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Foundations of Statistical Mechanics: in and out of Equilibrium
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Condensed Matter Physics · October 2005
DOI: 10.5488/CMP.9.2.219 · Source: arXiv
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Condensed Matter Physics, 200?, Vol. ?, No ?, pp. 1–
??
Foundations of Statistical Mechanics: inand out of Equilibrium
D. Karevski
Laboratoire de Physique des Mat´eriaux, UMR CNRS 7556,Universit´e Henri Poincar´e, Nancy 1,F-54506 Vandœuvre les Nancy Cedex, France
February 2, 2008
The ﬁrst part of the paper is devoted to the foundations, that is the math-ematical and physical justiﬁcation, of equilibrium statistical mechanics. It isa pedagogical attempt, mostly based on Khinchin’s presentation, which pur-pose is to clarify some aspects of the development of statistical mechanics. Inthe second part, we discuss some recent developments that appeared out of equilibrium, such as ﬂuctuation theorem and Jarzynski equality.
Key words:
Foundations of Statistical Mechanics, Fluctuation Theorem,Jarzynski Equality.
PACS:
05.20.Gb Classical ensemble theory 05.30.Ch Quantum ensemble theory 05.70.Ln Non-equilibrium and irreversible thermodynamics
1. Introduction
The main goal of statistical mechanics, at least from the point of view of theinitiators like Boltzmann, Maxwell, Gibbs
1
, Einstein, was the derivation of the ther-modynamical laws from the microscopic (atomistic) structure of matter. All theseattempts are subjected to start from some models of the structure of matter. Butit is a well known fact that thermodynamics was constructed independently or atleast following a parallel road on the basis of few foundamental laws that are viewedas empirical facts. Very recently, Lieb and Yngvason tried to clarify some aspects of the second law (entropy) of thermodynamics on the basis of the concept of adiabaticaccessibility. This work was mainly motivated by the fact that usual formulations
1
Gibbs had a pragmatic point of view which was somehow diﬀerent from Boltzmann’s view.Gibbs founded statistical mechanics as a branch of rational mechanics, no matter what physicalprocess generates the distribution in phase space. Contrary to that, Boltzmann’s view-point was toreally reduce thermodynamics to mechanics and consequently he necessitated an explanation of themechanism that lead to equilibrium a mechanical system which was initially in a nonequilibriumstate. To skech the diﬀerences one can say that the Boltzmann approach is more physical whereasthe Gibbsian is more rigourous.
D. Karevski
of the second law, such as Kelvin or Clausius, use concepts such as hot, cold orheat that are intuitive but not really well deﬁned nor precise before the theory isfully developed. Their basic derivation of the second law (that is the existence of the entropy state function) is based on some abstract postulates of a certain kind of ordering on a set of states.From our point of view, the problem of the foundations of statistical mechanicsis two-fold. One is: given a statistical theory, one has to extract quantities (averagesof phase functions) and laws that can be identiﬁed with thermodynamical quantitiesand fundamental laws. The identiﬁcation itself being of analogy type.
2
Once thosefundamental laws are recognized, one can logically develop the entire consequences of these laws. This logical enterprise was perfectly achieved by Gibbs in his celebratedtreaty[3]. The other, less easy task, is to justify the use of the statistical theory(precepts) itself from a realistic
3
point of view, that is, so to speak to justify the veryuse of ensembles. Diﬀerently stated, why canonical or microcanonical ensembles aresuitable to describe real physical systems ? This question arises since it is generallybelieved that the system to be studied is in a deﬁnite state and not distributed overa continuum of states. It is clear that this second task is more physically relatedto the very structure of matter, and it is within this perspective, that the work of Boltzmann has to be viewed. The (partial) answer to this question is related to thefact that real thermodynamical systems are constituted, at least approximately, of ahuge collection of particles. The discussion of these points will be largely developedin the next section. Section 3 deals with non-equilibrium aspects and we presenttheir relations, such as the ﬂuctuation theorem and Jarzynski equality, which seemto many physicists to be of fundamental interest. In the last section we present someresults obtained on the Ising model in the ﬂuctuation relation context.
