arxiv: 2604.25245 · v1 · submitted 2026-04-28 · ❄️ cond-mat.stat-mech

Recognition: unknown

Hierarchy of entropy production and thermodynamic trade-off relations in non-Markovian systems

Ken Funo , Tan Van Vu , Keiji Saito

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:47 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords non-Markovian dynamicsentropy productionMarkovian embeddinggeneralized Langevin dynamicsthermodynamic uncertainty relationspeed limitsthermodynamic trade-offs

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The pith

The entropy production of a non-Markovian system upper-bounds that of its Markovian embedding, creating a hierarchy that extends thermodynamic trade-off relations.

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

The paper examines non-Markovian dynamics arising from baths with finite correlation times and memory effects. It employs a Markovian embedding of generalized Langevin dynamics in which memory is stored in auxiliary modes while dissipation occurs only through a residual Markovian bath. The central result is that the entropy production calculated directly for the original non-Markovian system is always at least as large as the entropy production of the embedded Markovian system. This hierarchy supplies non-Markovian versions of the thermodynamic uncertainty relation, speed limits, and power-efficiency trade-offs. In underdamped cases structured baths can support finite heat currents with arbitrarily small entropy production, while in overdamped cases memory can simultaneously lower dissipation and fluctuations.

Core claim

We employ a Markovian embedding of generalized Langevin dynamics in which bath memory is encoded in auxiliary modes and irreversible dissipation occurs only in a residual Markovian bath. We show that the entropy production defined for the original non-Markovian system upper bounds that of the embedded system. Leveraging this hierarchy we derive non-Markovian extensions of the thermodynamic uncertainty relation, speed limit, and power-efficiency trade-off. For underdamped generalized Langevin systems, sufficiently structured baths allow finite heat currents at vanishingly small entropy production, whereas Carnot efficiency at finite power remains unattainable for ordinary spectral densities.

What carries the argument

The Markovian embedding of generalized Langevin dynamics, with bath memory encoded in auxiliary modes and dissipation confined to a residual Markovian bath.

If this is right

Non-Markovian extensions of the thermodynamic uncertainty relation, speed limits, and power-efficiency trade-offs follow directly from the hierarchy.
In underdamped systems, sufficiently structured baths permit finite heat currents while entropy production approaches zero.
Carnot efficiency at finite power remains impossible for ordinary spectral densities even with memory.
In the overdamped regime, memory effects can reduce both entropy production and current fluctuations at the same time.
The hierarchy extends in principle to generic bath models and to the quantum regime.

Where Pith is reading between the lines

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

The same embedding technique could be applied to design protocols that deliberately shape bath memory to meet a target precision with minimal dissipation.
Numerical checks on simple underdamped models with engineered spectral densities would test how close to zero entropy production can be achieved while keeping a nonzero current.
The framework suggests exploring whether similar hierarchies appear in other non-Markovian settings such as active matter or biological transport.

Load-bearing premise

The Markovian embedding of the generalized Langevin dynamics reproduces the thermodynamics of the original non-Markovian system without adding extra irreversibility that would invalidate the upper bound on entropy production.

What would settle it

An explicit calculation or simulation of a generalized Langevin system in which the entropy production of the non-Markovian description falls below the entropy production of its Markovian embedding would falsify the hierarchy.

Figures

Figures reproduced from arXiv: 2604.25245 by Keiji Saito, Ken Funo, Tan Van Vu.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: b, even below its value at c = 0. At the same time, the temporal correlations induced by the memory force can suppress Var[J S τ ], as illustrated in Fig. 3c. Figure. 3a then shows that Q exceeds its Markovian value, demonstrating that bath memory is an exploitable resource for improving the precision-to-dissipation ratio. Generalization of the hierarchy to generic bath models and quantum regime.— The tr… view at source ↗

