall forget about hyperbolicity, and discuss
the Markov case, which is realized by stochastic differential equations,
(see for instance Ikeda and Watanabe [14], Arnold [2]). This situation
gives simple formulae, which will again be derived only formally (in
particular we sail clear off the subtleties of stochastic integration).
Let thus ${\bf f}_\lambda^t=(\theta^t,f_{\omega\lambda}^t)$ where
$(f_{\omega\lambda}^t)$ is a family of diffeomorphisms obtained by
integrating the stochastic differential equation ({\it Langevin
equation}) $dx(t)/dt=F(\theta^t\omega,x)+\lambda X(x)$. Define
$$ P\Psi=\sum_iX_i\partial_iX $$
$$ Q_\omega^t\Psi=\Psi\circ f_{\omega0}^t $$
Then
$${\delta^r\over\delta\lambda^r}(\Phi\circ f_{\omega\lambda}^T)|_{\lambda=0}
=r!\int\ldots\int_{00$) such that
$$ (\int{\bf P}(d\omega)Q_\omega^\tau\Phi)(x)=(p^\tau\Phi)(x)
=\int p^\tau(x,y)\Phi(y)dy $$
We have
$$ p^{0+}=1 $$
$$ {d\over d\tau}p^\tau=p^\tau A=Ap^\tau\eqno{(3.2)} $$
where the second order elliptic differential operator $A$ (infinitesimal
generator of the diffusion process) is given in coordinates by
$$ A\Phi={1\over\sqrt{a}}\sum_{ij}\partial_i(\sqrt{a}\,a^{ij}\partial_j)\Phi
+\sum_ib^i\partial_i\Phi $$
Here the $a^{ij}$, $b^i$ are functions on $M$, $(a^{ij})$ is a positive
definite matrix, and $a$ is the inverse of the determinant of
$(a^{ij})$. The second term $(b\cdot{\rm grad})$ in the expression for
$A$ corresponds to a {\it drift}. The absence of term of order 0 means
that
$$ {d\over d\tau}\int\sqrt{a(y)}\prod dy^ip^\tau(x,y)=0 $$
because, writing $\Phi(\cdot)=p^\tau(x,\cdot)$ we have
$dp^\tau(x,\cdot)/d\tau=A^*\Phi$ where, in coordinates,
$$A^*\Phi={1\over\sqrt{a}}\sum_{ij}\partial_i(\sqrt{a}\,a^{ij}\partial_j)\Phi
-{1\over\sqrt{a}}\sum_i\partial_i(\sqrt{a}b^i\Phi) $$
We choose the metric on $M$ such that $(g_{ij})$ is proportional to the
inverse of $(a^{ij})$, obtaining
$$ A\Phi=\epsilon\triangle\Phi+b\cdot{\rm grad}\Phi $$
$$ A^*\Phi=\epsilon\triangle\Phi-{\rm div}(\Phi b) $$
where $\triangle$ is the {\it Laplace-Beltrami operator}, $\epsilon$ a
constant, and ${\rm div}$ the divergence with respect to the volume
element $dx=\sqrt{g}\prod dx^i$.
