arxiv: 2604.25095 · v1 · submitted 2026-04-28 · ❄️ cond-mat.stat-mech

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Markovian thermodynamics of non-Markovian Langevin equations

Andreas Dechant , Kiyoshi Kanazawa

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords non-Markovian dynamicsgeneralized Langevin equationMarkovian embeddingentropy productionstochastic thermodynamicshydrodynamic interactionsauxiliary variables

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The pith

Non-Markovian generalized Langevin equations embed into Markovian systems yielding unique monotonic entropy production.

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

The paper develops thermodynamics for non-Markovian generalized Langevin equations by embedding them into a larger Markovian system that includes auxiliary degrees of freedom. When the memory is linear and satisfies detailed balance with the noise, an explicit construction exists even for multiple particles and hydrodynamic interactions. Thermodynamic quantities including entropy production prove independent of the choice of embedding. This supplies a definition of entropy production for the original non-Markovian system that is guaranteed to increase with time. The apparent entropy decrease seen directly in the non-Markovian description is reinterpreted as heat and information exchange with the auxiliary variables.

Core claim

We develop the thermodynamics of non-Markovian generalized Langevin equations by embedding them in a high-dimensional Markovian representation involving auxiliary degrees of freedom. If the memory is linear and satisfies detailed balance with the noise, we provide an explicit construction of the embedding for non-Markovian dynamics with many degrees of freedom and hydrodynamic interactions. Moreover, while the embedding is generally not unique, we show that it results in unique values of thermodynamic quantities of the Markovian system. This allows us to define the Markovian entropy production of a non-Markovian system, which is guaranteed to increase monotonically with time. Moreover, the 1

What carries the argument

The explicit construction of a Markovian embedding with auxiliary degrees of freedom for linear memory kernels that satisfy detailed balance with the noise.

Where Pith is reading between the lines

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

The same embedding approach may supply fluctuation theorems for non-Markovian processes by transferring them from the Markovian representation.
Numerical checks on colloidal particles or other systems with hydrodynamic memory could directly test the uniqueness of the entropy production.
The auxiliary variables act as an information reservoir, suggesting links to information thermodynamics that the paper leaves open.
The construction might extend to certain classes of nonlinear memory if analogous balance conditions can be identified.

Load-bearing premise

The memory must be linear and must satisfy detailed balance with the noise.

What would settle it

A concrete counterexample in which the constructed Markovian entropy production decreases over time for a system obeying the linear-memory and detailed-balance conditions would disprove the monotonicity claim.

Figures

Figures reproduced from arXiv: 2604.25095 by Andreas Dechant, Kiyoshi Kanazawa.

**Figure 1.** Figure 1: FIG. 1. Markovian and non-Markovian EPR for a particle in a trap with periodically varying trap stiffness. Left: As a function view at source ↗

**Figure 2.** Figure 2: FIG. 2. Heat flow for a driven particle with power-law memory. Left: Position of the trap view at source ↗

read the original abstract

We develop the thermodynamics of non-Markovian generalized Langevin equations by embedding them in a high-dimensional Markovian representation involving auxiliary degrees of freedom. If the memory is linear and satisfies detailed balance with the noise, we provide an explicit construction of the embedding for non-Markovian dynamics with many degrees of freedom and hydrodynamic interactions. Moreover, while the embedding is generally not unique, we show that it results in unique values of thermodynamic quantities of the Markovian system. This allows us to define the Markovian entropy production of a non-Markovian system, which, in contrast to the definition based directly on the non-Markovian dynamics, is guaranteed to increase monotonically with time. Moreover, the Markovian representation allows us to identify the apparent decrease in the non-Markovian entropy with heat and information exchange between the system and the auxiliary degrees of freedom.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit embedding of linear-memory non-Markovian Langevin equations into Markovian dynamics that makes thermodynamic quantities unique and entropy production monotonic.