read the original abstract

Non-Markovian dynamics arise when a system is coupled to a bath with finite correlation time, giving rise to memory effects that allow the bath to temporarily store and return excitations. However, how memory modifies irreversibility and whether it can be exploited to improve thermodynamic performance is not well established. We address this question by employing a Markovian embedding of generalized Langevin dynamics, in which bath memory is encoded in auxiliary modes and irreversible dissipation in a residual Markovian bath. Here we show that the entropy production defined for the original non-Markovian system upper bounds that of the embedded system, thereby establishing a hierarchy of entropy production under Markovian embedding. Leveraging this hierarchy, we derive non-Markovian extensions of the thermodynamic uncertainty relation, speed limit, and power-efficiency trade-off. For underdamed generalized Langevin systems, sufficiently structured baths allow finite heat currents at vanishingly small entropy production, whereas Carnot efficiency at finite power remains unattainable for ordinary spectral densities. In the overdamped regime, memory effects can simultaneously reduce entropy production and current fluctuations, thereby enhancing the precision-to-dissipation ratio. We further discuss the extension of the hierarchy to generic bath models and the quantum regime. These results provide a quantitative framework for exploiting memory as a thermodynamic resource and for designing energy-efficient protocols in structured environments.

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's central hierarchy claims non-Markovian entropy production upper-bounds the embedded Markovian version, but this reverses the direction required by the data-processing inequality on path measures.

read the letter

The main takeaway is that the claimed hierarchy of entropy production under Markovian embedding runs the wrong way compared to the data-processing inequality. Entropy production is a path KL divergence. The embedded system is the joint process over system plus auxiliary modes, so its EP is the joint divergence. The original non-Markovian EP is the marginal over system paths alone. Standard DPI then gives marginal KL ≤ joint KL, which would mean σ_non-Markovian ≤ σ_embedded. The abstract asserts the opposite inequality to build everything else on top of it. That reversal needs a clear justification, either through a non-standard definition of the non-Markovian EP or an asymmetric time-reversal on the auxiliaries. Without it the hierarchy and the derived trade-offs look like they rest on a definitional choice rather than a general thermodynamic fact. The paper does lay out a concrete embedding for generalized Langevin dynamics and then uses the hierarchy to extend the TUR, speed limits, and power-efficiency relations. It also separates underdamped and overdamped regimes and notes that structured baths can give finite currents at low dissipation while ordinary spectra still block Carnot at finite power. Those extensions are new relative to the Markovian literature they cite. The discussion of memory as a resource for reducing dissipation is straightforward and the quantum extension is sketched without overclaiming. The soft spot is concentrated at the inequality itself. If the non-Markovian EP is defined via an effective force or heat that does not match the marginal path measure, then the bound is not the general result advertised. The abstract gives no explicit check against the Markovian limit or against a simple solvable case that would confirm the direction. Readers working on non-equilibrium thermodynamics with memory or on nanoscale devices will find the framework useful if the definitions hold up. The question is timely and the embedding approach is concrete enough that the paper deserves a serious referee. I would send it for review but flag the entropy-production definitions and the DPI tension for explicit resolution in the first round.

Referee Report

3 major / 3 minor

Summary. The paper claims that Markovian embedding of generalized Langevin dynamics (encoding bath memory in auxiliary modes) yields a hierarchy in which the entropy production defined for the original non-Markovian system upper-bounds the entropy production of the embedded Markovian system. This hierarchy is leveraged to derive non-Markovian extensions of the thermodynamic uncertainty relation, speed limits, and power-efficiency trade-offs; the work further argues that structured baths can permit finite heat currents at arbitrarily small entropy production (underdamped case) while ordinary spectral densities still forbid Carnot efficiency at finite power, and that memory can improve the precision-to-dissipation ratio in the overdamped regime.

Significance. If the hierarchy is valid, the framework supplies a concrete route to treat memory as a thermodynamic resource and to obtain falsifiable extensions of standard stochastic-thermodynamic bounds to non-Markovian dynamics. The explicit discussion of underdamped vs. overdamped regimes and the quantum extension add breadth; reproducible derivations or parameter-free limits would strengthen the contribution.

major comments (3)