\medskip
We assume now that at $\lambda=0$ we have equilibrium, {\it
i.e.}, the measure $dx$ is stationary for the diffusion process $(p^t)$,
or $A^*1=0$, or ${\rm div}b=0$, so that
$$ A^*\Phi=\epsilon\triangle\Phi-b\cdot{\rm grad}\Phi $$
In the present situation, we have $\rho_0(dx)=dx$; furthermore the
operator $A$ vanishes on constants and maps the space
$\{\Phi:\rho_0(\Phi)=0\}$ (orthogonal to constants with respect to
$\rho_0$) to itself. We have in $L^2(dx)$
$$ {d\over d\tau}\|p^\tau\Phi\|^2
=\rho_0((p^\tau\Phi)^*(A+A^*)p^\tau\Phi)
=2\epsilon((p^\tau\Phi)^*\triangle p^\tau\Phi) $$
Assuming $M$ to be a connected compact manifold, we have $-\triangle\ge
c>0$ on $\{\Phi:\rho_0(\Phi)=0\}$, so that
$$ {d\over d\tau}\|p^\tau\Phi\|^2\le-2\epsilon c\|p^\tau\Phi\|^2 $$
hence
$$ \|p^\tau\Phi\|\le e^{-\epsilon c\tau}\|\Phi\| $$
If $\pi_0$ denotes the orthogonal projection on constants in $L^2(dx)$,
we have thus $\|p^\tau(1-\pi_0)\|\le e^{-\epsilon c\tau}$ in the operator
norm on $L^2(dx)$ and we can thus define
$$ R=\int_0^\infty d\tau\int{\bf P}(d\omega)Q_\omega^\tau(1-\pi_0)
=\int d\tau p^\tau(1-\pi_0) $$
so that $R$ has operator norm $\le1/c\epsilon$ on $L^2(dx)$. From (3.2)
we see that
$$ 1+RA=1+AR=\pi_0 $$
hence
$$ P(1+RA)=0\eqno{(3.3)} $$
Since
$$ P\int_0^\infty d\tau\int{\bf P}(d\omega)Q_\omega^\tau=PR $$
(3.1) yields
$$ \rho_\lambda(\Phi)=\rho_0((1-\lambda PR)^{-1}\Phi) $$
[It is satisfactory to check that $\rho_\lambda$ is invariant under the
diffusion process $(p_\lambda^\tau)$ associated with the elliptic
operator
$$ A_\lambda=\epsilon\triangle+(b+\lambda X)\cdot{\rm grad}
=A+\lambda P $$
We have indeed
$$ {d\over dt}\rho_\lambda(p_\lambda^t\Phi)=\rho_0((1-\lambda
PR)^{-1}(A+\lambda X\cdot{\rm grad})p_\lambda^t\Phi)=0 $$
because, using (3.3),
$$ (1-\lambda PR)^{-1}(A+\lambda P)
=[1+(1-\lambda PR)^{-1}\lambda PR](A+\lambda P) $$
$$ =A+\lambda P+(1-\lambda PR)^{-1}(-\lambda P+\lambda PR\lambda P)
=A+\lambda P-(1-\lambda PR)^{-1}(1-\lambda PR)\lambda P=A $$
and $\rho_0(A\Phi)=0$].
\medskip
The formula expected for the entropy production is
$$ e_\lambda=\int{\bf P}(d\omega)\rho_\lambda
(-{\rm div}F_\omega(t)-\lambda{\rm div}X) $$
(the discrete time version of this formula is discussed in [22]). Since
the measure $dx$ is stationary for the process associated with
$F_\omega$ we write $\int {\bf P}(d\omega){\rm div}F_\omega=0$ so that
$$ e_\lambda=-\lambda\rho_\lambda({\rm div}X)
=-\lambda\rho_0((1-\lambda PR)^{-1}{\rm div}X) $$
$$ =-\sum_{r=1}^\infty\lambda^{r+1}\rho_0((PR)^r{\rm div}X)
=\sum_{r=1}^\infty\lambda^{r+1}\rho_0({\rm div}X.RP\ldots PR\,{\rm div}X) $$
$$ =\sum_{r=1}^\infty\tilde L^{(r+1)}\lambda^{r+1} $$
\par
Let us now replace $\lambda X$ by $\sum_\alpha\lambda_\alpha X_\alpha$
and define the fluxes
$$ {\cal J}_\alpha=-\rho_\lambda({\rm div}X_\alpha)
=\sum_\beta\tilde L_{\alpha\beta}^{(2)}\lambda_\beta
+\sum_{\beta\gamma}\tilde L_{\alpha\beta\gamma}^{(3)}
\lambda_\beta\lambda_\gamma+\ldots $$
We have then
$$ e_\lambda=\sum_\alpha\lambda_\alpha{\cal J}_\alpha $$
The coefficients of ${\cal J}_\alpha$ are given by
$$ \tilde L_{\alpha\beta}=\rho_0({\rm div}X_\beta.R{\rm div}X_\alpha)
\eqno{(3.4)}$$
$$ \tilde L_{\alpha\beta\gamma}^{(3)}=\rho_0({\rm div}X_\beta.RP_\gamma
R{\rm div}X_\alpha)\eqno{(3.5)} $$
and similarly for higher order.
