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Fluctuation Theorems and Large Deviations
Fluctuation Theorems and the Jarzynski equality are two closely related,very general
results pertaining to strongly out of equilibrium systems.
There are various ﬂuctuation theorems,but they all state that the probability of some
work (or entropy production) W satisﬁes an equation of the form
P(W)
P(−W)
∼ e
βW
(1)
where β is some quantity that can be interpreted as an inverse temperature.It was originally
stated by Evans,Cohen and Morris,but the ﬁrst proof for a stationary state was given by
Gallavotti and Cohen.
The Jarzynski equality concerns a system that starts from an equilibrium state,and is
taken out of equilibrium by changing a parameter (the pressure or magnetic ﬁeld,say) from
h
i
to h
f
not necessarily slowly.If the experiment is repeated many times,one obtains a
distribution for the work done,and one has that:
dW P(W)e
βW
= e
−β(F
h
i
−F
h
f
)
(2)
where F
h
i
and F
h
f
denote the equilibrium free energies at ﬁeld h
i
and h
f
,respectively.From
here,it is a simple mathematical step (via Jensen’s inequality) to deduce the Second Law.I
have written a rather personal review [68],where there are many references,including some
to more extensive reviews.
More generally,there has been in the last few years an intense activity on the physics of
large deviations,in particular of systems with conserved quantities.These large deviations
obey the two results above,but more can be said,at least in some cases.A formalism
to compute the large deviations was developed by the group of Bertini,DeSole,Gabrielli,
Landim and JonaLasinio – about which more below.
A.Fluctuation theorems for stochastic Dynamics
I became interested in the ﬂuctuation theorems because they are the generalisation for
strongly out of equilibrium systems of the ﬂuctuationdissipation theorem.At that time,
the only proof I knew (but did not understand) was the one of Gallavotti and Cohen,valid
for systems in contact with a deterministic thermostat obeying timereversal.Because the
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ﬂuctuationdissipation theorem itself works as well with a Langevin (stochastic) thermostat
– and is trivial to prove in that case – I was certain that its generalisation,the ﬂuctuation
theorem,should do too.This turned out to be the case (see [34]),and indeed,the ﬂuctu
ation theorem for stochastic dynamics is very simple and transparent.The derivation was
generalised in several ways by Lebowitz,Spohn and others.It has been also applied to glassy
physics,see for example [67].
These results where received with relief by a community who welcomed an elementary
proof.
B.A quantum ﬂuctuation theorem
The ﬂuctuation theorem,just as the Jarzynski equality,hold also for quantum systems.
In [46] I gave the ﬁrst proof of a quntum ﬂuctuation theorem,while Tasaki and Yukawa gave
the corresponding ones for Jarzynski’s equality at about the same time.There is at present
quite a lot of activity on this,especially on the subtler questions connected with quantum
measurement,trajectories,etc.
C.Chaotic Hypothesis and Stochastic Stability
Going back to the GallavottiCohen theorem,it should be mentioned that its diﬃculty
is not artiﬁcial.Most of it stems from the fact that in a deterministic system ergodicity
is not guaranteed.Indeed,the hypotheses of Gallavotti and Cohen are strong but quite
necessary:unlike the stochastic case,the ﬂuctuation theorem for deterministic thermostats
does not apply in some systems:whether it does so or not teaches us something about
the system itself.This inspired Gallavotti to propose a ‘Chaotic Principle’,generalising
the ergodic hypothesis of equilibrium,according to which one assumes that macroscopic
systems are such that they satisfy the chaoticity conditions (transitivity,Anosov) required
in the GallavottiCohen theorem.In order to render this more intuitive,in a recent work [72]
I considered a system with a thermostat just as the one of GallavottiCohen,plus a small
energyconserving noise of amplitude .For all the ﬂuctuation theoremholds trivially.The
whole mystery is in the limit → 0.One can analise this limit in very good detail,using
the fact that for small noises one can make use of the saddle point approximation.As an
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example,I applied this to the Lorentz gas under a ﬁeld (see Fig.1).