read the letter

The core move is to add auxiliary variables so that a non-Markovian generalized Langevin equation with linear memory becomes a higher-dimensional Markov process. When the memory kernel satisfies detailed balance with the noise, the authors give a concrete construction that works for many particles and includes hydrodynamic interactions. They then show that thermodynamic observables, including entropy production, do not depend on which valid embedding you pick. This lets them import the standard Markovian second law directly, so entropy production is guaranteed to increase. Apparent violations in the original non-Markovian description are reinterpreted as heat and information flow into the auxiliaries. That part is clean and addresses a real practical problem in soft-matter and biophysical modeling. The construction is new enough relative to earlier non-Markovian thermodynamics work, and the uniqueness result is the part that actually makes the framework usable. The limitation is stated up front: everything requires linear memory plus detailed balance. Systems with nonlinear memory or broken detailed balance fall outside the construction, and the paper does not claim otherwise. If the full text contains the explicit checks against equilibrium limits and the multi-particle hydrodynamic case, that would be solid evidence; the abstract alone leaves those steps implicit. For someone already working on memory effects in active or colloidal systems, the paper supplies a usable recipe rather than another abstract definition. It is worth sending to referees because the claim is concrete, the assumptions are clear, and the payoff is a well-defined entropy production that can be computed from observables.

Referee Report

0 major / 2 minor

Summary. The manuscript develops the thermodynamics of non-Markovian generalized Langevin equations by embedding them in a high-dimensional Markovian representation involving auxiliary degrees of freedom. Under the condition of linear memory that satisfies detailed balance with the noise, an explicit construction is provided for systems with many degrees of freedom and hydrodynamic interactions. The embedding is generally not unique, but the thermodynamic quantities of the Markovian system are shown to be unique. This allows defining the Markovian entropy production of a non-Markovian system that increases monotonically with time, and attributes the apparent decrease in non-Markovian entropy to heat and information exchange with the auxiliary degrees of freedom.

Significance. If the results hold, this work offers a valuable framework for assigning consistent thermodynamic properties, especially entropy production, to non-Markovian systems by reducing them to Markovian ones where standard results apply. The explicit construction for multi-degree-of-freedom systems including hydrodynamic interactions is a notable strength, as it applies to physically relevant cases like interacting particles in fluids. The demonstration of uniqueness of thermodynamic quantities across different embeddings is particularly useful, enabling unambiguous definitions. The paper is credited for providing the explicit construction and the uniqueness result under clearly stated conditions, which supports the central claims.

minor comments (2)

The introduction would benefit from a short paragraph contrasting the present embedding with prior Markovian representations of non-Markovian dynamics to clarify the advance.
Equation numbering for the memory kernel and its fluctuation-dissipation relation should be introduced early and used consistently in later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. The referee summary accurately reflects the scope and main results of our work. We note that no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper supplies an explicit embedding construction for non-Markovian generalized Langevin equations into a high-dimensional Markovian system, conditioned on linear memory that satisfies detailed balance with the noise. It then proves that thermodynamic quantities (including entropy production) remain invariant across the family of possible embeddings. The monotonicity of the resulting Markovian entropy production follows directly from standard Markovian thermodynamics once the embedding is established. No step reduces by definition to its inputs, no fitted parameter is relabeled as a prediction, and no load-bearing self-citation or uniqueness theorem imported from the authors' prior work is invoked. The derivation chain is therefore independent of the target result and rests on verifiable construction plus existing Markovian theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption of linear memory obeying detailed balance; auxiliary degrees of freedom are introduced without independent evidence outside the construction itself.

axioms (1)

domain assumption The memory is linear and satisfies detailed balance with the noise
Required for the explicit embedding construction and uniqueness of thermodynamic quantities.

invented entities (1)

auxiliary degrees of freedom no independent evidence
purpose: To embed the non-Markovian dynamics into a Markovian representation
Postulated to carry memory effects; no independent falsifiable prediction outside the embedding is given in the abstract.

pith-pipeline@v0.9.0 · 5440 in / 1216 out tokens · 56014 ms · 2026-05-07T14:53:59.792300+00:00 · methodology

discussion (0)

Reference graph

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