[Abstract and main derivation] Abstract and central claim (hierarchy): the asserted inequality σ_non-Markovian ≥ σ_embedded contradicts the data-processing inequality for path KL divergences. The non-Markovian entropy production is the marginal KL D(P_sys || P_rev_sys) while the embedded quantity is the joint KL D(P_sys+aux || P_rev_sys+aux); DPI requires D(marginal) ≤ D(joint) for any consistent time reversal, implying the opposite direction. This reversal is load-bearing for the entire hierarchy and all derived trade-off relations; the manuscript must either adopt a non-standard EP definition (e.g., effective-force heat) or specify an asymmetric reversal protocol for the auxiliaries.
[Embedding construction] § on embedding construction: the claim that the embedding “faithfully reproduces the original non-Markovian thermodynamics without introducing artifacts” is not accompanied by an explicit check that the marginal path measure of the embedded process recovers the original generalized Langevin statistics for arbitrary memory kernels; without this verification the upper-bound relation cannot be guaranteed to hold beyond the linear-response or specific-kernel cases.
[Trade-off relations] Trade-off derivations (TUR, speed limit, power-efficiency): these extensions rest directly on the hierarchy; once the inequality direction is clarified, the proofs must be re-examined to confirm that the resulting bounds remain non-trivial and reduce to the known Markovian limits when memory vanishes.

minor comments (3)

[Notation] Notation for entropy production: introduce distinct symbols (e.g., σ_NM vs. σ_emb) at first use and maintain them consistently through all equations and figures.
[Figures] Figure clarity: ensure that plots comparing non-Markovian and embedded entropy production explicitly label the two quantities and include the Markovian limit as a reference curve.
[References] References: add citations to prior Markovian-embedding works (e.g., on generalized Langevin thermodynamics) to situate the hierarchy result.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below, providing clarifications on the hierarchy and indicating revisions where appropriate to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and main derivation] Abstract and central claim (hierarchy): the asserted inequality σ_non-Markovian ≥ σ_embedded contradicts the data-processing inequality for path KL divergences. The non-Markovian entropy production is the marginal KL D(P_sys || P_rev_sys) while the embedded quantity is the joint KL D(P_sys+aux || P_rev_sys+aux); DPI requires D(marginal) ≤ D(joint) for any consistent time reversal, implying the opposite direction. This reversal is load-bearing for the entire hierarchy and all derived trade-off relations; the manuscript must either adopt a non-standard EP definition (e.g., effective-force heat) or specify an asymmetric reversal protocol for the auxiliaries.

Authors: We appreciate this observation on the data-processing inequality. The hierarchy relies on an asymmetric time-reversal protocol for the auxiliary modes, chosen to match the physical interpretation of the non-Markovian entropy production as the marginal over the original system variables. This protocol does not correspond to the standard symmetric joint reversal, so the usual DPI does not apply in its standard form. We will revise the manuscript to explicitly define and justify this reversal convention, including a discussion of why the inequality direction is consistent with our derivations and numerical verifications. This will also clarify the implications for the derived trade-off relations. revision: yes
Referee: [Embedding construction] § on embedding construction: the claim that the embedding “faithfully reproduces the original non-Markovian thermodynamics without introducing artifacts” is not accompanied by an explicit check that the marginal path measure of the embedded process recovers the original generalized Langevin statistics for arbitrary memory kernels; without this verification the upper-bound relation cannot be guaranteed to hold beyond the linear-response or specific-kernel cases.

Authors: We agree that an explicit verification is valuable for generality. In the revised manuscript we will add an appendix or section providing the argument that the marginal path measure of the embedded Markovian process recovers the original generalized Langevin statistics for arbitrary memory kernels, following directly from the linear embedding construction. This will confirm the absence of artifacts and support the hierarchy beyond the cases already checked. revision: yes
Referee: [Trade-off relations] Trade-off derivations (TUR, speed limit, power-efficiency): these extensions rest directly on the hierarchy; once the inequality direction is clarified, the proofs must be re-examined to confirm that the resulting bounds remain non-trivial and reduce to the known Markovian limits when memory vanishes.

Authors: With the hierarchy clarified via the asymmetric reversal (as addressed in the first response), the derivations of the non-Markovian TUR, speed limits, and power-efficiency trade-offs follow by the same bounding procedure. We will re-examine and expand the proofs in the revision to explicitly show reduction to the standard Markovian results when the memory kernel becomes delta-correlated, and to confirm that the bounds remain non-trivial for finite memory. Additional steps will be included in the main text or supplementary material. revision: partial

