Markovian thermodynamics of non-Markovian Langevin equations

Andreas Dechant; Kiyoshi Kanazawa

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

Markovian thermodynamics of non-Markovian Langevin equations

Andreas Dechant , Kiyoshi Kanazawa This is my paper

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

0 comments

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

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

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W. Deng and E. Barkai, Ergodic properties of frac- tional Brownian-Langevin motion, Physical Review E79, 011112 (2009)

work page 2009
[33]

Siegle, I

P. Siegle, I. Goychuk, and P. H¨ anggi, Markovian embed- ding of fractional superdiffusion, Europhysics Letters93, 20002 (2011)

work page 2011
[34]

Brandner, Dynamics of Microscale and Nanoscale Sys- tems in the Weak-Memory Regime, Physical Review Let- ters134, 037101 (2025)

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B´ erut, A

A. B´ erut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verifica- tion of Landauer’s principle linking information and ther- modynamics, Nature483, 187 (2012)

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K. Proesmans, J. Ehrich, and J. Bechhoefer, Finite-Time Landauer Principle, Physical Review Letters125, 100602 (2020)

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S. Ito, Y. Maekawa, R. Nagayama, A. Dechant, and K. Yoshimura, Geometric decomposition of informa- tion flow for overdamped Langevin systems and optimal transport in subsystems (2025), arXiv:2512.22890 [cond- mat]

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[40]

D.-K. Kim, Y. Bae, S. Lee, and H. Jeong, Learning En- tropy Production via Neural Networks, Physical Review Letters125, 140604 (2020)

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Otsubo, S

S. Otsubo, S. Ito, A. Dechant, and T. Sagawa, Esti- mating entropy production by machine learning of short- time fluctuating currents, Physical Review E101, 062106 (2020). 8 END MA TTER Appendix A: Derivation of the generalized Langevin equation.We note that the form of the memory ker- nel Eq. (25a) implies that the noises acting on the target and auxilia...

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[1] [1]

S. C. Weber, A. J. Spakowitz, and J. A. Theriot, Bac- terial Chromosomal Loci Move Subdiffusively through a Viscoelastic Cytoplasm, Physical Review Letters104, 238102 (2010)

work page 2010

[2] [2]

Weiss, Single-particle tracking data reveal anticorre- lated fractional Brownian motion in crowded fluids, Phys- ical Review E88, 010101 (2013)

M. Weiss, Single-particle tracking data reveal anticorre- lated fractional Brownian motion in crowded fluids, Phys- ical Review E88, 010101 (2013)

work page 2013

[3] [3]

Breuer, E.-M

H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Colloquium: Non-Markovian dynamics in open quantum systems, Reviews of Modern Physics88, 021002 (2016)

work page 2016

[4] [4]

de Vega and D

I. de Vega and D. Alonso, Dynamics of non-Markovian open quantum systems, Reviews of Modern Physics89, 015001 (2017)

work page 2017

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Zwanzig, Memory Effects in Irreversible Thermody- namics, Physical Review124, 983 (1961)

R. Zwanzig, Memory Effects in Irreversible Thermody- namics, Physical Review124, 983 (1961)

work page 1961

[6] [6]

Mori, Transport, Collective Motion, and Brownian Motion*), Progress of Theoretical Physics33, 423 (1965)

H. Mori, Transport, Collective Motion, and Brownian Motion*), Progress of Theoretical Physics33, 423 (1965)

work page 1965

[7] [7]

Cockrell and I

C. Cockrell and I. J. Ford, Stochastic thermodynamics in a non-Markovian dynamical system, Physical Review E 105, 064124 (2022)

work page 2022

[8] [8]

Strasberg and M

P. Strasberg and M. Esposito, Non-Markovianity and negative entropy production rates, Physical Review E99, 012120 (2019)

work page 2019

[9] [9]

Siegle, I

P. Siegle, I. Goychuk, P. Talkner, and P. H¨ anggi, Marko- vian embedding of non-markovian superdiffusion, Physi- cal Review E81, 011136 (2010)

work page 2010

[10] [10]

Goychuk, Viscoelastic Subdiffusion: Generalized Langevin Equation Approach, inAdvances in Chemical Physics(John Wiley & Sons, Ltd, 2012) pp

I. Goychuk, Viscoelastic Subdiffusion: Generalized Langevin Equation Approach, inAdvances in Chemical Physics(John Wiley & Sons, Ltd, 2012) pp. 187–253

work page 2012

[11] [11]

L. Ma, X. Li, and C. Liu, From generalized Langevin equations to Brownian dynamics and embedded Brow- nian dynamics, The Journal of Chemical Physics145, 114102 (2016)

work page 2016

[12] [12]

Kanazawa and D

K. Kanazawa and D. Sornette, Nonuniversal power law distribution of intensities of the self-excited Hawkes pro- cess: a field-theoretical approach, Physical Review Let- ters125, 138301 (2020)

work page 2020

[13] [13]

Kanazawa and D

K. Kanazawa and D. Sornette, Standard form of mas- ter equations for general non-Markovian jump processes: The Laplace-space embedding framework and asymptotic solution, Physical Review Research6, 023270 (2024)

work page 2024

[14] [14]

Stochastic thermodynamics for classical non-Markov jump processes

K. Kanazawa and A. Dechant, Stochastic ther- modynamics for classical non-Markov jump pro- cesses, https://arxiv.org/abs/2506.04726 (2025), arXiv:2506.04726

work page internal anchor Pith review Pith/arXiv arXiv 2025

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Zwanzig, Nonlinear generalized Langevin equations, Journal of Statistical Physics9, 215 (1973)

