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arxiv: 2603.12184 · v2 · submitted 2026-03-12 · ❄️ cond-mat.stat-mech · physics.bio-ph

Non-Markovian Entropy Dynamics in Living Systems from the Keldysh Formalism

Feiyi Liu , Min Guo , Hongwei Tan , Yang Wang This is my paper

Pith reviewed 2026-05-15 11:39 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-ph
keywords non-Markovian entropy dynamicsKeldysh formalismentropy production ratefluctuation-dissipation relationactive biological fluctuationscolored noisestochastic thermodynamicsliving systems
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The pith

The Keldysh formalism yields an exact frequency-domain expression for entropy production in non-Markovian living systems.

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

This paper builds a non-Markovian framework for entropy dynamics in living systems by extending the Keldysh functional formalism and stochastic thermodynamics. The approach incorporates colored environmental noise, memory-dependent dissipation, and many-body interactions to produce generalized Langevin dynamics and non-Markovian master equations. Its central result is an exact frequency-domain formula for the entropy production rate, with violations of the fluctuation-dissipation relation acting as a direct thermodynamic marker of active biological fluctuations. Environmental memory is shown to amplify low-frequency fluctuations and entropy production, producing critical slowing down near dynamical instabilities. The work supplies a microscopic physical basis for the entropy bathtub picture of living systems and ties entropy evolution to development, aging, and death.

Core claim

The authors establish a non-Markovian framework for entropy dynamics in living systems by extending the Keldysh formalism and stochastic thermodynamics. This yields generalized Langevin equations with memory and colored noise, and non-Markovian master equations. They obtain an exact frequency-domain expression for the entropy production rate, demonstrating that violations of the fluctuation-dissipation relation serve as a thermodynamic indicator of active fluctuations in biological systems. Furthermore, memory effects in the environment amplify low-frequency fluctuations and entropy production, resulting in critical slowing down near dynamical instabilities.

What carries the argument

The Keldysh functional formalism extended via stochastic thermodynamics, which generates the frequency-domain entropy production rate from non-Markovian dynamics with colored noise and memory-dependent dissipation.

If this is right

  • Environmental memory enhances low-frequency fluctuations and entropy production.
  • Critical slowing down appears near dynamical instabilities.
  • The framework supplies a microscopic foundation for the entropy bathtub picture of living systems.
  • Entropy evolution links to development, aging, and death in nonequilibrium dynamics.

Where Pith is reading between the lines

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

  • The frequency-domain approach could be used to model memory effects in specific processes such as gene expression or cellular signaling.
  • It may allow tests of aging models by predicting entropy production changes from measured fluctuation spectra.
  • The same machinery might extend to other memory-bearing nonequilibrium systems like active matter or ecological networks.

Load-bearing premise

The Keldysh functional formalism and stochastic thermodynamics can be applied directly to living systems with memory-dependent dissipation, colored noise, and many-body interactions without extra biological constraints.

What would settle it

A measurement of entropy production spectra in a living cell or tissue that deviates from the derived frequency-domain expression while active fluctuations and memory effects are present.

Figures

Figures reproduced from arXiv: 2603.12184 by Feiyi Liu, Hongwei Tan, Min Guo, Yang Wang.

Figure 1
Figure 1. Figure 1: Time evolution of system entropy Ssys(t) for different memory times τc = 0 (blue), 0.5 (brown), 1 (green), and 2 (red). Gray bands indicate fluctuation ranges for τc = 0 and τc = 2. The vertical dotted lines mark the boundaries between life stages (development, maturity, aging, death). A reversible perturbation (disease event) is shown for τc = 1 and 2, illustrating non-Markovian recovery. The horizontal d… view at source ↗
Figure 2
Figure 2. Figure 2: Frequency-dependent fluctuation-dissipation ratio [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time evolution of the damage variable D(t) from Eq. (49) with a power-law memory kernel (θ = 0.5). Below the critical point (µ = 0.8, blue), damage decays to zero. At the critical point (µ = 1.0, green), damage grows extremely slowly, exhibiting critical slowing down. Above the critical point (µ = 1.1, purple; µ = 1.2, red), damage accelerates and eventually saturates at a finite steady state. The gray das… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of system entropy Ssys(t) (blue) and system–environment mutual information IS:E(t) (red) during the lifespan. The vertical dotted lines separate the devel￾opmental, maturity, aging, and death phases. During development, Ssys decreases while IS:E increases, reflecting the buildup of structured correlations with the environment. In the mature phase both quantities stabilize, while during aging the… view at source ↗
read the original abstract

Living systems are open nonequilibrium systems that continuously exchange energy, matter, and information with their environments, leading to stochastic dynamics with memory and active fluctuations. In this study, we develop a non-Markovian theoretical framework for the entropy dynamics of living systems based on the Keldysh functional formalism and stochastic thermodynamics. The approach naturally incorporates colored environmental noise, memory-dependent dissipation, and many-body interactions, yielding generalized Langevin dynamics and non-Markovian master equations. Within this framework we derive an exact frequency-domain expression for the entropy production rate and show that violations of the fluctuation-dissipation relation provide a direct thermodynamic signature of active biological fluctuations. We further demonstrate that environmental memory enhances low-frequency fluctuations and entropy production, leading to critical slowing down near dynamical instability. These results provide a microscopic physical foundation for the entropy "bathtub" picture of living systems and connect entropy evolution with development, aging, and death in nonequilibrium dynamics.

