Lie groups in nonequilibrium thermodynamics: Geometric structure behind viscoplasticity

TL;DR

This paper explores the geometric structure of nonequilibrium thermodynamics using Lie groups, providing a classification of rheological models and distinguishing viscoplasticity from viscoelasticity.

Contribution

It introduces a Lie group-based classification framework for rheological models, highlighting the geometric differences between viscoplasticity and viscoelasticity.

Findings

Classification of rheological models via Lie groups

Deep distinction between viscoplasticity and viscoelasticity

Illustration with a crystal viscoplasticity toy model

Abstract

Poisson brackets provide the mathematical structure required to identify the reversible contribution to dynamic phenomena in nonequilibrium thermodynamics. This mathematical structure is deeply linked to Lie groups and their Lie algebras. From the characterization of all the Lie groups associated with a given Lie algebra as quotients of a universal covering group, we obtain a natural classification of rheological models based on the concept of discrete reference states and, in particular, we find a clear-cut and deep distinction between viscoplasticity and viscoelasticity. The abstract ideas are illustrated by a naive toy model of crystal viscoplasticity, but similar kinetic models are also used for modeling the viscoplastic behavior of glasses. We discuss some implications for coarse graining and statistical mechanics.