Thermodynamics in Terms of a Sequence of $n-$chains Derived from a Martingale Decomposition of the Energy Process

TL;DR

This paper explores the connection between thermodynamics and algebraic topology by representing a system's energy process at equilibrium as a sequence of $n$-chains derived from a martingale decomposition, revealing new structural insights.

Contribution

It introduces a novel algebraic framework linking energy processes in thermodynamics with topological structures via martingale decomposition and $n$-chains.

Findings

Establishes a correspondence between energy processes and $n$-chains.

Provides a new perspective on the topological structure of thermodynamic systems.

Bridges algebraic topology with stochastic processes in statistical mechanics.

Abstract

The role of the algebraic method has long been understood in shedding light on the topological structure of sets. However, when the set is a simplicial complex and host to a dynamical process, in particular the trajectory of a canonically distributed system in thermal equilibrium with a heat bath, the algebra re-enters. Via a theorem of Levy and Dynkin, there is a correspondence between a system's energy process at equilibrium and a sequence of $n-$chains on the state space.