Dissipation Function: Nonequilibrium Physics and Dynamical Systems

Abstract

An exact response theory has recently been developed within the field of Nonequilibrium Molecular Dynamics. Its main ingredient is known as the Dissipation Function, Ω. This quantity determines nonequilbrium properties like thermodynamic potentials do with equilibrium states. In particular, Ω can be used to determine the exact response of particle systems obeying classical mechanical laws, subjected to perturbations of arbitrary size. Under certain conditions, it can also be used to express the response of a single system, in contrast to the standard response theory, which concerns ensembles of identical systems. The dimensions of Ω are those of a rate, hence Ω can be associated with the entropy production rate, provided local thermodynamic equilibrium holds. When this is not the case for a particle system, or generic dynamical systems are considered, Ω can equally be defined, and it yields formal, thermodynamic-like, relations. While such relations may have no physical content, they may still constitute interesting characterizations of the relevant dynamics. Moreover, such a formal approach turns physically relevant, because it allows a deeper analysis of Ω and of response theory than possible in case of fully fledged physical models. Here, we investigate the relation between linear and exact response, pointing out conditions for the validity of the response theory, as well as difficulties and opportunities for the physical interpretation of certain formal results.

Exact Response Theory for Time-Dependent and Stochastic Perturbations

“…and for the exact response, we have the following: Again, the same response is obtained only for the fixed initial condition (θ 0 = 1, ϕ 0 = 0). This is not strange; it happens when the exact response amounts to a first order perturbation, which is the case in special situations [39]. However, in general, this condition is not verified.…”

Exact Response Theory and Kuramoto dynamics

“…Its investigation has greatly progressed with the works of Callen, Green, Kubo, and Onsager, in particular, who contributed to the development of linear response theory [31,35]. In the '90s, the derivation of the Fluctuation Relations [18,20,25] provided the framework for a more general response theory, applicable to both Hamiltonian as well as dissipative deterministic particle systems [8,9,11,12,13,23,35,39]. The study of response in stochastic processes, with a special focus on diffusion and Markov jump processes, has also been inspired by fluctuation relations, and has been studied e.g.…”