\medskip
{\bf 3.2. Small random perturbations.}
\medskip
Various results are known showing that the SRB states for
uniformly hyperbolic systems are stable under small random perturbations
(see [15], [16]). Comparison of the formulae in Sections 1.3 and 3.1 yield a
result of this type for the coefficients $L$, $\tilde L$ of the
$\lambda$-expansions of $e_\lambda$ and the ${\cal J}_\alpha$. In fact,
inspection of (1.4), (1.5) and (3.4), (3.5) shows that the coefficients $L$
corresponding to a deterministic evolution are obtained from the coefficients
$\tilde L$ corresponding to a random evolution by the replacement
$$ R=\int_0^\infty d\tau\int{\bf P}(d\omega)Q_\omega^\tau(1-\pi_0)\qquad
\longrightarrow\qquad\int_0^\infty d\tau Q^\tau(1-\pi_0) $$
This means that the limit $\epsilon\to0$ (zero noise limit) of the
coefficients $\tilde L$ reproduces the coefficients $L$. At least this
holds formally, and assuming that the integrals defining the
coefficients $L$ converge, as discussed in Section 1. The introduction
of the term $\epsilon\triangle$ in $A$ is however a singular
perturbation, and the analysis of the $\epsilon$-dependence of the
coefficients $\tilde L$ near $\epsilon=0$ would require more definite
mathematical assumption than we have chosen to make.
\vfill\eject
{\bf 4. Diffeomorphism case.}
\bigskip
In this Section we derive the formulae which correspond for
diffeomorphisms to those established in Section 1 for flows. While it
was natural in Section 1 to assume an affine dependence on the parameter
$\lambda$, this is no longer possible in the discrete time situation
studied here. The formulae obtained will therefore be somewhat more
complicated.
\medskip
Let the diffeomorphism $f:M\to M$ depend on the real parameter
$\lambda$, and let $X_\lambda$ be the vector field on $M$ such that $\delta
f\circ f^{-1}=X_\lambda\ldotp\delta\lambda$. We write
$X_{\lambda k}={\delta^{k-1}X_\lambda/\delta\lambda^{k-1}}$, and define
operators $P_{\lambda k}$, $Q_\lambda$ acting on smooth functions
$M\to{\bf R}$ by
$$ P_{\lambda k}\Psi=\sum_i X_{\lambda k}^i\partial_i\Psi $$
$$ Q_\lambda\Psi=\Psi\circ f $$
[We have used local coordinates $x_i$ to define
$\partial_i=\partial/\partial x_i$ and the components $X_{\lambda k}^i$
of $X_{\lambda k}$; also $\delta$ denotes differentiation with respect to
$\lambda$]. By induction on $k$, we define operators $R_{\lambda k}$
such that $R_{\lambda 1}=P_{\lambda 1}$ and
$$ (k+1)R_{\lambda(k+1)}=P_{\lambda 1}\ldotp R_{\lambda k}
+[{\delta\over\delta\lambda},R_{\lambda k}] $$
Since $[{\delta\over\delta\lambda},P_{\lambda k}]=P_{\lambda(k+1)}$, we
see that the $R_{\lambda k}$ are polynomials in the $P_{\lambda k}$.
\medskip
We claim that, if $\Phi:M\to{\bf R}$ is smooth and independent
of $\lambda$,
$$ {\delta^r\over\delta\lambda^r}(\Phi\circ f^N) $$
$$ =r!\sum_{s=1}^r\quad\sum_{k_1,\ldots,k_s\ge1}^*
\quad\sum_{n_1,\ldots,n_s\ge1}^{**}
\quad Q_\lambda^{n_s}R_{\lambda k_s}Q_\lambda^{n_{s-1}}\ldots
Q_\lambda^{n_1}R_{\lambda k_1}
Q_\lambda^{N-\sum n_i}\Phi\eqno(4.1)$$
where $\sum^*$ is subjected to the condition $\sum k_i=r$, while
$\sum^{**}$ is subjected to the condition $\sum n_i\le N$.
