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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 Jona-Lasinio – 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 ﬂuctuation-dissipation 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 time-reversal.Because the

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ﬂuctuation-dissipation 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 Gallavotti-Cohen 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 Gallavotti-Cohen 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 Gallavotti-Cohen,plus a small

energy-conserving 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 coarse-grained 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 large-deviation 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 large-deviation 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

large-deviation 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

large-deviation 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 Jona-Lasinio devel-

oped over the years a ‘Hamilton-Jacobi’ 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 ‘Hamilton-Jacobi’

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 path-integral approach.

The output of such an approach is a Hamiltonian system consisting of a one-dimensional

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 non-local 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 large-deviation functions previously derived analytically by Bodineau and Derrida,

and reproduced with our program.

FIG.3 Left:shock (traﬃc-jam) 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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