Corrections and enhancements of quasi-equilibrium states

Abstract

We give a compact non-technical presentation of two basic principles for reducing the description of nonequilibrium systems based on the quasi-equilibrium approximation. These two principles are: construction of invariant manifolds for the dissipative microscopic dynamics, and coarse-graining for the entropy-conserving microscopic dynamics. Two new results are presented: first, an application of the invariance principle to hybridization of micro-macro integration schemes is introduced, and is illustrated with non-linear dumbbell models; second, Ehrenfest's coarse-graining is extended to general quasi-equilibrium approximations, which gives the simplest way to derive dissipative equations from the Liouville equation in the short memory approximation. © 2001 Elsevier Science B.V. All rights reserved.Keywords: Non-equilibrium thermodynamics; Quasi-equilibrium approximation; Invariant manifold; Coarse-graining Quasi-equilibrium approximations and their place in the problem of transition from microscopic to macroscopic variables Most of the works on non-equilibrium thermodynamics deal with corrections to quasi-equilibrium approximations or with applications of these approximations (with or without corrections).This viewpoint is not only possible but it proves very efficient for the construction of a variety of useful models, approximations and equations, as well as methods to solve them.

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

“…In the limit b → ∞, the Hookean spring is recovered. Recently, it has been demonstrated that FENE-P model appears as first approximation within a systematic self-confident expansion of nonlinear forces [28].…”

Legendre integrators, post-processing and quasiequilibrium

“…The methods of the first order, based on this idea, were suggested and tested in Refs. [12][13][14]. The method of the thermodynamic projector lets us represent every ansatz-manifold as the solution to the variational problem (2) with specially chosen constraints.…”

“…Let us assume that in the interesting domain of initial conditions x 0 the solutions x(t) of Eq. (14) are developing in the following way: the vector x(t) is going rapidly to the value that is defined by the slow variables M; after that x can be represented as a function of M with good accuracy, and this function is unique for every initial condition. So, (A) for each value of the slow variables…”

“…(14) remains the Lyapunov function for the macroscopic system (17), and if the microscopic system is conservative, then its quasiequilibrium projection to the space of macroscopic variables remains conservative.…”

“…We only mention that in the projection on ∆ one gets a step of the relaxation method for the invariant manifold construction. For the static post-processing with frozen parameters the "naive" estimation given by the invariance defect (67) makes sense [14]. Serious problems for reduced description can arise if the approximate invariant manifold is unstable in the sense that after small perturbations the perturbed motion can go far away.…”