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\title{\bf Fluctuation Theorem}
\author{\small\textsc{Giovanni Gallavotti}}%
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\maketitle
{\it Fluctuation Theorem}:\\ a simple consequence of a time reversal
symmetry; it deals with motions which are chaotic in the strong
mathematical sense of being hyperbolic and transitive (ie are
generated by smooth hyperbolic evolutions on a smooth compact surface
(the ``phase space'') and with a dense trajectory, also called {\bf
Anosov systems}) and ``furthermore are time reversible''. In such
systems any initial data, with the exception of a set of zero volume
in phase space, have the same statistical properties in the sense that
all smooth observables admit a time average independent of the initial
data and expressed as an integral with respect to a probability
distribution on phase space, called the ''natural stationary state'',
or simply the ``stationary state''. The theorem provides,
asymptotically in the observation time, a quantitative and parameter
free relation between the stationary state probability of observing a
value of the average entropy production rate and its
opposite. Although there are quite a few examples of mechanical
systems which are hyperbolic and transitive in the above mathematical
sense, the fluctuation theorem acquires physical interest only in
connection with the {\bf chaotic hypothesis}. Under the latter general
assumption, combined with time reversal, it predicts a ''universal
relation'' between an entropy creation rate value and its opposite,
accessible to simulations and possibly to laboratory experiments. The
basis for the physical interpretation of the theorem as a property of
stationary states in nonequilibrium statistical mechanics is developed
here.
\*\*
{\it Statistical Mechanics of Nonequilibrium Stationary States. Thermostats}
\*
In nonequilibrium statistical mechanics the molecules of a system are
subject to nonconservative forces whose work is dissipated in the form
of heat supplied to other systems kept at constant temperature: the
''thermostats'' with which the system is in contact. Under such
conditions the systems statistical properties usually reach, after a
transient, a ``{\bf stationary state}'', i.e. they are described by a
probability distribution on phase space which is invariant under time
evolution. The study of this situation is a natural extension of
''{\bf equilibrium thermodynamics}'' where the probability distribution is quite
generally simply proportional to the Liouville volume on the energy
surface.
Mathematical models for thermostats often involve equations of motion
with velocity dependent forces acting on the molecules of the
thermostats.
\*\*
{\it Volume Contraction}
\*
Therefore the equations of motion generate evolutions on the phase
space $X$, `` i.e.'' the space of the points representing
the microscopic configurations of the molecules of the system ``and''
of the thermostats. Such motions do not conserve the phase space
volume (unlike the case of equilibrium statistical mechanics, where
the evolution is Hamiltonian and volume preserving, by ``{\bf Liouville's
Theorem}''). This is manifested by the non vanishing of the phase
space volume contraction rate $\sigma(x)$, which is defined
as minus the divergence of the equations of motion for all particles
including those of the thermostats, evaluated at the microscopic state
$x\in X$ (and it could be $>0$ or
$\le0$).
As a rule motion $t\to S_tx$ is observed through
{\bf Chaotic Hypothesis/Ti\-ming Events$|$timing events} (also called
{\it Poincar\'e sections}) in a subset $\Xi\subset X$ of
phase space, so that the time evolution of the point
$\xi\in\Xi$ takes place in discrete time and is described
by a map $\xi\to S\xi$. Then the volume contraction rate is:
$\sigma(\xi){def\atop=}-\log| \det (\frac{\partial(S
\xi)_j}{\partial_{\xi_i}} |$, where $i,j$ label the
coordinates of $\xi$.
\*\*
{\it Dissipation}
\*
In systems that are really out of equilibrium (''i.e.'' subject to
nonconservative forces and/or to thermostats at different
temperatures) and dissipative the phase space contraction
$\sigma(S_t x)$, or $\sigma(S^n\xi)$ in discrete
evolution models, is not only not identically $0$ but it
has an average $\sigma_+$ over $t>0$ or,
respectively, $n>0$, which is positive.