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

1 major / 1 minor

Summary. The paper develops a non-Markovian theoretical framework for entropy dynamics in living systems by combining the Keldysh functional formalism with stochastic thermodynamics. It incorporates colored environmental noise and memory-dependent dissipation to obtain generalized Langevin dynamics and non-Markovian master equations. An exact frequency-domain expression for the entropy production rate is derived, with violations of the fluctuation-dissipation relation identified as a thermodynamic signature of active biological fluctuations. Memory effects are shown to enhance low-frequency fluctuations and entropy production, producing critical slowing down near dynamical instability, thereby providing a microscopic foundation for the entropy 'bathtub' picture and linking entropy evolution to development, aging, and death.

Significance. If the central derivations hold, the work supplies a field-theoretic route to non-Markovian stochastic thermodynamics that cleanly separates equilibrium memory effects from active drive. The frequency-domain entropy-production formula and the explicit demonstration that memory kernels alone do not generate net entropy production would constitute a useful technical advance for the field, with direct implications for interpreting fluctuation spectra in biological systems.

major comments (1)
  1. [Derivation of entropy production rate (frequency-domain expression)] In the derivation of the frequency-domain entropy production rate, the manuscript must explicitly verify that the residual entropy production vanishes identically when the active drive is switched off, using the memory-adjusted (frequency-dependent) FDR appropriate to the generalized Langevin equation. If the derivation instead invokes the standard Markovian FDR form without re-deriving the contour-integral identities for the colored-noise correlators, the claimed isolation of active fluctuations from memory-induced FDR modifications is not guaranteed.
minor comments (1)
  1. The connection between the derived entropy-production formula and the 'entropy bathtub' picture should be made explicit, with a short paragraph or figure showing how the low-frequency enhancement maps onto the bathtub shape.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the presentation of our non-Markovian framework. We address the single major comment below and will revise the manuscript to incorporate the requested explicit verification.

read point-by-point responses

  1. Referee: In the derivation of the frequency-domain entropy production rate, the manuscript must explicitly verify that the residual entropy production vanishes identically when the active drive is switched off, using the memory-adjusted (frequency-dependent) FDR appropriate to the generalized Langevin equation. If the derivation instead invokes the standard Markovian FDR form without re-deriving the contour-integral identities for the colored-noise correlators, the claimed isolation of active fluctuations from memory-induced FDR modifications is not guaranteed.

    Authors: We appreciate the referee's emphasis on rigorously confirming the equilibrium limit. Our derivation proceeds from the Keldysh contour integrals over the full non-Markovian response and correlation functions of the generalized Langevin equation; these functions are constructed to obey the memory-adjusted, frequency-dependent FDR in the absence of active drive. Consequently, the entropy-production integrand cancels identically when the active term is set to zero. Nevertheless, to make this cancellation fully transparent, we will add a dedicated paragraph (and accompanying appendix identities) that explicitly re-derives the contour-integral relations for the colored-noise correlators and demonstrates the vanishing residual entropy production in the passive limit. This addition will strengthen the separation between memory-induced FDR modifications and active fluctuations without altering the central results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives an exact frequency-domain expression for the entropy production rate from the Keldysh functional formalism applied to generalized Langevin dynamics and non-Markovian master equations. No equations or steps are shown that reduce the claimed result to a fitted parameter, a self-citation load-bearing premise, or an ansatz smuggled via prior work by the same authors. The link between FDR violations and active fluctuations is presented as following directly from the contour integration and noise correlators within the formalism, without evidence that the residual entropy production is forced to vanish only by re-using the input assumptions. The framework is self-contained against external benchmarks of stochastic thermodynamics and Keldysh techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard assumptions from stochastic thermodynamics and the applicability of Keldysh methods to classical biological noise; no new free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Keldysh functional formalism applies to classical stochastic dynamics with memory in living systems
    Used to derive generalized Langevin dynamics and non-Markovian master equations from colored noise and memory-dependent dissipation.
  • domain assumption Violations of the fluctuation-dissipation relation directly indicate active biological fluctuations
    Invoked to interpret the thermodynamic signature of activity.

pith-pipeline@v0.9.0 · 5466 in / 1298 out tokens · 39412 ms · 2026-05-15T11:39:53.075815+00:00 · methodology

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