FIG.1 Trajectories in periodic Sinai billiard with a ﬁeld.
This work lead to a reformulation of the Chaotic Principle as follows:‘macroscopic
systems are such that their observables are unchanged by the addition of a small,energy
conserving noise’.In other words,they can be considered stochastically stable.In this form,
the Chaotic Principle is quite appealing and testable.
D.Large Deviations
When a systemis coarsegrained in boxes of size N,the probability P(A) of an observable
taking a value Ais of the formP(A) ∼ e
Nf(A)
.This deﬁnes the largedeviation function f(A).
Typical observables of interest are the particle or energy currents and density,magnetisation,
etc.
When a system is in equilibrium,large excursions away from the typical values follow
the same paths as the relaxations back to them.This is the Onsager reciprocity principle.
Because relaxations are easy to compute,this immediately allows to ﬁnd the excursions by
reversing the time,and thus the largedeviation functions (which are just the probability of
excursions leading to a conﬁguration).Once the system is driven out of equilibrium,the
symmetry between relaxations and excursions is lost,and,although the former are still easy
to compute,the latter are highly nontrivial.This is why there are no general results for the
largedeviation functions out of equilibrium.
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In the last years some intriguing and beautiful exact solutions have been obtained for
simple,but non trivial models.In particular,many exact results have been obtained for the
largedeviation functions of the simple symmetric exclusion process (SSEP),where particles
jump without overlapping,driven by sources at the extremes (see Fig.2)
FIG.2 The simple symmetric exclusion process (SSEP).Particles jump to the right or left with
equal probability,but double occupancy is forbidden.Below:the hydrodynamic limit.
In a parallel line of research,Bertini,DeSole,Gabrielli,Landim and JonaLasinio devel
oped over the years a ‘HamiltonJacobi’ theory of large deviations.Armed with their theory,
they attacked the problem of the large deviations of density of the SSEP,and managed to
reproduce the solution of Bodineau and Derrida
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.To do this,they made use of a series of
ingenious and rather miraculous changes of variables.The question arises as to the reason
for,or the generality of these changes.In a very recent work [74],in collaboration with
my thesis student J.Tailleur and V.Lecomte,we set out to study the ‘HamiltonJacobi’
approach of Bertini et al.in order to understand it in our own terms.In an eﬀort to
make their development intelligible for us,we rederived it with the methods of theoretical
physics – as opposed to their Probabilitic language.We have now a derivation that is quite
straightforward for anybody familiar with the pathintegral approach.
The output of such an approach is a Hamiltonian system consisting of a onedimensional
classical ﬁeld.The miracle mentioned above,is that Bertini et al.managed to ﬁnd solutions
of such equations of motions by a succession of clever and surprising changes of variables.
Next,we set about to understand why these changes are possible.The striking answer we
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L.Bertini,D.Gabrielli and J.L.Lebowitz,J.Stat.Phys.121 843 (2005),and references therein.
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found is the following:in the SSEP and all other models that have been explicitely solved
this way,there is a very nonlocal change of variables that maps the driven system into a
system in equilibrium.Once the system is mapped into equilibrium,its large deviations are
easily obtained using the Onsager reciprocity property.
Large deviations are rare ﬂuctuations,by deﬁnition hard to observe.Some form of biased
sampling is needed in order to be able to check the results numerically.Together with C.
Giardin`a and L.Peliti [69] we devised such a program,which is inspired in the Diﬀussion
QuantumMechanical methods of electron systems.Figures 3 show the ‘shock’ conﬁgurations
and the largedeviation functions previously derived analytically by Bodineau and Derrida,
and reproduced with our program.
FIG.3 Left:shock (traﬃcjam) conﬁgurations obtained when one conditions the current to be
abnormally small.Right:the large deviation function,simulation and analytic results of Bodineau
and Derrida.
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