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Research Interests 2012

The Construction of Nonequilibrium Statistical Mechanics

Christian MAES

From astrophysical to life processes nature is full of open systems driven by their contact with

conflicting reservoirs causing transient or much longer lived (steady) currents to flow in the system.

Important questions of transport, of stability and relaxation and of dissipation and creation are

intimately connected with fundamental issues of nonequilibrium theory. But also human affairs and

complex systems in general connect with nonequilibrium questions. Climatology, ion channel

research and environmental science are just three arbitrary but very interesting and rather diverse

interdisciplinary examples where new ideas from nonequilibrium physics will be very welcome.

The study of nonequilibria starts from the moment the idea of equilibrium is contemplated. That is

why we find it already in the beginnings of standard thermodynamics. Of course, the true revolution

starts with the formulation of the second law of thermodynamics. Clausius introduced the ever non-

decreasing thermodynamic entropy, which is related to heat, work and machine efficiency. Later,

still in equilibrium, that entropy was interpreted statistically, which marked the beginning offluctuation theory. The latter refers to the birth of statistical mechanics via kinetic gas theory with

the pioneering work of Maxwell, Boltzmann and Gibbs. The major theme then in nonequilibrium

physics was the understanding of relaxation to equilibrium. That has continued to be an important

problem up to these days, especially in connection with quantum processes, e.g under quantum

quenches, or in small systems or in systems with long range forces (such as gravity, and where

black hole physics wants to understand the transition from a unitary to a dissipative dynamics in the

so called information paradox). Other current themes include the rigorous derivation of irreversible

transport equations following the example of the Boltzmann equation but extending it to include

more dense fluids and other transport laws. Finally, important efforts are made in the understanding

of metastability, ageing and glassy dynamics, with new studies connecting it with kinematically

constrained models and dynamical phase transitions. Such aspects do however also play animportant role in the steady regime. Recent efforts have mostly gone to the nonequilibrium

statistical mechanics of mesoscopic to macroscopic systems in contact with stationary

thermodynamic reservoirs.

Steady nonequilibrium refers to a time-scale where the considered systems properties arestationary

in time. As the system can be small (say micrometer-scale) its constituents are often subject to

fluctuations, as usual in studies of mesoscopic systems. That could involve colloidal systems,

polymers, macro-molecules, oscillators or other particles moving in thermal and chemical

environments subject to nonconservative forces. The nonequilibrium results from weak contacts

with different reservoirs making up the environment. Each reservoir is an equilibrium bath

characterized by some given temperature, chemical potential or pressure. There then exist steadyenergy or particle currents maintained in the open system from/into these (local) equilibrium baths,

changing their entropy. During the steady regime the dissipated heat is sufficiently small and

rapidly diffusing within the large environment. Such abstraction finds realizations in a variety of

Markov stochastic or thermostated models. Then, especially for classical systems, the reservoirs are

only present in the form of intensive quantities or parameters of the system dynamics. In some cases

covered by a wide nonequilibrium literature they are known as driven diffusive lattice gases. Their

study has entered physics mostly from the study of steady transport, from steady chemical reactions

and from biological processes. As a physical theory however we lack a systematic understanding

and a sufficiently robust mathematically formulated framework.

In many cases, by lack of success, such an ambition was limited to the close-to-equilibrium regime.

It is often summarized as irreversible thermodynamics with an underlying fluctuation theory along

the lines of Onsager and Onsager-Machlup and with linear response theory as developed by Kubo

and others. All of that already contains many important insights. For example, the microscopic

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time-reversal invariance penetrates fluctuation-response theory closetoequilibrium in the form of

