Stochastic Dynamics of Extended Objects in Driven Systems II: Current Quantization in the Low-Temperature Limit

TL;DR

This paper demonstrates that in driven stochastic systems at low temperature, higher-dimensional currents become rationally quantized, using topological and Markov chain models to analyze non-equilibrium steady states.

Contribution

It introduces a topological framework for analyzing stochastic motion of extended objects, revealing rational quantization of currents in the low-temperature adiabatic limit.

Findings

Higher-dimensional currents are rationally quantized under generic driving.

Reduction to a discrete Markov chain on a CW complex simplifies analysis.

Utilizes higher-dimensional Kirchhoff theorems to establish quantization results.

Abstract

Driven Langevin processes have appeared in a variety of fields due to the relevance of natural phenomena having both deterministic and stochastic effects. The stochastic currents and fluxes in these systems provide a convenient set of observables to describe their non-equilibrium steady states. Here we consider stochastic motion of a (k - 1)-dimensional object, which sweeps out a k-dimensional trajectory, and gives rise to a higher k-dimensional current. By employing the low-temperature (low-noise) limit, we reduce the problem to a discrete Markov chain model on a CW complex, a topological construction which generalizes the notion of a graph. This reduction allows the mean fluxes and currents of the process to be expressed in terms of solutions to the discrete Supersymmetric Fokker-Planck (SFP) equation. Taking the adiabatic limit, we show that generic driving leads to rational quantization of the generated higher dimensional current. The latter is achieved by implementing the recently developed tools, coined the higher-dimensional Kirchhoff tree and co-tree theorems. This extends the study of motion of extended objects in the continuous setting performed in the prequel to this manuscript.