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

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.