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\centerline{GENERAL LINEAR RESPONSE FORMULA IN STATISTICAL MECHANICS,}
\centerline{AND THE FLUCTUATION-DISSIPATION THEOREM FAR FROM EQUILIBRIUM.}
\bigskip
\centerline{by David Ruelle}
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\centerline{IHES, 91440 Bures sur Yvette, France. $<$ruelle@ihes.fr$>$.}
\bigskip
Abstract. {\it Given a nonequilibrium steady state $\rho$ we derive
formally the linear response formula given by equation (6) in the text for
the variation of an expectation value at time $t$ under a time-dependent
infinitesimal perturbation $\delta_\tau F$ of the acting forces. This leads
to a form of the fluctuation-dissipation theorem valid far from equilibrium:
the complex singularities of the susceptibility are in part those of the
spectral density, and in part of a different nature to be discussed.}
\bigskip
PACS: 05.45.+b, 05.70.Ln, 47.70.-n.
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Keywords: {\sl fluctuation-dissipation, linear response,
nonequilibrium, statistical mechanics}\bigskip.
\bigskip
Serious attention is now starting to be paid to the chaotic
dynamics underlying nonequilibrium statistical mechanics. (See for instance
Chernov et al. [1]). A statistical mechanical system is kept far away from
equilibrium by nonhamiltonian forces, and ``cooled'' by a ``Gaussian
thermostat'' ( Hoover [2], Evans and Morriss [3]). What this amounts to is
that the phase space $M$ of the system is taken to be compact, and its
time evolution given by
$$ {dx\over dt}=F(x,t)\eqno{(1)} $$
where no particular assumption is made on $F$ (except smoothness, and often
$t$-indepen- dence).
\medskip
It would at first appear that the framework provided by
$$ {dx\over dt}=F(x)\eqno{(2)} $$
is much too general to provide results of interest for nonequilibrium
statistical mechanics. It is reasonable however to assume that (2)
defines a chaotic time evolution, and that we may exclude a set of
Lebesgue measure 0 of initial conditions in the distant past. These
assumptions have surprisingly strong consequences. If we translate {\it
chaos} mathematically by {\it uniform hyperbolicity}, then time averages
are uniquely determined and given by a so-called SRB measure (see
below). Physical time evolutions are often hyperbolic in the weaker
sense that most Lyapunov exponents are different from 0. A natural idea
is thus to proceed as if physical systems were uniformly hyperbolic (and
then compare the results with experiments). This has been called the
{\it chaotic hypothesis} by Gallavotti and Cohen (who also assume
microscopic reversibility). The approach just outlined has been
vindicated in the case of the Gallavotti-Cohen fluctuation theorem [4],
which agrees with numerical experiments far from equilibrium [5], [6]. From
the chaotic principle one also recovers near equilibrium the Onsager
reciprocity relations [7], [8]. The present letter follows the same
philosophy for the study of linear response far from equilibrium. Our
calculations will be formal and easy. They can be made rigorous if uniform
hyperbolicity holds [9], but this is harder. A rigorous analysis in a
more general setup seems difficult and one may have to untroduce new ideas
like the limit of a large system (thermodynamic limit). Such a limit will
be used anyway to compute transport coefficients like the viscosity.
What may be said now is that if linear response for physical systems has
a simple expression, it must be the one given by our formal calculation.
The formulae that we obtain appear thus unavoidable, and should be
fundamental for nonequilibrium statistical mechanics far from equilibrium.
\medskip
Integrating (1) with initial condition $x$ at time $s$
gives at time $t$ a point $x(s,t)$. Suppose now that we are in the time
independent situation of (2). We may thus write
$$ x(s,t)=x(t-s)=f^{t-s}x\eqno{(3)} $$
If $m$ is a probability measure absolutely continuous with respect to the
volume element $dx$ on the phase space $M$, and if $m$ under the time
evolution $f^t$ tends weakly to a limit $\rho$, then $\rho$ is a good
candidate to describe a nonequilibrium steady state. Using the notation
$$ m(\Phi)=\int m(dx)\Phi(x)\qquad,\qquad\rho(\Phi)=\int\rho(dx)\Phi(x) $$
we have by assumption, for every continuous $\Phi$,
$$ \lim_{s\to-\infty}\int m(dx)\Phi(f^{-s}x)=\int\rho(dx)\Phi(x)\eqno{(4)} $$
We call $\rho=\rho_F$ an SRB measure, it is usually not absolutely continuous
with respect to $dx$. Such measures were introduced by Sinai, Ruelle, and
Bowen [10], [11], [12] for uniformly hyperbolic dynamical systems, where it
was shown that there is a unique SRB measure on each mixing attractor. A
much more general study of SRB measures was then made by Ledrappier, Strelcyn,
and Young [13], [14] (see Young [15] for recent results). The use of SRB measures to describe nonequilibrium
steady states in statistical mechanics was advocated early by Ruelle [16],
but only recently did it lead to useful results with the {\it Fluctuation
Theorem} of Gallavotti and Cohen [4].