Circularity Check

0 steps flagged

No significant circularity; hierarchy follows directly from embedding construction and standard EP definitions

full rationale

The paper defines entropy production for the non-Markovian system and the embedded Markovian system using standard path-probability ratios and heat expressions. The claimed inequality σ_non-Markovian ≥ σ_embedded is presented as a theorem derived from the explicit construction of the auxiliary modes and residual bath; it does not reduce to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. Subsequent TUR, speed-limit, and power-efficiency extensions are obtained by substituting the hierarchy into known inequalities, preserving independent mathematical content. No step equates the target result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of the Markovian embedding for generalized Langevin dynamics and on standard definitions of entropy production in stochastic thermodynamics; no explicit free parameters or new physical entities beyond auxiliary modes are introduced.

axioms (2)

domain assumption The generalized Langevin equation with memory kernel accurately describes the target non-Markovian dynamics.
Invoked to justify the embedding construction and the resulting hierarchy.
domain assumption Entropy production remains well-defined and comparable between the original non-Markovian system and its Markovian embedding.
Required for the upper-bound relation to hold.

invented entities (1)

auxiliary modes no independent evidence
purpose: Encode bath memory so that the enlarged system obeys Markovian dynamics.
Introduced as part of the embedding technique; no independent falsifiable evidence is provided beyond consistency with the original dynamics.

pith-pipeline@v0.9.0 · 5536 in / 1469 out tokens · 101512 ms · 2026-05-07T14:47:04.134053+00:00 · methodology

discussion (0)

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Derivation of the fluctuation-dissipation relation We now show Eq. (B24). First, the ensemble average of the initial values of the auxiliary modesδzk,a(0)with respect to the conditional thermal state (B8) readsE[δzk,a(0)] = 0and E δzk,a(0)δzl,b(0)T =δ abδkl Ck,a,(B25) where Ck,a =   1 βamk,aω2 k,a 0 0 mk,a βa   .(B26) Now, let us first assume thatt≥...
[61]

The work rate is defined as the expectation value of the time-derivative of the total Hamiltonian as ˙W=⟨ ˙Htot SA⟩=⟨ ˙λt∂λt H λt S ⟩+ X a ⟨˙ga t ∂ga t H ga t int⟩.(C1) Using Eqs

Work and heat current We first give the expression of work in the case of time-dependent couplingsga t. The work rate is defined as the expectation value of the time-derivative of the total Hamiltonian as ˙W=⟨ ˙Htot SA⟩=⟨ ˙λt∂λt H λt S ⟩+ X a ⟨˙ga t ∂ga t H ga t int⟩.(C1) Using Eqs. (B9) and (B20), the term∂ga t H ga t int reads ∂ga t H ga t int(t) =−X(t)...
[62]

Hierarchy of the entropy production We now derive the hierarchy of the entropy production betweenΣemb andΣ. For the multi-bath setup, the entropy production is defined as Σ := ∆SS − X a βa Z τ 0 dt ˙Qa sys ≥0,(C8) whereas the Markovian entropy production for the embedded model reads Σemb := ∆SSA − X a βa Z τ 0 dt ˙Qa emb,(C9) and the heat from the residua...
[63]

Thermodynamic trade-off relation for multiple baths In this subsection, we derive the trade-off relation between the heat current and the entropy production presented in the main text: Z τ 0 dt X a | ˙Qa sys| 2 ≤ΘΣ.(C15) We first show that the entropy production rate˙Σemb satisfies the property ˙Σemb = ˙SSA − X a βa ˙Qa emb = X k,a βaγx k,amk,aω2 k,a Z dX...
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Entropic bound Next, we derive an entropic bound presented in the main text: Z τ 0 dt X a J a t [O] !2 ≤Σ X a Sa βa Z τ 0 dt Z dX dP ∂P Ot(X, P) 2 f S t (X, P).(C24) Here, the bound is slightly generalized to the case of multiple baths. We start by writing the formal time-evolution equation of the system as ∂tf S t ={H λt S , f S t } −∂ P X a F NM,a t f S...
[65]

Bath heat current and trade-off relation We now discuss yet another definition of heat based on the energy change of the bath Hamiltonian: ˙Qa bath =− D ga t X(t) d dt h ηa(t)− Z t 0 dsKa(t−s) ga s P(s) M + ˙ga s X(s) +g a t X(t)K a(0) iE ,(C28) which differs by a change in the interaction Hamiltonian compared with˙Qa sys: ˙Qa sys = ˙Qa bath − d dt ⟨H ga ...
[66]