\*\*
{\it SRB Statistics}
\*
If the ``{\bf Chaotic Hypothesis}'' is accepted then motions have a well
defined statistics $\mu$ in the sense that time averages of
a generic observable $F$ exist, aside from a set of
$0$ volume in the phase space $X$, and are
expressed as the $x$-independent limit:
$$\lim_{T\to\infty} \frac1T \int_0^T F(S_t x)dt=\int_X
F(y)\mu(dy){def\atop=} F_+$$
where $\mu$ is the {\bf SRB distribution}. In particular the
time average of the phase space contraction rate will be:
$$\sigma_+=\int_X\sigma(y)\mu(dy).$$
\*\*
{\it Fluctuation Theorem for Hyperbolic Systems}
\*
Consider a ``transitive hyperbolic systems'' (see also ``{\bf Chaotic
Hypothesis}''), whose evolution admits a ``{\bf time reversal
symmetry}'', ``i.e.'' there is a smooth {\bf isometry} $I$
of phase space with the property $I S_t=S_{-t}I$ or, in the
discrete case, $IS=S^{-1}I$. Assuming
$\sigma_+>0$, `` i.e.'' supposing the system dissipative
in the average, the ``time reversal'' symmetry is reflected by a
symmetry property that can be proved for the large deviations rate
(see ``{\bf Chaotic Hypothesis}'') $\zeta(p)$ of the variable
$$p=\frac1T\int_0^T
\frac{\sigma(S_tx)}{\sigma_+}\,dt$$
or, in the discrete
time case,
$$p=\frac1N\sum_{j=0}^{N-1}\frac{\sigma(S^j\xi)}{\sigma_+},
$$
regarded as a random variable with respect to the {\bf SRB
probability distribution} $\mu$ of the motion; this
variable is called ``average entropy creation rate'', see
below. Namely
'''Fluctuation Theorem''' (for Anosov systems):
$\zeta(-p)=\zeta(p)-p\sigma_+$, for all $p$'s
within the domain of definition $(-p^*,p^*)$ of
$\zeta(p)$.
For the domain of definition see ``{\bf Chaotic Hypothesis}''. The above
theorem is very different from other formally similar relations which
have been given the same name (at later times, see
''Fluctuation Theorem'' in Wikipedia for a glimpse of such relations).
\*\*
{\it The Fluctuation Theorem and Entropy production rate}
\*
The physical interest of the above theorem can be seen by considering
that, in models of nonequilibrium statistical mechanics systems, the
phase space contraction rate has the form:
$$\sigma(x)=\sum_j \frac{Q_j(x)}{k_B T_j}+\dot R(x){def\atop
=}\varepsilon(x)+\dot R(x)$$
where $Q_j(x)$ is the amount of work that the system
molecules perform per unit time on the molecules of the
$j$-th thermostat whose constant temperature is
$T_j$, $k_B$ is ``{\bf Boltzmann constant}'', and
$\dot R(x)$ is a suitable ``total time derivative''.
Interpreting $Q_j(x)$ as the heat that the molecules of the
system inject into the $j$--th thermostat, the statistics
of the observable:
$p=\frac1T\int_0^T
\frac{\sigma(S_tx)}{\sigma_+}dt$
approaches as
$T\to\infty$ that of:
$p'=\frac1T\int_0^T
\frac{\varepsilon(S_tx)}{\varepsilon_+}dt$
because
$$
\frac1T\int_0^T \sigma(S_tx)dt\equiv\frac1T\int_0^T
\varepsilon(S_tx)dt +\frac1T(R(S_Tx)-R(x))$$
the two averages as well as $\varepsilon_+$ and $\sigma_+$
are asymptotically the same, at least if
$R(x)$ is bounded.
The quantity called $\varepsilon(x)$
above has time averages which are physically measurable, in principle,
because they are the heat received per unit time by the thermostats:
this is a quantity which can be defined and measured without really
knowing the equations of motion: in sharp contrast with the time
averages of $\sigma(x)$ which can be measured directly only
in simulations.
\*\*
{\it Fluctuation Relation}
\*
Hence if the system evolution is time reversible, which is certainly
true in many models but subtle and delicate to establish in
experiments, see [BGGZ06], the ``fluctuation relation'' for the statistical
properties of the finite time averages :$p'=\frac1T\int_0^T
\frac{\varepsilon(S_tx)}{\varepsilon_+}dt$ of the ``physically
observable'' dimensionless entropy production rate $p'$,
$\zeta(-p')=\zeta(p')-p'\varepsilon_+$, can be expected to
hold, for $T$ large with corrections of
$O(T^{-1})$, on the basis of the ``Chaotic Hypothesis'' and
of the fluctuation theorem for the (usually not observable) quantity
$\frac{\sigma(x)}{\sigma_+}$. Its interest is therefore
that it gives an explicit and parameter free relation for the (large)
fluctuations of the finite time averages of the entropy production
rate $\varepsilon(x)$ in a stationary nonequilibrium state
(with statistics given by the {\bf SRB distribution}).
