Classical stochastic dynamics and continuous matrix product states: gauge transformations, conditioned and driven processes, and equivalence of trajectory ensembles

TL;DR

This paper introduces a formalism using continuous matrix product states to analyze classical stochastic trajectories, enabling the study of conditioned processes, gauge transformations, and ensemble equivalences with applications to fluctuation theorems.

Contribution

It develops a novel cMPS-based framework for classical stochastic dynamics, connecting gauge transformations to conditioned processes and proving ensemble equivalences.

Findings

Gauge transformations correspond to Doob transforms in conditioned processes.

The framework simplifies proofs of trajectory ensemble equivalence.

Fluctuation theorems are naturally derived within this formalism.

Abstract

Borrowing ideas from open quantum systems, we describe a formalism to encode ensembles of trajectories of classical stochastic dynamics in terms of continuous matrix product states (cMPSs). We show how to define in this approach "biased" or "conditioned" ensembles where the probability of trajectories is biased from that of the natural dynamics by some condition on trajectory observables. In particular, we show that the generalised Doob transform which maps a conditioned process to an equivalent "auxiliary" or "driven" process (one where the same conditioned set of trajectories is generated by a proper stochastic dynamics) is just a gauge transformation of the corresponding cMPS. We also discuss how within this framework one can easily prove properties of the dynamics such as trajectory ensemble equivalence and fluctuation theorems.