2. Foundations of statistical mechanics
2.1. Interpretation of physical quantities
The state of our system (classically a point in the phase space or a Hilbert spec-tral ray quantum mechanically) fully determines the physical (dynamical) quantitieswhich caracterize the given system. We will generally call such a quantity a phasefunction (classical case
f
(
q,p
), quantum case
f
(
ψ
) = (
ψ,
Q
ψ
) where
Q
is the opera-tor associated to the quantity
f
). In order to have a suitable theoretical description,one has to identify such phase functions with the various physical quantities ob-tained experimentally from measurement processes and compare their respectivevalue. However, in order to compare the empirical data with the theoretical predic-tions, one has to know the actual state of the system, that is, for example classically,to determine 2
s
(
∼
10
23
) coordinates. But in general, the empirical (macroscopic)description of what is called a (equilibrium) thermodynamical state is fully speciﬁed
2
As it is very explicitly emphasised in Gibbs treaty[3]
3
Given that the basic ontology is a single mechanical system composed of many subsystems(particles).
2
Foundations of Statistical Mechanics
by a very small set of independant variables, such as the energy, volume, pressure,... So that the question that rises is which state should we choose in order to eval-uate the relevant phase functions and compare their value with their experimentalcounterpart ? and obviously no one has or can have any reasonable answer to sucha question. Nevertheless, if one realizes that the measurement of a physical quantityis performed during a ﬁnite time, which in general is very large compared to someinternal time scale, one realizes that the actual empirical datas are given as averagesof the quantities over long time periods. But the initial question still survives, that is,which (part of the)
4
trajectory the system is actually following ? In order to answersuch a question one has to know 2
s
−
1 independant integrals of motion and it seemsthat a very small path has been done toward the solution since our starting puzzlingproblem of ﬁnding the 2
s
coordinates. At this step, as it is well known, in order toavoid the average over an unknown trajectory, normally one invoques ergodic theo-rems or hypothesis to replace time averages by phase averages. However, in generalvery few systems are known to be ergodic and it seems really unprobable that ina realistic case one will ever prove ergodicity. But the requirement of ergodicity istoo strong. For instance one has simply to require that only few (corresponding tothe empirical ones) phase functions should have time average equal to their phaseaverage. So, it would be an unnecessary hypothesis to demand the validity of suchan equality for all phase functions. Another objection that has to be emphasizedis the fact that ergodicity is a requirement that involves average over recurrencetime, which is too long to have any physical relevance (several astronomical ordersof magnitude), but in actual experiment, the times involved to obtain the averagesare by far shorter than the reccurence time, see for example for a discussion of this point Ref.[6]. The reason of this is lying in the fact that the majority of phasefunctions describing physical quantities exhibit a very peculiar behaviour. They areapproximately constant on almost all the points of the constant energy manifold(since we are talking here of an isolated system). Why is it so is linked to the factthat the mechanical systems, considered here, are breaking up into a large numberof components and the fact that the interesting phase functions are sum functions,that is sum of functions depending on the dynamical coordinates of the componentsubsystems alone.If we suppose, for some reasons, that we can replace time-averages by phase-averages, then the remaining problem is to determine the suitable phase averageprocedure. In the case of an isolated mechanical system, it is usually argued thatone has to restrict the phase average to the constant energy manifold since the actualtrajectory is taking place into this subset of the phase space. Indeed, if one consider
4
One problem that arises is the fact that time average of a phase function on a given trajectorymay have very diﬀerent values for diﬀerent time intervals. This diﬃculty is overcome thanks toa theorem due to Birkhoﬀ, which states that for almost all trajectories, the time averages of agiven phase function tend to a deﬁnite limit when the time interval tends to inﬁnity. It means inparticular that the averages over ﬁnite time intervals on a given trajectory (a typical one) will takeapproximately the same value for suﬃciently large time-periods. This remark, basically, is at theheart of the time average procedure used widely to start an exposition of statistical mechanics, seefor example ref.[5].
3

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