The entropy of a thermodynamic graph

TL;DR

This paper introduces a graph-based model for heat conduction that uses recurrence relations and defines an entropy measure to analyze stability, offering a flexible alternative to traditional differential equation methods.

Contribution

The paper presents a novel algorithmic model of heat conduction using thermodynamic graphs and introduces an entropy concept to determine stability boundaries.

Findings

Maximum stable time step length determined by entropy non-decrease

Entropy provides a measure of thermodynamic process stability

Model offers a flexible, differential-equation-free approach to heat conduction

Abstract

We introduce an algorithmic model of heat conduction, the thermodynamic graph. The thermodynamic graph is analogous to meshes in the finite difference method in the sense that the calculation of temperature is carried out at the vertices of the graph, and the edges indicate the direct heat exchange between the vertices. Recurrence relations of heat conduction in graph are derived without using of differential equations and based on the coefficients of thermal conductivity and heat capacity. This approach seems to be more direct and flexible from the point of view of algorithmic modeling of thermodynamic process than the derivation of difference schemes from differential equations. We introduce also the notion of entropy of thermodynamic graph. We find the maximum length of the time step at which the entropy does not decrease in the general case. As a result, this give us the accurate boundary of the model stability.