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arxiv: 2412.08432 · v2 · submitted 2024-12-11 · ❄️ cond-mat.stat-mech · cs.IT· math.IT

Generalized free energy and excess/housekeeping decomposition in nonequilibrium systems: from large deviations to thermodynamic speed limits

Artemy Kolchinsky , Andreas Dechant , Kohei Yoshimura , Sosuke Ito This is my paper

Pith reviewed 2026-05-23 07:31 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.ITmath.IT
keywords nonequilibrium thermodynamicslarge deviationsexcess entropy productionhousekeeping dissipationthermodynamic speed limitsmetabolic networksgeneralized free energystochastic master equations
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The pith

A generalized free energy from large deviations governs nonequilibrium dynamics and decomposes dissipation into excess and housekeeping parts.

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

In genuine nonequilibrium systems with continuous driving, nonconservative forces mean no standard free energy potential exists. The paper establishes that a generalized free energy nonetheless governs the dynamics, obtained from a large-deviations variational principle applied to the governing equations. This principle produces a decomposition of fluxes, forces, and entropy production into a conservative excess component and a nonconservative housekeeping component. The excess entropy production obeys a thermodynamic speed limit that constrains the rate of state evolution or external fluxes. The results hold for stochastic master equations, deterministic chemical reaction networks, and open systems, and yield dissipation bounds when applied to metabolic networks.

Core claim

The dynamics of genuine nonequilibrium systems that undergo continuous driving are governed by a generalized free energy derived from a large-deviations variational principle. This variational principle also yields a decomposition of fluxes, forces, and dissipation (entropy production) into a conservative excess part and a nonconservative housekeeping part. The excess entropy production obeys a thermodynamic speed limit, a fundamental thermodynamic constraint on the rate of state evolution and/or external fluxes. The approach is universally applicable to stochastic master equations, deterministic chemical reaction networks, and open systems, and is empirically accessible for thermodynamic 2n

What carries the argument

Large-deviations variational principle that defines the generalized free energy and produces the excess/housekeeping decomposition of entropy production.

If this is right

  • The excess entropy production obeys a thermodynamic speed limit that bounds the rate of state evolution and external fluxes.
  • The decomposition applies universally to stochastic master equations, deterministic chemical reaction networks, and open systems.
  • Fundamental dissipation bounds follow for real-world metabolic networks and can identify futile cycles.
  • The generalized free energy and decomposition are empirically accessible for thermodynamic inference in stochastic and deterministic systems.
  • The framework connects to information geometry and Onsager theory as well as prior excess/housekeeping decompositions.

Where Pith is reading between the lines

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

  • The speed limit could be tested in laboratory-driven systems to quantify how closely real processes approach the bound.
  • The decomposition might supply new observables for inferring hidden driving forces in experimental data.
  • Extensions to quantum or spatially extended systems would require checking whether the same variational principle still closes.
  • The approach could unify earlier housekeeping/excess splits by showing they arise from one large-deviations principle.

Load-bearing premise

The large-deviations variational principle can be applied to the stochastic master equations, deterministic chemical reaction networks, and open systems under consideration to define the generalized free energy and produce the excess/housekeeping split.

What would settle it

A measurement or simulation on a continuously driven system in which the observed excess entropy production rate violates the bound given by the thermodynamic speed limit would falsify the central claim.

Figures

Figures reproduced from arXiv: 2412.08432 by Andreas Dechant, Artemy Kolchinsky, Kohei Yoshimura, Sosuke Ito.

Figure 1
Figure 1. Figure 1: Formalism illustrated on a simple MJP, a two-level system coupled to a pair of heat baths. Transitions between the two levels occur with rates R c 21 and R c 12 when exchanging energy with the cold bath (at inverse temperature βc), and with rates R h 21 and R h 12 when exchanging energy with the hot bath (at inverse temperature βh). This gives four one-way transitions, here characterized by the incidence m… view at source ↗
Figure 3
Figure 3. Figure 3: Information-geometric interpretation of ex￾cess/housekeeping decomposition. (a) The EPR σ = D(j∥ ˜j) is the relative entropy from the forward fluxes j to the reverse fluxes ˜j. Excess EPR σex is defined by the projection of ˜j onto the set of fluxes that give the correct time evolution (orange line), Eq. (46). Excess/housekeeping provide an orthogonal decomposition of the EPR in flux space, Eq. (64). (b) I… view at source ↗
Figure 4
Figure 4. Figure 4: Large deviations interpretation of the irreversibility measure L(ϕ). The change of state observable ϕ is measured in n independent stochastic systems over time [t, t + dt]. The empirical mean change is captured by the random variable ∆ϕn. Here we show schematically the probability of different outcomes ∆ϕn = z ∈ R. For large n, the probability distribution is peaked at z = E[∆ϕ]. The probability that the e… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of results on unicyclic MJP with d = 21 states. (a): Time evolution of the probability distribution for three driving strengths (γ = 0, 1, 4). The initial distribution is concentrated on three initial microstates i ∈ {10, 11, 12}. (b) Generalized free energy ϕ ⋆ and steady-state potential ϕ ss for different driving strengths, evaluated at t = 0. (c) Wasserstein speed at t = 0 increases with dr… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of our approach on the Brusselator, a nonlinear CRN previously shown [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left: a model of a discrete system with odd variables, consisting a particle on a ring with position r ∈ {1, . . . , k} and odd velocity v ∈ {−1, +1} [67–69]. Right: overall EPR σ, our excess EPR σex, and HS excess EPR σ HS ex for this model. η ̸= 0 means forces are nonconservative, γ ̸= 0 means steady-state distribution is not symmetric under conjugation of odd variables. HS decomposition can give unphysi… view at source ↗
read the original abstract