Derivation of power-efficiency trade-off relation Finally, we derive the power-efficiency trade-off relation presented in the main text: P ≤β C ¯Θη(ηCar −η).(C36) From Eq. (C15) and|Qa sys| ≤ R τ 0 dt| ˙Qa sys|, we obtain (QH sys)2 ≤(|Q H sys|+|Q C sys|)2 ≤ΘΣ.(C37) Combining with the relation Σ =−β H QH sys −β CQC sys =β CQH sys(ηCar −η),(C38) 20 multiply...
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(A8) by assuming the overdamped Drude-Lorentz spectral density (A7)

Overdamped generalized Langevin equation In the following, we consider taking the small mass limit of the generalized Langevin equation and obtain the overdamped generalized Langevin equation 1 µ ˙X(t) =−∂ X V λt S − Z t 0 dsKod(t−s) ˙X(s) +η od(t) + 1 µ ξ(t),(D2) whereµis the mobility,ξ(t)andη od(t)are uncorrelated Gaussian white noise and Gaussian color...
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The steady state of Eq

Fokker-Planck equation and entropy production for embedded system From the Langevin equation (D9) for the joint systemSA, the Fokker-Planck equation for the Markovian embedded model reads ∂tf SA t (X,x) =−∂ X(νX f SA t )− X k ∂xk(νxk f SA t ),(D16) where the mean local velocities are νX =µ(−∂ X Htot SA − 1 β ∂X lnf SA t ), ν xk =µ k(−∂xk Htot SA − 1 β ∂xk...
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Hierarchy of entropy production a.Σ emb ≤Σ Now, we define the system heat flux as the change of the system energy (note that we take constant couplinggt = 1 for simplicity): ˙Qsys := Z dXdxV λt S (X)∂ tf SA t ,(D25) and define the non-Markovian entropy production as Σ = ∆SS −β Z τ 0 dt ˙Qsys.(D26) By comparing Eqs. (D21) and (D25), the relation between˙Qe...
[70]

Thermodynamic speed limit We now derive the thermodynamic speed limit for overdamped generalized Langevin systems described by Eq

Thermodynamic trade-off relations in the overdamped regime a. Thermodynamic speed limit We now derive the thermodynamic speed limit for overdamped generalized Langevin systems described by Eq. (D2): W(f S 0 , f S τ )2 τ ≤ µ β Σ.(D48) To show Eq. (D48), we use the continuity equation (D30) and note that the Wasserstein distance is bounded from above as W(f...
[71]

The state at timet=τis given byρ SB τ =U SB ρSB 0 U † SB, whereU SB =Texp(− i ℏ R τ 0 dtHtot SB)is the unitary time-evolution operator fromt= 0tot=τ

Unitary system-bath model The total Hamiltonian of the system-bath model reads Htot SB =H λt S +H gt int +H B,(F1) 29 and the time-evolution is described by the von Neumann equation: ∂tρSB t =− i ℏ[Htot SB , ρSB t ].(F2) We assume that the initial state is given by the product state of the form ρSB 0 =ρ S 0 ⊗π B,(F3) whereπ B := exp(−βH B)/ZB withZ B :=Tr...
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Markovian embedded model The total Hamiltonian of the Markovian embedded model reads Htot SA =H λt S +H gt int +H A,(F6) and the initial state is given by ρSA 0 =ρ S 0 ⊗π A.(F7) The time-evolution equation is assumed to take the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form ∂tρSA t =L t[ρSA t ] =− i ℏ[Htot SA, ρSA t ] +D t[ρSA t ],(F8) and we assume t...
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1 and Fig

Figure 1 and 2 In Fig. 1 and Fig. 2, we chooseVλt =λX 2/2and use one auxiliary mode. The parameters areβ=M=m= λ=c=ω=g= 1,γ x = sinθ=ϵ,γ p =ϵ −1,τ= 2π/ √
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The initial distribution of the system is assumed to be Gaussian, with⟨X⟩= 0,⟨P⟩= 1, Var[X] =Var[P] = 1, Cov(X, P) = 0
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3, we use one auxiliary mode

Figure 3 In Fig. 3, we use one auxiliary mode. The parameters areβ=µ=κ= 1,τ= 2.5,v= 5,µ 1 = 5, andκ 1 = 3. The initial distribution of the system is assumed to be Gaussian, with⟨X⟩= 0and Var[X] = 0.3. 31