We have indeed
$$ {\delta\over\delta\lambda}(\Phi\circ f^N)
=\sum_{n=0}^{N-1}[X_\lambda^i\partial_i(\Phi\circ f^n)](f^{N-n}x) $$
$$ =\sum_{n=0}^{N-1}Q_\lambda^{N-n}P_{\lambda 1}Q_\lambda^n
=\sum_{n=1}^NQ_\lambda^nP_{\lambda 1}Q_\lambda^{N-n} $$
which proves (4.1) for $r=1$. Note the identities
$$ [{\delta\over\delta\lambda},Q_\lambda^n]
=\sum_{n'=1}^{n-1}Q_\lambda^{n-n'}R_{\lambda 1}Q_\lambda^{n'}
+Q_\lambda^nR_{\lambda 1} $$
$$ [{\delta\over\delta\lambda},R_{\lambda k}]
=(k+1)R_{\lambda(k+1)}-R_{\lambda 1}R_{\lambda k} $$
and therefore also
$$ [{\delta\over\delta\lambda},Q_\lambda^nR_{\lambda k}]
=\sum_{n'=1}^{n-1}Q_\lambda^{n-n'}R_{\lambda 1}Q_\lambda^{n'}R_{\lambda k}
+(k+1)Q_\lambda^nR_{\lambda(k+1)} $$
The proof of (4.1) for higher $r$ is then readily obtained by induction.
\medskip
Note that, to third order, we have
$$ R_{\lambda 1}=P_{\lambda 1}\quad,\quad
R_{\lambda 2}={1\over2}(P_{\lambda 1}P_{\lambda 1}+P_{\lambda 2})\eqno(4.2) $$
$$ R_{\lambda 3}={1\over6}(P_{\lambda 1}P_{\lambda 1}P_{\lambda 1}
+2P_{\lambda 1}P_{\lambda 2}+P_{\lambda 2}P_{\lambda 1}+P_{\lambda 3}) $$
\par
We shall now compute formally the derivatives of the SRB measure
$\rho_\lambda$ associated with $f$. We assume that $f^{*N}\mu\to\rho_\lambda$
when $N\to\infty$, for a probability measure $\mu$ with smooth
density\footnote{*}{In the uniformly hyperbolic Axiom A case, one would
assume that $\rho_\lambda$ has its support on an attractor, and that $\mu$ has
support close to this attractor. For this case, a rigorous proof of
differentiability with respect to $\lambda$ has been given in [23].}. Thus
(formally)
$$ ({\delta^r\over\delta\lambda^r}\rho_\lambda)(\Phi)
=\lim_{N\to\infty}\mu({\delta^r\over\delta\lambda^r}(\Phi\circ f^N))$$
and, using (4.1),
$$ \rho_\lambda(\Phi)=\sum_{r=0}^\infty{\lambda^r\over {r!}}
[({\delta^r\over\delta\lambda^r}\rho_\lambda|_{\lambda=0})(\Phi)] $$
$$ =\sum_{s=0}^\infty\sum_{k_1=1}^\infty\lambda^{k_1}
\ldots\sum_{k_s=1}^\infty\lambda^{k_s}
\sum_{n_0=0}^\infty\sum_{n_1=1}^\infty\ldots\sum_{n_{s-1}=1}^\infty
\rho_0(R_{0k_s}Q_0^{n_{s-1}}\cdots Q_0^{n_1}R_{0k_1}Q_0^{n_0}\Phi) $$
$$ =\rho_0((1-\sum_{k=1}^\infty
\lambda^kQ_0R_{0k}\sum_{n=0}^\infty Q_0^n)^{-1}\Phi)\eqno(4.3) $$
\medskip
As in Section 1, one could further rewrite the above formula by
separating the stable and unstable directions. Let us write
$X_{\lambda k}=X_k^{(s)}+X_k^{(u)}$ where $X_k^{(s)}$ and $X_k^{(u)}$ are
