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

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.