In genuine nonequilibrium systems that undergo continuous driving, the thermodynamic forces are nonconservative, meaning they cannot be described by any free energy potential. Nonetheless, we show that the dynamics of such systems are governed by a "generalized free energy" that is derived from a large-deviations variational principle. This variational principle also yields a decomposition of fluxes, forces, and dissipation (entropy production) into a conservative "excess" part and a nonconservative "housekeeping" part. Our decomposition is universally applicable to stochastic master equations, deterministic chemical reaction networks, and open systems. We also show that the excess entropy production obeys a thermodynamic speed limit (TSL), a fundamental thermodynamic constraint on the rate of state evolution and/or external fluxes. We demonstrate our approach on several examples, including real-world metabolic networks, where we derive fundamental dissipation bounds and uncover "futile" metabolic cycles. Our generalized free energy and decomposition are empirically accessible to thermodynamic inference in both stochastic and deterministic systems. We discuss important connections to several theoretical frameworks, including information geometry and Onsager theory, as well as previous excess/housekeeping decompositions.

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 manuscript claims that a large-deviations variational principle yields a generalized free energy governing the dynamics of continuously driven nonequilibrium systems despite nonconservative forces. This principle induces a universal decomposition of fluxes, forces, and entropy production into conservative excess and nonconservative housekeeping components, applicable to stochastic master equations, deterministic chemical reaction networks, and open systems. The excess entropy production obeys a thermodynamic speed limit. The framework is illustrated on examples including metabolic networks, shown to be empirically accessible for inference, and connected to information geometry, Onsager theory, and prior decompositions.

Significance. If the variational construction is rigorously justified and the decomposition remains well-defined, the result would offer a unifying variational foundation for nonequilibrium thermodynamics in driven systems, delivering new speed limits, dissipation bounds, and inference tools with direct applicability to biological networks.

major comments (1)
  1. [Abstract and sections on deterministic CRNs/open systems] The central claim that the large-deviations variational principle applies to deterministic chemical reaction networks and continuously driven open systems (including those with nonconservative cycles) is load-bearing, yet the abstract provides no explicit rate-function construction or large-volume limit argument showing that the resulting functional acts as a potential for the macroscopic flow when driving is time-dependent.
minor comments (1)
  1. [Abstract] The abstract states that the approach is demonstrated on real-world metabolic networks but does not specify which networks, what observables were inferred, or how the excess/housekeeping split was validated against data.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment concerns the presentation of the large-deviations construction for deterministic and open systems. We address it directly below and will revise the manuscript to improve clarity.

read point-by-point responses

  1. Referee: [Abstract and sections on deterministic CRNs/open systems] The central claim that the large-deviations variational principle applies to deterministic chemical reaction networks and continuously driven open systems (including those with nonconservative cycles) is load-bearing, yet the abstract provides no explicit rate-function construction or large-volume limit argument showing that the resulting functional acts as a potential for the macroscopic flow when driving is time-dependent.

    Authors: The explicit rate-function construction, large-volume limit, and resulting variational principle are derived in the main text for stochastic master equations and then extended to the deterministic limit of chemical reaction networks (including open systems with nonconservative cycles). For time-dependent driving the generalized free energy is obtained from the instantaneous large-deviations rate functional, which acts as a potential for the macroscopic flow at each instant; this is shown to induce the excess/housekeeping decomposition and the thermodynamic speed limit. We agree that the abstract is too concise on these technical points and will revise it to include a brief statement of the rate-function construction and the large-volume argument for time-dependent driving. This revision will be made without changing the scope or claims of the work. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation anchored in external large-deviations variational principle

full rationale

The paper starts from an external large-deviations variational principle applied to stochastic master equations, deterministic chemical reaction networks, and open systems to define the generalized free energy and produce the excess/housekeeping decomposition of fluxes, forces, and entropy production. This principle is presented as an independent starting point from large-deviations theory rather than being constructed from the target quantities. The thermodynamic speed limit on excess entropy production follows directly from the same variational application. No equations reduce the claimed results to fitted parameters, self-definitions, or a load-bearing self-citation chain; prior excess/housekeeping work is referenced for context but is not required to establish the new generalized free energy. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of large-deviations theory to the dynamics of driven systems and the existence of a variational principle that isolates the excess component.

axioms (1)
  • domain assumption Large-deviations principle holds for the stochastic and deterministic systems under continuous driving
    Invoked to derive the generalized free energy and the decomposition from the variational principle.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Geometric decomposition of information flow: New insights into information thermodynamics

    cond-mat.stat-mech 2025-09 unverdicted novelty 7.0

    Information flow in bipartite Markov systems is split into a housekeeping part that maintains correlations through cyclic modes and an excess part that changes mutual information through conservative forces.

  2. Duality between dissipation-coherence trade-off and thermodynamic speed limit based on thermodynamic uncertainty relation for stochastic limit cycles

    cond-mat.stat-mech 2025-09 unverdicted novelty 7.0

    Derives duality between dissipation-coherence trade-off and thermodynamic speed limit for general stochastic limit cycles via dual observables substituted into the thermodynamic uncertainty relation in the weak-noise limit.

  3. Infinite variety of thermodynamic speed limits with general activities

    cond-mat.stat-mech 2024-12 unverdicted novelty 7.0

    A unified framework based on generalized means derives an infinite family of thermodynamic speed limits for Markov jump processes and chemical reaction networks, each giving a lower bound on entropy production.

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