Onsager reciprocity (symmetry) of the linear response coefficients. The Kubo theory unifies

response relations such as those of Sutherland-Einstein between mobility and diffusion constant and

the Johnson-Nyquist relation for noise. The resulting fluctuation-dissipation theorem is a corner

stone of nonequilibrium statistical mechanics close-to-equilibrium and constitutes the best general

framework so far. All of that however involves entropy production as central physical concept. The

linear response picks up the entropy flux that is released while relaxing (back) to equilibrium. Onecould indeed argue that the present state of nonequilibrium statistical mechanics centers around

entropy production as it was understood in irreversible thermodynamics. There are nowadays

important additions related to the fluctuations of that entropy production, but it remains entropic

nevertheless. Other immediate applications are in the formulation of McLennan-Zubarev ensembles

which in a sense lifts the Green-Kubo relations to the level of the stationary distribution close-to-

equilibrium. One sees it also in the many discussions on the minimum and the maximum entropy

production principles. Various information-theoretic attempts have been made there to make sense

of these principles, but the upshot remains to formulate variational characterizations of the steady

nonequilibrium regime in entropic terms. We have added another perspective, one that adds

frenetic aspects, frenesy being the new word to denote specific aspects of dynamical activity

incorporating for example how the reactivity gets changed by steady nonequilibrium drivingconditions.

I. STATE OF THE ART: FLUCTUATION THEORY

Various important and also mathematically interesting developments have taken place these last

decades. On the level of exact solutions we have seen beautiful work on one-dimensional

nonequilibrium lattice gases. For example, the matrix representation has enabled a variety of results

with explicit information on static and dynamical fluctuation behavior. On the level of

hydrodynamic scaling and corresponding large deviation theory, diffusive systems have found an

extended Onsager-Machlup description. Important contributions were made there in infinite-dimensional versions of Freidlin-Wentzel theory and associated Hamilton-Jacobi equations. The

latter are used to determine the effective potential as in static fluctuation theory. Successes were

also obtained in dynamical fluctuation theory with of course the fluctuation theorems for the

entropy production, the Jarzynski equalities and their application to determine free energy

landscapes using (nonequilibrium) work measurements. All of that is basically non-perturbative

and hence marks a true departure from irreversible thermodynamics and linear or close-to-

equilibrium regimes. At the same time many new studies have been launched, numerically and

analytically, treating models of heat and particle conduction and verifying parts of that general

theory. As is very normal and useful, most attention went to the behavior of the currents and the

entropy production. The many investigations on fluctuation symmetries, transient and steady, are

all around entropy fluxes and dissipative work. We conclude that most (and often very successful)

investigations in steady nonequilibrium statistical mechanics have concentrated on entropic aspects.

Much less time was spent with changes in the time-symmetric fluctuation sector. Kinetic aspects

that reveal the importance of changing escape and reaction rates do of course exist but they are

either treated on a more phenomenological level or they are restricted to long transient regimes.

The latter correspond to undriven dynamics with such kinetic constraints that metastable and ageing

regimes get realized.

II. DYNAMICAL ACTIVITY: GENERAL PICTURE AND INTUITION

The usual formulation of statistical physics in its derivation of thermodynamics emphasizes thegreat disparity in phase space volumes corresponding to different macroscopic behavior. Away

from thermodynamic equilibrium, phase space volume relations play a lesser role and moreover, as

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volumes get smaller, the surface area (in terms of exit and entrance rates) of phase space regions

becomes more important.

Ever since thermodynamics was consistently formulated in terms of first and second laws, energy

and entropy have been its principal couple. While their main role there originates in the discussion

of machine efficiency and dissipation, soon enough, heat was recognized as a form of energy (flow)

and also entropy was understood in the context of classical mechanics and got a statisticalmechanical interpretation. The main idea then in the microscopic derivation of the second law and

for establishing H-theorems such as for the Boltzmann equation, got engraved as S = k logW.