\medskip
One problem with SRB measures is that their characterizations [10],
[11], [12], [13], [14] are difficult to use. It is however relatively easy
to expand $\rho_{F+\delta F}$ with respect to $\delta F$ [17]. This has been
done rigorously in a special case [9]. Here we proceed instead formally, and to
first order with respect to a {\it time dependent} perturbation $\delta_tF$
(we keep $F$ {\it time independent}). We shall thus recover in a new manner
some classical results of nonequilibrium statistical mechanics, for
which see for instance [18], and also obtain new results.
\medskip
Using (4), we have formally
$$ \delta_t\rho(\Phi)=\delta\lim_{s\to-\infty}\int m(dx)\,\Phi(x(s,t))
=\lim_{s\to-\infty}\int m(dx)\,\delta x(s,t)\cdot\nabla_{x(s,t)}\Phi $$
and we may assume $s0$.
>From this one can deduce the Kramers-Kronig {\it dispersion relations} [21].
The dispersion relations therefore also hold far from equilibrium. We
discuss now the singularities of $\hat\kappa_\omega$ for ${\rm Im}\,\omega<0$.
For the sake of clarity, we consider the situation first at equilibrium, then
away from equilibrium. Our discussion will not make use of microscopic
reversibility.
\medskip
{\it At equilibrium}, the SRB measure $\rho_F$ is absolutely
continuous with respect to the volume element of $M$, and we simply
write $\rho_F(dx)=dx$. From (7) and (8), it follows that the singularities
of $(\hat\kappa_\omega^{s,u}X)\Phi$ for ${\rm Im}\,\omega<0$ are the same
as those of
$$ \int_{-\infty}^\infty e^{i\omega\sigma}d\sigma
\rho_F((\Phi\circ f^\sigma).\Psi)=S_\omega(\Phi,\Psi)\eqno{(9)} $$
for $\Psi={\rm div}^s X^s$, or $F\cdot\nabla\phi+{\rm div}^u X^u$.
The right-hand side of (9) is the Fourier transform of the correlation function
$\sigma\mapsto\rho_F((\Phi\circ f^\sigma).\Psi)$, and $S_\omega$ is called
the {\it spectral density}. For sufficiently regular $\Phi$, $\Psi$,
one can extend $S_\omega(\Phi,\Psi)$ to complex $\omega$. The singularities
of $\hat\kappa_\omega^s$ and $\hat\kappa_\omega^u$ (and $\hat\kappa_\omega$)
are thus expected to be the same as those of $S_\omega$ for
${\rm Im}\,\omega<0$. This connection between $\hat\kappa_\omega$ and
$S_\omega$ is basically the fluctation-dissipation theorem.
\medskip
{\it Far away from equilibrium}, the singularities of
$\hat\kappa_\omega^u$ are (in view of (8)) again the same as the singularities
with ${\rm Im}\,\omega<0$ of the spectral density $S_\omega$. For
instance a simple pole of $S_\omega$ at $\omega_0$ (with ${\rm
Im}\,\omega_0<0$) corresponds to a simple pole $a(\omega-\omega_0)^{-1}$
of $\hat\kappa_\omega^u$ (even if it is not clear how to determine the
residue $a$ in practice). This is what remains here of the
fluctuation-dissipation theorem. The singularities of
$\hat\kappa_\omega^s$, however, become different from those of
$\hat\kappa_\omega^u$. Define ${\cal T}^\sigma$ on the vector fields
$X^s$ (in the stable direction) by
$$ ({\cal T}^\sigma X^s)(x)
=(T_{f^{-\sigma}x}f^\sigma)X^s(f^{-\sigma}x)\eqno{(10)} $$
Then, $({\cal T}^\sigma)_{\sigma\ge0}$ is a contraction semigroup and, if
$-H$ is its infinitesimal generator, we have by the Hille-Yosida theorem [22]
$$ \int_0^\infty e^{i\omega\sigma}d\sigma((Tf^\sigma)X^s)\circ f^{-\sigma}
=(H-i\omega)^{-1} $$
so that $(\hat\kappa_\omega^sX)(\Phi)
=\rho_F[(H-i\omega)^{-1}X^s\cdot\nabla\Phi]$. The singularities of
$\hat\kappa_\omega^s$ are thus related to the spectrum of $H$.
