Non-Equilibrium Thermodynamics and Topology of Currents

TL;DR

This paper applies large deviations theory to stochastic systems with topological currents, deriving principles and variational relations, and illustrating with a spin-chain example under weak noise conditions.

Contribution

It develops a framework for analyzing large deviations of topological currents in stochastic maps, including explicit calculations for spin-chain models.

Findings

Large deviations principle established for time-averaged currents.

Explicit Cramér functional derived for topological current in spin-chain.

Reduction of stochastic models to effective Markov chains in weak-noise limit.

Abstract

In many experimental situations, a physical system undergoes stochastic evolution which may be described via random maps between two compact spaces. In the current work, we study the applicability of large deviations theory to time-averaged quantities which describe such stochastic maps, in particular time-averaged currents and density functionals. We derive the large deviations principle for these quantities, as well as for global topological currents, and formulate variational, thermodynamic relations to establish large deviation properties of the topological currents. We illustrate the theory with a nontrivial example of a Heisenberg spin-chain with a topological driving of the Wess-Zumino type. The Cram\'er functional of the topological current is found explicitly in the instanton gas regime for the spin-chain model in the weak-noise limit. In the context of the Morse theory, we discuss a general reduction of continuous stochastic models with weak noise to effective Markov chains describing transitions between stable fixed points.