Bernard Gaveau

Theory of nonequilibrium first-order phase transitions for stochastic dynamics

A dynamic definition of a first-order phase transition is given. It is based on a master equation description of the time evolution of a system. When the operator generating that time evolution has an isolated near degeneracy there is a first-order phase transition. Conversely, when phenomena describable as first-order phase transitions occur in a system, the corresponding operator has near degeneracy. Estimates relating degree of degeneracy and degree of phase separation are given. This approach harks back to early ideas on phase transitions and degeneracy, but now enjoys greater generality because it involves an operator present in a wide variety of systems. Our definition is applicable to what have intuitively been considered phase transitions in nonequilibrium systems and to problematic near equilibrium cases, such as metastability.

Coarse Grains: The Emergence of Space and Order

The emergence of macroscopic variables can be effected through coarse graining. Despite practical and fundamental benefits conveyed by this partitioning of state space, the apparently subjective nature of the selection of coarse grains has been considered problematic. We provide objective selection methods, deriving from the existence of relatively slow dynamical time scales. Using a framework for nonequilibrium statistical mechanics developed by us, we show the emergence of both spatial variables and order parameters. Although significant objective criteria are introduced in the coarse graining, we do not provide a unique prescription. Most significantly, the grains, and by implication entropy, are only defined modulo a characteristic time scale of observation.

A general framework for non-equilibrium phenomena: the master equation and its formal consequences

Abstract We consider non-equilibrium systems defined by a state space, and by a stochastic dynamics and its stationary state. The dynamics need not satisfy detailed balance. In this abstract framework we do the following: (1) define and analyze “relative entropy”, (2) study dissipation in the relaxation to the stationary state, as well as the extra dissipation to maintain the system in its stationary state against some detailed balance dynamics, (3) extend the fluctuation-dissipation theorem and the Onsager relations, and (4) give a formula for the stationary state in terms of a summation over trees.

Master equation based formulation of nonequilibrium statistical mechanics

For a nonequilibrium system characterized by its state space, by a dynamics defined by a transfer matrix and by a reference equilibrium dynamics given by a detailed‐balance transfer matrix, we define various nonequilibrium concepts: relative entropy, dissipation during the relaxation to the stationary state, path entropy, cost for maintaining the system in a nonequilibrium state, fluctuation‐dissipation theory, and finally a tree integral formula for the stationary state.

Multiple phases in stochastic dynamics: Geometry and probabilities

Stochastic dynamics is generated by a matrix of transition probabilities. Certain eigenvectors of this matrix provide observables, and when these are plotted in the appropriate multidimensional space the phases (in the sense of phase transitions) of the underlying system become manifest as extremal points. This geometrical construction, which we call an observable representation of state space, can allow hierarchical structure to be observed. It also provides a method for the calculation of the probability that an initial points ends in one or another asymptotic state.

Slow relaxation, confinement, and solitons.

Millisecond crystal relaxation has been used to explain anomalous decay in doped alkali halides. We attribute this slowness to Fermi-Pasta-Ulam solitons. Our model exhibits confinement of mechanical energy released by excitation. Extending the model to long times is justified by its relation to solitons, excitations previously proposed to occur in alkali halides. Soliton damping and observation are also discussed.

Spectral signatures of hierarchical relaxation

Abstract Using dynamical concepts of phase transitions developed in earlier work, we exhibit the characteristic features of the spectrum and eigenfunctions to be expected when a hierarchy of phase transitions is present in a system, such as is expected to occur for spin glasses.

Entropy, extropy and information potential in stochastic systems far from equilibrium

The relations between information, entropy and energy, which are well known in equilibrium thermodynamics, are not clear far from equilibrium. Moreover, the usual expression of the classical thermodynamic potentials is only valid near equilibrium. In previous publications, we showed for a chemical system maintained far from equilibrium, that a new thermodynamic potential, the information potential, can be defined by using the stochastic formalism of the Master Equation. Here, we extend this theory to a completely general discrete stochastic system. For this purpose, we use the concept of extropy, which is defined in classical thermodynamics as the total entropy produced in a system and in the reservoirs during their equilibration. We show that the statistical equivalent of the thermodynamic extropy is the relative information. If a coarse-grained description by means of the thermodynamic extensive variables is available (which is the case for many macroscopic systems) the coarse-grained statistical extropy allows one to define the information potential in the thermodynamic limit. Using this potential, we study the evolution of such systems towards a non-equilibrium stationary state. We derive a general thermodynamic inequality between energy dissipation and information dissipation, which sharpens the law of the maximum available work for non-equilibrium systems.

Creation, Dissipation and Recycling of Resources in Non-Equilibrium Systems

In this article, we define stochastic dynamics for a system coupled to reservoirs. The rules for forward and backward transitions are related by a generalized detailed balance identity involving the system and its reservoirs. We compare the variation of information and of entropy. We define the Carnot dissipation and prove that it can be expressed in terms of cyclic transformations. Lower bounds for partial dissipations are also studied, as well as the effect of switching off certain reservoirs. We also study the near degeneracy of the stochastic matrix, relate it to phase transitions and we show that the reduced dynamics on the set of phases is a permutation. Finally, we relate these concepts to heat, work and more generally to the dissipation and creation of resources, in general systems.

A stochastic theory of chemical reaction rates. I. Formalism

A class of stochastic processes is studied that can be used to model elementary and complex chemical reactions composed of a series of several distinct steps. Formal correlation function expressions are directly computed for the stochastic model to yield the overall rate constant for the reaction. One of the main results is a formula connecting the overall rate constant to the rate constants characterizing the elementary steps of the reaction.