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arxiv: 2506.04726 · v3 · submitted 2025-06-05 · ❄️ cond-mat.stat-mech

Stochastic thermodynamics for classical non-Markov jump processes

Kiyoshi Kanazawa , Andreas Dechant This is my paper

Pith reviewed 2026-05-19 11:45 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords stochastic thermodynamicsnon-Markov processesjump processesFourier embeddingsecond lawtime-reversal symmetrymemory effects
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0 comments X

The pith

Fourier embedding converts non-Markov jump processes into Markovian field dynamics to derive the second law and time-reversal conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to build a thermodynamic framework that works for classical jump processes whose future depends on their full history rather than only the present state. It introduces Fourier embedding to map any such process that allows the mapping onto an equivalent Markov process in an expanded space of auxiliary modes. This mapping makes standard thermodynamic relations available again, including necessary and sufficient conditions for time-reversal symmetry and a second-law inequality. The result matters because most real small systems exhibit memory, yet existing stochastic thermodynamics has been limited to memoryless cases.

Core claim

The authors show that any classical non-Markov jump process that admits the Fourier embedding can be exactly represented as Markovian field dynamics of auxiliary Fourier modes. From this representation they obtain necessary and sufficient conditions for the process to be time-reversal symmetric and derive the second law for the entire class of strong-memory dynamics that satisfy the embedding condition. They illustrate the construction with two explicit models: a history-dependent two-level system and a history-dependent random walk.

What carries the argument

The Fourier embedding, which converts non-Markov jump processes into Markovian field dynamics of auxiliary Fourier modes without loss of statistics.

If this is right

  • The second law holds for any jump process whose statistics are preserved under Fourier embedding.
  • Time-reversal symmetry of the original non-Markov process is equivalent to a symmetry condition on the embedded Markovian field.
  • History-dependent models can be built systematically while guaranteeing thermodynamic consistency.
  • Energetic and entropic bounds become available for a wide range of memoryful systems observed in experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same embedding idea may extend to continuous-state non-Markov processes if an analogous auxiliary-field representation can be found.
  • Numerical simulation of the embedded Markov process could provide an efficient way to sample rare events in the original memoryful dynamics.
  • If many experimental systems turn out to admit the embedding, thermodynamic inference methods developed for Markov cases could be reused after a change of variables.

Load-bearing premise

The non-Markov jump processes must admit an exact conversion into Markovian field dynamics of auxiliary Fourier modes.

What would settle it

Construct a concrete non-Markov jump process that cannot be represented by any Fourier embedding yet still satisfies the second law and time-reversal symmetry in its original variables.

Figures

Figures reproduced from arXiv: 2506.04726 by Andreas Dechant, Kiyoshi Kanazawa.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic of the non-Markov jump process. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Numerical results for the non-Markov two-level system with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Fourier vs. Laplace embeddings by assuming the time-reversal symmetry. (a) For the Fourier embedding, when the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Schematic path of the Laplace embedding [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Stochastic thermodynamics investigates energetic and entropic bounds in small systems. Foundational results, e.g., the first and second laws, predominantly rely on the Markov (memoryless) assumption. Although physicists recognise that the Markov assumption is questionable in real experimental setups, extending stochastic thermodynamics to general non-Markov systems has proven challenging. Fundamentally, it has been elusive how to model memory-dependent non-Gaussian fluctuations consistently with thermodynamic laws. Here we establish a general theory of stochastic thermodynamics for classical non-Markov jump processes. We introduce a key technique, called the Fourier embedding, which converts non-Markov jump processes into Markovian field dynamics of auxiliary Fourier modes. This yields necessary and sufficient conditions for time-reversal symmetry and enables the derivation of the second law for a broad class of strong-memory dynamics that admit the Fourier embedding. We demonstrate the power of our framework by presenting two novel non-Markov models: (i) a history-dependent two-level system and (ii) a history-dependent random walk. Our work accommodates diverse non-Markov dynamics in realistic experimental settings and offers a guiding principle for physics-informed modelling of history-dependent fluctuations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to establish a general theory of stochastic thermodynamics for classical non-Markov jump processes. It introduces the Fourier embedding technique, which converts such processes into Markovian field dynamics involving auxiliary Fourier modes. This construction is used to obtain necessary and sufficient conditions for time-reversal symmetry and to derive the second law for the subclass of strong-memory dynamics that admit the embedding. The framework is illustrated with two concrete models: a history-dependent two-level system and a history-dependent random walk.