To remind ourselves of Boltzmanns ideas, we think of a classical closed and isolatedmechanical

system consisting of a huge number N of mass-points with states x = (q 1, ... ,qN ; p1, ,pN) on a

fixed energy surface. Total energy, particle number, and other quantities like volume or charge can

be supposed fixed. Other extensive macroscopic variables or even more general functions X(x) of x,

are not globally conserved and can change in time, such as the position of a big test object or

profiles of particle or energy density or of charge distribution etc. Having thus made the choice of

reduced variables whose values we denote by X, we associate to each a region with volume W(X),

the Liouville measure of the phase region containing all states x producing X= X(x). We skip the

more subtle physical and mathematical considerations to define this partitioning. The upshot is thatthe phase surface of constant energy is partitioned into very very unequal volumes each

corresponding to a different macroscopic state X with size W(X) = exp S(X)/k and S(X) is the

entropy of the macro-state X. We get the typical macroscopic outlook of our system by finding the

X that maximizes that entropy for given energy, volume and particle number. These are the

equilibrium values Xeq, and X(x) = Xeq is constant over almost all of phase space with

corresponding thermodynamic entropy Seq = k logW(Xeq) extensive in the particle number N. In

classical mechanics, the physical coarse graining above most often preserves a continuum structure

with reduced macroscopic states labeled by elements of Rn. For the purpose of visualization it is

however helpful to imagine quite simply that the reduced states are finite in number. We then see a

network in which each node represents a particular reduced state. The nodes visualize the joint

reduced description of system plus environment, imagining then the state cof the system itself andthe state eof the environment. Together X= (c; e) where for example cwould indicate the chemo-

mechanical configuration of a polymer or the positions of test particles and ewould tell us about the

chemical, geometric and energetic contents of the environment. The phase volume has disappeared

from this network picture, but clearly is a property of the nodes; they remain having a certain total

entropy S(X) = S(c; e). Something else has however been added to the picture and now stands at the

forefront: these are the connections in the network between the nodes. In fact, these connections or

edges in the network are typically directed and XYmust be clearly distinguished from YX.

But also the openness of the connection in this or that direction clearly matters, referring to the

degree of unobstructed entrance and exit of a state. Clearly fluctuation-response behavior not only

depends on the size of the states (their entropy), but also on the available roads; not only volume but

also surfaceof the reduced states matters and that especially so when volumes get smaller as for

nonequilibrium states. The impression arises that the architecture of the graph of states is much

more important. Yet, it is not straightforward to imagine a unique measure of openness of the

connections because it strongly depends on the nature of the reduced states. Within chemical

thermodynamics and chemical kinetics the notion of reactivity comes closest. More generally we

think of exit and entrance rates out and in the reduced state. That constitutes the dynamical activity.

III.STATE OF THE ART: DYNAMICAL ACTIVITYDynamical activity can in general be described as a measure of the systems reactivity or of its

escape rates, which can significantly change under driving conditions. Very recently we havelearned that dynamical activity matters in nonequilibrium fluctuation-response theory, but no major

observational consequences have been found yet.

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equilibrium correlation function involving the desired observable and the perturbation, nothing else.

In particular, in equilibrium the original dynamics has disappeared from the linear response

formula; all that remains from it is implicit in the equilibrium correlation functions. Not so for our

nonequilibrium formul: the specific dynamics remains explicit, especially because the second term

involving the excess in dynamical activity. It is probably not so surprising that the dynamics

matters more in nonequilibrium, and the mathematics cannot be changed of course. Yet, what we

can hope is to be able to guess that dynamical contribution from other aspects of the system. Innature the system does not present itself with a complete mathematical model included. We would

hope that from knowing some simple enough kinetic aspects we can estimate the response to

basically all small perturbations. As an example, for granular gases we can easily imagine various

small perturbations but, in the absence of a uniquely accepted Markovian mathematical description

of the systems dynamics, it is not clear what to import as DA in spite of the good experimental

access. As conclusion, the main thing we have is that traces of DA appear in a variety of formulae

related to fluctuation and response theory away from equilibrium. We wish to understand better

how to apply these formulae in natural cases where no specific detailed mathematical model is

a pri oriassumed.

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