respectively in the stable and unstable subbundles of $TM$. Then
$P_{\lambda k}=P_k^{(s)}+P_k^{(u)}$ where $P_k^{(s)}=X_k^{(s)}\cdotp{\rm grad}
=X_k^{(s)}\cdotp{\rm grad}^s$, $P_k^{(u)}=X_k^{(u)}\cdotp{\rm grad}=
X_k^{(u)}\cdotp{\rm grad}^u$, and ${\rm grad}^s$, ${\rm grad}^u$ are
the gradients in the stable and unstable directions. We have for example
$$ P_1^{(s)}Q_\lambda^n\Phi=X^{(s)}\cdotp{\rm grad}^s(\Phi\circ f^n)
=(Tf^nX^{(s)})\cdot({\rm grad}^s\Phi)\circ f^n $$
Using the fact that $\rho_\lambda$ is SRB we also have
$$ \rho_\lambda(\Psi Q_\lambda^nP_1^{(u)}\ldots)
=\rho_\lambda((\Psi\circ f^{-n})X^{(u)}\cdotp{\rm grad}^u\ldots)
=-\rho_\lambda({\rm div}^u((\Psi\circ f^{-n})\cdot X^{(u)})\ldots) $$
$$ =-\rho_\lambda((\Psi\circ f^{-n}){\rm div}^u X^{(u)}\ldots)
-\rho_\lambda((({\rm grad}^u\Psi)\circ f^{-n})\cdot (Tf^{-n}X^{(u)})\ldots) $$
\medskip
We evaluate now the derivatives of $\log J_\lambda(x)$, where
$J_\lambda(x)$ is the Jacobian of f. We have
$$ J_\lambda(x)=|\det(A_j^i)| $$
where $A_j^i(x)=\partial_j f^i(x)$, hence
$$ {\delta\over\delta\lambda}\log J_\lambda(x)
={\delta\over\delta\lambda}{\rm tr}\,\log(\partial_jf^i(x))
=\sum_{i,j}(A^{-1})_j^i{\delta\over\delta\lambda}
(\partial_if^j(x)) $$
$$ =\sum_{i,j}(A^{-1})_j^i\partial_iX_\lambda^k(f(x))\partial_kf^j(x)
=\sum_i(\partial_iX_\lambda^i)(f(x))=(Q_\lambda{\rm div}X_\lambda)(x) $$
We have (using (1) with $N=1$)
$$ {\delta^k\over\delta\lambda^k}\log J_\lambda
={\delta^{k-1}\over\delta\lambda^{k-1}}Q_\lambda{\rm div}X_\lambda
=Q_\lambda{\rm div}X_{\lambda k}+\sum_{r=1}^{k-1}{k-1\choose r}r!Q
R_{\lambda r}{\delta^{k-r-1}\over\delta\lambda^{k-r-1}}{\rm div}X_\lambda $$
$$ =Q_\lambda{\rm div}X_{\lambda k}
+\sum_{r=1}^{k-1}{(k-1)!\over(k-r-1)!}Q_\lambda
R_{\lambda r}{\rm div}X_{\lambda,k-r} $$
so that
$$ {\delta\over\delta\lambda}\log J_\lambda
=\sum_{k=1}^\infty{\lambda^{k-1}\over(k-1)!}(Q_0{\rm div}X_{0k}+
\sum_{r=1}^{k-1}{(k-1)!\over(k-r-1)!}Q_0R_{0r}{\rm div}X_{0,k-r}) $$
$$ =\sum_{s=0}^\infty{\lambda^s\over s!}Q_0{\rm div}X_{0,s+1}
+\sum_{r=1}^\infty\sum_{s=0}^\infty{\lambda^{r+s}\over s!}
Q_0R_{0r}{\rm div}X_{0,s+1} $$
$$ =(Q_0+\sum_{r=1}^\infty\lambda^rQ_0R_{0r})
\sum_{s=0}^\infty{\lambda^s\over s!}{\rm div}X_{0,s+1}\eqno(4.4) $$
\par
We shall now for simplicity drop the indices 0, so that $Q$,
$X$, $X_k$, $P_k$ are taken at $\lambda=0$ in accordance with the