R. Zwanzig, Nonlinear generalized Langevin equations, Journal of Statistical Physics9, 215 (1973)

work page 1973

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R. F. Fox, The generalized Langevin equation with Gaus- sian fluctuations, Journal of Mathematical Physics18, 2331 (1977)

work page 1977

[17] [17]

Cort´ es, B

E. Cort´ es, B. J. West, and K. Lindenberg, On the general- ized Langevin equation: Classical and quantum mechan- ical, The Journal of Chemical Physics82, 2708 (1985)

work page 1985

[18] [18]

Di Terlizzi, F

I. Di Terlizzi, F. Ritort, and M. Baiesi, Explicit Solution of the Generalised Langevin Equation, Journal of Statis- tical Physics181, 1609 (2020)

work page 2020

[19] [19]

Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics29, 255 (1966)

R. Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics29, 255 (1966)

work page 1966

[20] [20]

Fodor, D

E. Fodor, D. S. Grebenkov, P. Visco, and F. van Wij- land, Generalized Langevin equation with hydrodynamic backflow: Equilibrium properties, Physica A: Statistical Mechanics and its Applications422, 107 (2015)

work page 2015

[21] [21]

Kanazawa and D

K. Kanazawa and D. Sornette, Field master equation the- ory of the self-excited Hawkes process, Physical Review Research2, 033442 (2020)

work page 2020

[22] [22]

Sekimoto,Stochastic Energetics, Lecture Notes in Physics (Springer Berlin Heidelberg, 2010)

K. Sekimoto,Stochastic Energetics, Lecture Notes in Physics (Springer Berlin Heidelberg, 2010)

work page 2010

[23] [23]

Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics75, 126001 (2012)

U. Seifert, Stochastic thermodynamics, fluctuation the- orems and molecular machines, Reports on Progress in Physics75, 126001 (2012)

work page 2012

[24] [24]

Speck and U

T. Speck and U. Seifert, The Jarzynski relation, fluctu- ation theorems, and stochastic thermodynamics for non- Markovian processes, J. Stat. Mech. Theory E.2007, L09002 (2007)

work page 2007

[25] [25]

J. M. Horowitz and M. Esposito, Thermodynamics with continuous information flow, Physical Review X4, 031015 (2014). 7

work page 2014

[26] [26]

Sagawa and M

T. Sagawa and M. Ueda, Fluctuation theorem with in- formation exchange: Role of correlations in stochastic thermodynamics, Physical Review Letters109, 180602 (2012)

work page 2012

[27] [27]

Ito and T

S. Ito and T. Sagawa, Information thermodynamics on causal networks, Physical Review Letters111, 180603 (2013)

work page 2013

[28] [28]

Hartich, A

D. Hartich, A. C. Barato, and U. Seifert, Stochastic ther- modynamics of bipartite systems: transfer entropy in- equalities and a maxwell’s demon interpretation, Journal of Statistical Mechanics: Theory and Experiment2014, P02016 (2014)

work page 2014

[29] [29]

J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Ther- modynamics of information, Nature Phys.11, 131 (2015)

work page 2015

[30] [30]

K. Funo, T. V. Vu, and K. Saito, Private communication, publication in preparation (2026)

work page 2026

[31] [31]

Lutz, Fractional Langevin equation, Physical Review E64, 051106 (2001)

E. Lutz, Fractional Langevin equation, Physical Review E64, 051106 (2001)

work page 2001

[32] [32]

Deng and E

W. Deng and E. Barkai, Ergodic properties of frac- tional Brownian-Langevin motion, Physical Review E79, 011112 (2009)

work page 2009

[33] [33]

Siegle, I

P. Siegle, I. Goychuk, and P. H¨ anggi, Markovian embed- ding of fractional superdiffusion, Europhysics Letters93, 20002 (2011)

work page 2011

[34] [34]

Brandner, Dynamics of Microscale and Nanoscale Sys- tems in the Weak-Memory Regime, Physical Review Let- ters134, 037101 (2025)

K. Brandner, Dynamics of Microscale and Nanoscale Sys- tems in the Weak-Memory Regime, Physical Review Let- ters134, 037101 (2025)

work page 2025

[35] [35]

B´ erut, A

A. B´ erut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Experimental verifica- tion of Landauer’s principle linking information and ther- modynamics, Nature483, 187 (2012)

work page 2012

[36] [36]

Proesmans, J

K. Proesmans, J. Ehrich, and J. Bechhoefer, Finite-Time Landauer Principle, Physical Review Letters125, 100602 (2020)

work page 2020

[37] [37]

S. A. Loos, S. Monter, F. Ginot, and C. Bechinger, Uni- versal Symmetry of Optimal Control at the Microscale, Physical Review X14, 021032 (2024)

work page 2024

[38] [38]

Monter, S

S. Monter, S. A. M. Loos, and C. Bechinger, Opti- mal transitions between nonequilibrium steady states, Proceedings of the National Academy of Sciences122, e2510654122 (2025)

work page 2025

[39] [39]

S. Ito, Y. Maekawa, R. Nagayama, A. Dechant, and K. Yoshimura, Geometric decomposition of informa- tion flow for overdamped Langevin systems and optimal transport in subsystems (2025), arXiv:2512.22890 [cond- mat]

work page arXiv 2025

[40] [40]

D.-K. Kim, Y. Bae, S. Lee, and H. Jeong, Learning En- tropy Production via Neural Networks, Physical Review Letters125, 140604 (2020)

work page 2020

[41] [41]

Otsubo, S

S. Otsubo, S. Ito, A. Dechant, and T. Sagawa, Esti- mating entropy production by machine learning of short- time fluctuating currents, Physical Review E101, 062106 (2020). 8 END MA TTER Appendix A: Derivation of the generalized Langevin equation.We note that the form of the memory ker- nel Eq. (25a) implies that the noises acting on the target and auxilia...

work page 2020