Time Asymmetry in Nonequilibrium Statistical Mechanics

Abstract

An overview is given on the recent understanding of time asymmetry in nonequilibrium statistical mechanics. This time asymmetry finds its origin in the spontaneous breaking of the time-reversal symmetry at the statistical level of description. The relaxation toward the equilibrium state can be described in terms of eigenmodes of fundamental Liouville's equation of statistical mechanics. These eigenmodes are associated with Pollicott-Ruelle resonances and correspond to exponential damping. These eigenmodes can be explicitly constructed in dynamical systems sustaining deterministic diffusion, which shows their singular character. The entropy production expected from nonequilibrium thermodynamics can be derived ab initio thanks to this construction. In the escape-rate theory, the transport coefficients can be related to the leading Pollicott-Ruelle resonance of open systems and to the characteristic quantities of the microscopic dynamics. In nonequilibrium steady states, the entropy production is shown to result from a time asymmetry in the dynamical randomness of the nonequilibrium fluctuations. Furthermore, the generalizations of Onsager reciprocity relations to nonlinear response can be deduced from the fluctuation theorem for the currents. Furthermore, the principle of temporal ordering in nonequilibrium steady states is formulated and its perspectives for the generation of biological information are discussed.

Thermodynamic Time Asymmetry and Nonequilibrium Statistical Mechanics

“…Similar results have been obtained for an RC electric circuit, showing the time asymmetry down to fluctuations of about a few thousands electronic charges [12]. The link between the formula (20) and the escape-rate theory has been discussed elsewhere [4]. The connection can also be established with nonequilibrium work relations [16,17].…”

“…A fortiori, this breaking can also happen in a chaotic system with a spectrum of positive Lyapunov exponents indicating many stable and unstable directions in phase space. These directions are mapped onto each other but physically distinct so that the time reversal symmetry will be broken if one specific direction is selected by the initial condition [3,4]. In statistical mechanics, each initial condition -and thus each phasespace trajectory -is weighted with a probability giving its statistical frequency of occurrence in a sequence of repeated experiments.…”

Multivariate fluctuation relations for currents

“…Remarkably, the relation (3) shows that the current fluctuations are larger in the direction of the affinities, which thus control the directionality. Therefore, the knowledge of the affinities in the frame defined by the reservoirs determines the direction of the most probable current fluctuations [66,67]. In this regard, the question arises whether it is possible to guess the direction of the affinities A imposed by the reservoirs from the sole observation of a current fluctuation J in some macroscopic steady state.…”

Information Entropy Production of Maximum Entropy Markov Chains from Spike Trains

“…Previous studies in the context of spike train statistics have measured the dynamical entropy production in spiking neuron networks using a deterministic approach based on the Pesin identity (sum of positive Lyapunov exponents) [ 47 ]. There are relationships between the deterministic and stochastic dynamics [ 48 ], and some interpretations of deterministic dynamical entropy production with information loss which should be investigated in more detail, in particular, if these relationships bring new knowledge in the field of computational neuroscience.…”