notation of Section 1. We shall however write $\rho_0$, $f_0$, $J_\lambda$
for $\rho_\lambda$, $f_\lambda$, $J_\lambda$ at $\lambda=0$. To second
order (4.4) gives then, using (4.2),
$$ {\delta\over\delta\lambda}\log J_\lambda $$
$$ =Q{\rm div}X+\lambda Q(P_1{\rm div}X+{\rm div}X_2)
+{1\over2}\lambda^2Q((P_1P_1+P_2){\rm div}X+2P_1{\rm div}X_2+{\rm div}X_3) $$
and therefore to third order
$$ \log J_\lambda=\log J_0+\lambda Q_0{\rm div}X
+{\lambda^2\over2}Q(P_1{\rm div}X+{\rm div}X_2) $$
$$ +{\lambda^3\over6}Q((P_1P_1+P_2){\rm div}X
+2P_1{\rm div}X_2+{\rm div}X_3)\eqno(4.5) $$
\par
We assume now that $\rho_0(dx)=dx$, so that $J_0=1$, and
$\rho_0({\rm divergence})=0$. To third order (4.3) and (4.5) give thus
$$ \rho_\lambda(\log J_\lambda)
=\lambda^2[{1\over2}\rho_0(P_1{\rm div}X)
+\sum_{n=1}^\infty\rho_0(P_1Q^n{\rm div}X)] $$
$$ +\lambda^3[{1\over6}\rho_0((P_1P_1+P_2){\rm div}X+2P_1{\rm div}X_2)
+\sum_{n=1}^\infty\rho_0(P_1Q^n{1\over2}(P_1{\rm div}X+{\rm div}X_2)) $$
$$ +\sum_{n=1}^\infty\rho_0({1\over2}(P_1P_1+P_2)Q^n{\rm div}X)
+\sum_{n=1}^\infty\sum_{n'=1}^\infty\rho_0(P_1Q^nP_1Q^{n'}{\rm div}X)] $$
Therefore the entropy production, to third order, is
$$ e_\lambda=\rho_\lambda(-\log J_\lambda)
=\lambda^2[{1\over2}\rho_0({\rm div}X.{\rm div}X)
+\sum_{n=1}^\infty\rho_0({\rm div}X.Q^n{\rm div}X)] $$
$$ +\lambda^3[{1\over6}\rho_0({\rm div}X.P_1{\rm div}X
+{\rm div}X_2.{\rm div}X+2{\rm div}X.{\rm div}X_2) $$
$$+\sum_{n=1}^\infty{1\over2}\rho_0({\rm div}X.Q^nP_1{\rm div}X
+{\rm div}X.Q^n{\rm div}X_2)+{\rm div}X.P_1Q^n{\rm div}X
+{\rm div}X_2.Q^n{\rm div}X $$
$$+\sum_{n=1}^\infty\sum_{n'=1}^\infty
\rho_0({\rm div}X.Q^nP_1Q^{n'}{\rm div}X)] $$
This can be rewritten as
$$ e_\lambda=L^{(2)}\lambda^2+L^{(3)}\lambda^3+\ldots $$
where
$$ L^{(2)}={1\over2}\sum_{n=-\infty}^\infty
\rho_0\bigl({\rm div}X.(({\rm div}X)\circ f^n)\bigr) $$
$$ L^{(3)}=[{1\over6}\rho_0\bigl({\rm div}X.P_1{\rm div}X\bigr)
+{1\over2}\sum_{n=1}^\infty
\rho_0\bigl((({\rm div}X)\circ f^{-n}).P_1{\rm div}X\bigr) $$
$$ +{1\over2}\sum_{n=1}^\infty
\rho_0\bigl({\rm div}X.P_1(({\rm div}X)\circ f^n)\bigr)
+\sum_{n=1}^\infty\sum_{n'=1}^\infty
\rho_0\bigl((({\rm div}X)\circ f^{-n}).
P_1(({\rm div}X)\circ f^{n'})\bigr) $$
$$ +{1\over2}\sum_{n=-\infty}^\infty
\rho_0\bigl(({\rm div}X_2).(({\rm div}X)\circ f^n)\bigr) $$
\vfill\eject
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\end