Construction and Analysis of a Discrete Heat Equation Using Dynamic Consistency: The Meso-scale Limit

TL;DR

This paper derives a meso-scale heat transfer model from a microscopic discrete system using dynamic consistency, correcting previous models and avoiding zero-limit parameters, highlighting challenges in formulating physically valid heat equations.

Contribution

It introduces a new derivation method for meso-scale heat equations based on dynamic consistency, improving upon prior models and addressing limitations related to parameter limits.

Findings

Reproduces and corrects existing heat PDE modifications

Avoids zero limiting values for microscopic parameters

Provides insights into the construction of physically valid heat equations

Abstract

We present and analyze a new derivation of the meso-level behavior of a discrete microscopic model of heat transfer. This construction is based on the principle of dynamic consistency. Our work reproduces and corrects, when needed, all the major previous expressions which provide modifications to the standard heat PDE. However, unlike earlier efforts, we do not allow the microscopic level parameters to have zero limiting values. We also give insight into the difficulties of constructing physically valid heat equations within the framework of the general mathematically inequivalent of difference and differential equations.