Significance. If the embedding is shown to map thermodynamic quantities (heat, work, entropy production) exactly onto the original jump process without extraneous contributions from the auxiliary modes, the result would meaningfully extend stochastic thermodynamics beyond the Markov assumption to a broad class of memory-dependent systems relevant to experiments. The provision of explicit, solvable non-Markov models is a concrete strength that allows direct verification of the claims.

major comments (2)
  1. [§4] §4 (derivation of the second law): The central claim that the second law holds for the original non-Markov process rests on the assertion that thermodynamic observables are preserved under the Fourier embedding. However, the text does not explicitly demonstrate that the entropy-production functional defined on the auxiliary-field dynamics reduces exactly to the jump-process entropy production (no additive term arising from the auxiliary modes). This correspondence is load-bearing and requires a direct proof or explicit calculation.
  2. [§3.1] §3.1, definition of the embedding map: While the embedding is stated to reproduce the original jump statistics exactly, the identification of heat and work along individual jumps in the embedded dynamics is not spelled out. Without this identification, it is unclear whether the first-law balance and the fluctuation theorems derived in the Markovian field theory apply verbatim to the physical non-Markov process.
minor comments (2)
  1. The notation distinguishing the original jump process from the auxiliary Fourier field could be made more uniform across equations and figures to avoid reader confusion.
  2. A brief comparison with existing embedding or auxiliary-variable techniques in the non-Markov literature would help situate the novelty of the Fourier embedding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments, which have helped us strengthen the presentation of our results. We address each major comment below and have revised the manuscript to provide the requested clarifications and proofs.

read point-by-point responses

  1. Referee: [§4] §4 (derivation of the second law): The central claim that the second law holds for the original non-Markov process rests on the assertion that thermodynamic observables are preserved under the Fourier embedding. However, the text does not explicitly demonstrate that the entropy-production functional defined on the auxiliary-field dynamics reduces exactly to the jump-process entropy production (no additive term arising from the auxiliary modes). This correspondence is load-bearing and requires a direct proof or explicit calculation.

    Authors: We agree that an explicit demonstration of this reduction is essential for the central claim. The auxiliary Fourier modes are introduced as a mathematical device to encode memory and do not represent additional physical degrees of freedom. In the revised manuscript we have added a direct calculation in §4 showing that the entropy-production functional of the embedded Markovian field dynamics, after integration over the auxiliary modes, reduces exactly to the entropy production of the original jump process. The auxiliary modes contribute no additive term because their evolution is fully determined by the jump history and carries no independent energetic cost. This establishes the second law for the non-Markov process. revision: yes

  2. Referee: [§3.1] §3.1, definition of the embedding map: While the embedding is stated to reproduce the original jump statistics exactly, the identification of heat and work along individual jumps in the embedded dynamics is not spelled out. Without this identification, it is unclear whether the first-law balance and the fluctuation theorems derived in the Markovian field theory apply verbatim to the physical non-Markov process.

    Authors: We concur that the mapping of thermodynamic quantities along jumps should be stated explicitly. In the revised manuscript we have expanded §3.1 to clarify that each jump of the original process corresponds one-to-one with a transition event in the embedded field dynamics. Heat is identified with the energy change of the system at that jump (following the standard stochastic-thermodynamics convention), and work is defined analogously from any external protocol. Because the embedding reproduces the jump statistics exactly and the auxiliary modes do not participate in the energy balance, the first-law balance and fluctuation theorems derived for the Markovian field theory apply directly to the physical non-Markov process. revision: yes

Circularity Check

0 steps flagged

Fourier embedding is an independent construction; no reduction to inputs by definition or self-citation

full rationale

The paper introduces the Fourier embedding as a novel technique that converts non-Markov jump processes into Markovian auxiliary-field dynamics, from which time-reversal symmetry conditions and the second law are then derived. This construction is presented as a new modeling step rather than a fit to data or a renaming of prior results. No load-bearing step reduces by the paper's own equations to a self-citation chain, a fitted parameter relabeled as a prediction, or a self-definitional loop. The derivation remains self-contained against external benchmarks because the embedding is defined to preserve the original statistics exactly, after which standard Markov stochastic-thermodynamic identities are applied without circular redefinition of heat, work, or entropy production.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the domain assumption that the target processes admit an exact Fourier embedding into Markovian auxiliary-mode dynamics; no explicit free parameters are introduced in the abstract, and the embedding itself functions as the central invented construct.

axioms (1)
  • domain assumption Non-Markov jump processes admit Fourier embedding into Markovian field dynamics of auxiliary Fourier modes.
    This is the key technical step that converts memory dependence into an extended Markov process.
invented entities (1)
  • Fourier embedding technique no independent evidence
    purpose: Converts non-Markov jump processes into Markovian dynamics of auxiliary Fourier modes to enable thermodynamic derivations.
    New modeling construct introduced to handle strong memory while preserving jump-process statistics.

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Forward citations

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