Christian Maes

The Fluctuation Theorem as a Gibbs Property

Common ground to recent studies exploiting relations between dynamical systems and nonequilibrium statistical mechanics is, so we argue, the standard Gibbs formalism applied on the level of space-time histories. The assumptions (chaoticity principle) underlying the Gallavotti–Cohen fluctuation theorem make it possible, using symbolic dynamics, to employ the theory of one-dimensional lattice spin systems. The Kurchan and Lebowitz–Spohn analysis of this fluctuation theorem for stochastic dynamics can be restated on the level of the space-time measure which is a Gibbs measure for an interaction determined by the transition probabilities. In this note we understand the fluctuation theorem as a Gibbs property, as it follows from the very definition of Gibbs state. We give a local version of the fluctuation theorem in the Gibbsian context and we derive from this a version also for some class of spatially extended stochastic dynamics.

The random geometry of equilibrium phases

Publisher Summary This chapter discusses the random geometry of equilibrium phases. Percolation will come into play here on various levels. Its concepts like clusters, open paths, connectedness etc. will be useful for describing certain geometric features of equilibrium phases, allowing characterizations of phases in percolation terms. Examples are presented where the (thermal) phase transition goes hand in hand with a phase transition in an associated percolation process. Percolation techniques can be used to obtain specific information about the phase diagram of the system. For example, equilibrium correlation functions are sometimes dominated by connectivity functions in an associated percolation problem which is easier to investigate. Further, representations in terms of percolation models yield explicit relations between certain observables in equilibrium models and some corresponding percolation quantities.

Time-Reversal and Entropy

There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time-reversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phase-space contraction and time-reversal starting from a deterministic dynamics. Both closed and open systems, in the transient and in the steady regime, are considered. The main result identifies under general conditions the statistical mechanical entropy production as the source term of time-reversal breaking in the path space measure for the evolution of reduced variables. This provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful (in)equalities. We also discuss the Markov approximation. Important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior.

Fluctuations and response of nonequilibrium states.

A generalized fluctuation-response relation is found for thermal systems driven out of equilibrium. Its derivation is independent of many details of the dynamics, which is only required to be first order. The result gives a correction to the equilibrium fluctuation-dissipation theorem, in terms of the correlation between observable and excess in dynamical activity caused by the perturbation. Previous approaches to this problem are recovered and extended in a unifying scheme.

On the definition of entropy production, via examples

We present a definition of entropy production rate for classes of deterministic and stochastic dynamics. The point of departure is a Gibbsian representation of the steady state path space measure for which “the density” is determined with respect to the time-reversed process. The Gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems. Our definition is motivated by recent work on the Gallavotti–Cohen (local) fluctuation theorem and it is illustrated via a number of examples.

Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states

We give the explicit structure of the functional governing the dynamical density and current fluctuations for a mesoscopic system in a nonequilibrium steady state. Its canonical form determines a generalised Onsager-Machlup theory. We assume that the system is described as a Markov jump process satisfying a local detailed balance condition such as typical for stochastic lattice gases and for chemical networks. We identify the entropy current and the traffic between the mesoscopic states as extra terms in the fluctuation functional with respect to the equilibrium dynamics. The density and current fluctuations are coupled in general, except close to equilibrium where their decoupling explains the validity of entropy production principles.

Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions

Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory out-of-equilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions.

The Gacs-Kurdyumov-Levin automaton revisited

We study a one-dimensional cellular automaton that was originally proposed as a candidate for exhibiting nonergodic behavior under noise. We prove that the deterministic model has the eroder property for two and only two invariant states. Moreover, we give the best possible estimates for the corresponding erosion times. We then review the results we have obtained from extensive computer simulations for the stochastic model and for a “mixed” model and argue that they suggest numerical and heuristic evidence in favour of ergodic behavior for all nonzero values of the noise parameter.

Steady state statistics of driven diffusions

We consider overdamped diffusion processes driven out of thermal equilibrium and we analyze their dynamical steady fluctuations. We discuss the thermodynamic interpretation of the joint fluctuations of occupation times and currents; they incorporate respectively the time-symmetric and the time-antisymmetric sector of the fluctuations. We highlight the canonical structure of the joint fluctuations. The novel concept of traffic complements the entropy production for the study of the occupation statistics. We explain how the occupation and current fluctuations get mutually coupled out of equilibrium. Their decoupling close-to-equilibrium explains the validity of entropy production principles.

An update on the nonequilibrium linear response

The unique fluctuation–dissipation theorem for equilibrium stands in contrast with the wide variety of nonequilibrium linear response formulae. Their most traditional approach is ‘analytic’, which, in the absence of detailed balance, introduces the logarithm of the stationary probability density as observable. The theory of dynamical systems offers an alternative with a formula that continues to work even when the stationary distribution is not smooth. We show that this method works equally well for stochastic dynamics, and we illustrate it with a numerical example for the perturbation of circadian cycles. A second ‘probabilistic’ approach starts from dynamical ensembles and expands the probability weights on path space. This line suggests new physical questions, as we meet the frenetic contribution to linear response, and the relevance of the change in dynamical activity in the relaxation to a (new) nonequilibrium condition.