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

2016/02/29 by Juan P. Garrahan · 3 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Equivalence (formal languages) #Formalism (music) #Mathematics #Observable #Physics #Pure mathematics #Quantum Mechanics and Applications #Quantum many-body systems #Quantum mechanics #Statistical physics #Stochastic process #Trajectory #cond-mat.stat-mech

paper · pdf · doi:10.1088/1742-5468/2016/07/073208

published as J. Stat. Mech. 073208 (2016) · 12 pages; example added, proof of equivalence extended

arxiv created 2016/05/29 · openalex publication_date 2016/07/26 · arxiv updated 2016/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems
• Stochastic thermodynamics, fluctuation theorems and molecular machines
• Computation of Current Cumulants for Small Nonequilibrium Systems
• Space–time thermodynamics of the glass transition

Cited by

• Rare Event Analysis of Large Language Models
• Detailed Fluctuation Theorems: A Unifying Perspective
• A Tensor Network Approach to Finite Markov Decision Processes