\medskip
To summarize, we see that at equilibrium the singularities of
the susceptibility $\hat\kappa_\omega$ are the singularities of the
spectral density $S_\omega$ with ${\rm Im}\,\omega<0$. Outside of
equilibrium, the singularities of the susceptibility $\hat\kappa_\omega$
bifurcate into those of $\hat\kappa_\omega^u$ which are again the
singularities of the spectral density $S_\omega$ with ${\rm Im}\,\omega<0$,
and those of $\hat\kappa_\omega^s$ which are different (and related to the
spectrum of the infinitesimal generator of the semigroup
$({\cal T}^\sigma)_{\sigma\ge0}$ defined by (10)). It is thus in
principle possible to distinguish the singularities of $\hat\kappa_\omega^u$
and of $\hat\kappa_\omega^s$, and it would be interesting to see them
in an experimental study bifurcating from each other as one moves away
from equilibrium [23].
\medskip
{\it Acknowledgements.}
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Part of this work was performed in the stimulating atmosphere of
the Santa Fe Institute under a grant from the Sloan Foundation.
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{\it References.}
\medskip
[1] N.I. Chernov et al. Phys. Rev. Letters {\bf 70}, 2209-2212(1993);
Commun. Math. Phys. {\bf154},569-601(1993).
[2] W.G. Hoover. {\it Molecular dynamics}. Lecture Notes in
Physics {\bf 258}. Springer, Heidelberg, 1986.
[3] D.J. Evans and G.P.Morriss. {\it Statistical mechanics of
nonequilibrium fluids}. Academic Press, NY, 1990.
[4] G. Gallavotti and E.G.D. Cohen. Phys. Rev. Letters {\bf 74},
2694-2697(1995); J. Statist. Phys. {\bf 80},931-970(1995).
[5] D.J. Evans, E.G.D. Cohen, and G.P.Morriss. Phys. Rev.
Letters {\bf 71},2401-2404 (1993).
[6] F. Bonetto et al. Physica D, {\bf 105},226-252(1997).
[7] G. Gallavotti. Phys. Rev. Letters, {\bf 77},4334-4337(1996).
[8] G. Gallavotti and D. Ruelle. Commun. Math. Phys. {\bf 190},
279-285(1997).
[9] D. Ruelle. Commun. Math. Phys. {\bf 187},227-241(1997).
[10] Ya.G. Sinai. Russian Math. Surveys {\bf 27},No 4,21-69(1972).
[11] D. Ruelle. Am. J. Math. {\bf 98},619-654(1976).
[12] R. Bowen and D. Ruelle. Invent. Math. {\bf 29},181-202(1975).
[13] F. Ledrappier and J.-M. Strelcyn. Ergod. Th. and Dynam. Syst.
{\bf 2},203-219(1982).
[14] F. Ledrappier and L.S. Young. Ann. of Math. {\bf 122},509-539,
540-574(1985).
[15] L.-S. Young. ``Ergodic theory of chaotic dynamical systems.'' Lecture at the International Congress of Mathematical Physics 1997.
[16] G. Gallavotti mentions a seminar talk in Zurich in 1973. See
also D. Ruelle. Ann. N.Y. Acad. Sci. {\bf 316},408-416(1978).
[17] A first version of this idea, near equilibrium, appears in
G. Gallavotti. J.Stat. Phys. {\bf 84},899-926(1996).
[18] S.R. de Groot and P. Mazur. {\it Nonequilibrium
thermodynamics.} Dover, New York, 1984.
[19] D. Ruelle. "Nonequilibrium statistical mechanics near
equilibrium: computing higher order terms." Nonlinearity, to appear.
[20] The names of H. Nyquist, M.S. Green, H.B. Callen,
and R. Kubo are attached to this result. For a general discussion see
for instance [18] Ch. VIII.
[21] See [18] Ch. VIII.3.
[22] See for example Section X.8 of M. Reed and B. Simon. {\it Methods of Mathematical Physics II.} Academic Press, New York, 1975. A concise presentation is also given in Section {\bf 378B} of the {\it Encyclopedic Dictionary of Mathematics (2nd English ed.)} MIT Press, Cambridge MA, 1987.
[23] Eric Siggia pointed out to the author that a change in the
nature of singularities of the susceptibility when one moves away from
equilibrium is implicit in P.C. Martin et al. Phys. Rev. A.{\bf
8},423-437(1973).
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