New fluctuation relations have been derived for systems subject to
noise, [Ku98,LS99]. It can also be extended to cases in which the
quantities $\varepsilon(x)$ and $R(x)$ are not
bounded, but its form can change in such cases, [BGG06], [CV03].
The above theorem arose from a new interpretation of the experimental
results in [ECM93], given in [GC95], (see [Ga95], [Ru99] for more
mathematical versions); the stochastic version was developed in
[Ku98], [LS99], see also [CDG06], [BGG06]. A discussion of the
relation between the above fluctuation theorem and other results that,
later, have been given the same name see [CG99].
The theorem provides, when time reversibility is satisfied, also a
criterion to test the {\bf chaotic hypothesis}: the precision with which
the relation predicted by the theorem for {\bf Anosov systems} is
satisfied becomes a measure of the correctness of the hypothesis. So
far this test has been possible only in simulations, given the
difficulty of observing very large fluctuation in stationary states.
Historically the concept, and the name of ``fluctuation theorem'',
were introduced in [CG95] and the name has been subsequently adopted,
in the literature, to designate other kinds of fluctuations (and some
confusion followed).
\*\*
{\it References}
{\small\def\*{\vskip2mm}
\*
[ECM93] D. J. Evans, E. G. D. Cohen, G. P. Morriss, {\it Probability
of second law violations in shearing steady flows}, Physical Review
Letters, {\bf 71}, 2401--2404, 1993
\* [GC95] G. Gallavotti, E. G. D. Cohen, {\it Dynamical ensembles in
nonequilibrium statistical mechanics}, Physical Review Letters, {\bf
74}, 2694--2697, 1995; and {\it Dynamical ensembles in stationary
states}, Journal of Statistical Physics, {\bf 80}, 931--970, 1995,
\*
[Ru99] D. Ruelle, {\it Smooth dynamics and new theoretical ideas in
non-equilibrium statistical mechanics},
Journal of Statistical Physics,
{\bf 95}, 393--468, 1999.
\*
[Ga95b] G. Gallavotti,
{\it Reversible Anosov diffeomorphisms and large deviations},
Mathematical Physics Electronic Journal (MPEJ), {\bf 1}, 1--12,
1995.
\*
[BGGZ06] F. Bonetto, G. Gallavotti, A. Giuliani, F. Zamponi,
{\it Fluctuations relation and external thermostats: an
application to granular materials},
Journal of Statistical Mechanics, P05009, 2006.
\*
[Ku98] J. Kurchan,
{\it Fluctuation theorem for stochastic dynamics},
Journal of Physics A, {\bf 31}, 3719--3729, 1998.
\*
[LS99] J. Lebowitz, H. Spohn,
{\it A Gallavotti--Cohen type symmetry
in large deviation functional for stochastic dynamics},
Journal of
Statistical Physics, {\bf 95}, 333--365, 1999.
\*
[CV03] R. Van Zon, E. G. D. Cohen,
{\it Extended heat-fluctuation
theorems for a system with deterministic and stochastic forces},
Physical Review E, {\bf 69}, 056121 (+14), 2004.
\*
[CDG06] R. Chetrite, J. Y. Delannoy, K.Gawedzki,
{\it Kraichnan flow in a square: an example of integrable chaos},
Journal of Statistical Physics, {\bf 126}, 1165-1200,
2007.
\*
[BGG06] F. Bonetto, G. Gallavotti, G. Gentile, {\it A fluctuation theorem
in a random environment}, Ergodic Theory and Dynamical Systems,
{\bf 28}, 21--47,2008,\\
doi:10.1017/S0143385707000417.
\*
[CG99] E. G. D. Cohen, G. Gallavotti,
{\it Note on Two Theorems in Nonequilibrium Statistical Mechanics},
Journal of Statistical Physics,
{\bf 96}, 1343--1349, 1999.
}
\*\*
{\rm See also}
\*
{\it Anosov Diffeomorphism}, {\it Chaos}, {\it Chaotic Hypothesis},
{\it Fluctuations}, {\it Entropy}, {\it Ergodic Theory}, {\it Smooth Dynamics}
\*
{\it Category:Dynamical Systems $|$ Statistical Mechanics}
\end{document}