Thermodynamic hierarchies of evolution equations

P Ván,T Fülöp,R Kovács

doi:10.3176/proc.2015.3s.09

Abstract

Non-equilibrium thermodynamics with internal variables introduces a natural hierarchical arrangement of evolution equations. Three examples are shown: a hierarchy of linear constitutive equations in thermodynamic rhelogy with a single internal variable, a hierarchy of wave equations in the theory of generalized continua with dual internal variables and a hierarchical arrangement of the Fourier equation in the theory of heat conduction with current multipliers.

Highlights

Theories and material models of multiscale phenomena in space and time treat the scale changes either as a step from a micro- or mesoscopic statistical level to the phenomenological one or as a reduction of the degrees of freedom by averaging over a field variable or spatial dimension
We show that a hierarchical arrangement of evolution equations is apparent in thermodynamics with internal variables where the different levels of the hierarchy are regulated by material parameters
A hierarchy of ordinary differential equations is presented in the thermodynamic rheology of solids with a single internal variable

Summary

Introduction

Theories and material models of multiscale phenomena in space and time treat the scale changes either as a step from a micro- or mesoscopic statistical level to the phenomenological one or as a reduction of the degrees of freedom by averaging over a field variable or spatial dimension. We show that a hierarchical arrangement of evolution equations is apparent in thermodynamics with internal variables where the different levels of the hierarchy are regulated by material parameters. The building block of the hierarchy is the basic constitutive equation of elasticity This is a hierarchy between the different time scales of the evolution, a time hierarchy. A hierarchy of hyperbolic partial differential equations is shown in the thermodynamic theory of generalized continua with dual internal variables. The building block of the hierarchy is the wave equation This is a dynamic hierarchy between different time and length scales of the evolution, a space-time hierarchy. It is a dynamic hierarchy between different time and length scales of the evolution, a space-time hierarchy

The hierarchy of rheological bodies and the Kluitenberg–Verhas model

Hierarchy of wave equations in the theory of dual internal variables

Hierarchical resonance: the example of the Guyer–Krumhansl equation