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arxiv: 2604.15000 · v1 · submitted 2026-04-16 · ❄️ cond-mat.stat-mech · physics.class-ph· quant-ph

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Thermodynamic Geometry of Relaxation

Hao Wang , Li Zhao , Shuai Deng , Yu-Han Ma
Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.class-phquant-ph
keywords thermodynamic geometryrelaxation dynamicsRayleigh quotientcritical slowing downvan der Waals gasentropic stiffnessfrictional dissipation
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The pith

A Rayleigh-quotient measure recasts relaxation as the competition between entropic stiffness and frictional dissipation.

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

The paper introduces a geometric quantity for the approach to equilibrium by applying the Rayleigh quotient to thermodynamic geometry. This quantity expresses the relaxation rate as the ratio of entropic restoring force to dissipative drag. When applied to a van der Waals gas possessing two distinct dissipation channels, the measure yields an explicit relaxation landscape whose slowest eigenvalue vanishes linearly with distance from the critical temperature. A reader cares because the construction supplies the missing dynamical half of thermodynamic geometry and directly accounts for the well-known phenomenon of critical slowing down without extra parameters.

Core claim

We define a thermo-geometric relaxation measure via the Rayleigh quotient that reformulates the return to equilibrium as a direct contest between entropic stiffness and frictional dissipation. For the van der Waals gas with two dissipation channels the measure produces a relaxation landscape in which the slow-mode eigenvalue satisfies λ_s ∝ (T − T_c)/T_c as the critical temperature is approached from above.

What carries the argument

The Rayleigh-quotient measure on the thermodynamic metric, which extracts the ratio of entropic stiffness to frictional dissipation and thereby determines the eigenvalues of the relaxation dynamics.

If this is right

  • Relaxation trajectories can be read off from the eigenvalues of the Rayleigh quotient on the thermodynamic metric.
  • Critical slowing down appears as a linear vanishing of the slowest rate exactly at the critical temperature.
  • The same construction applies to any system whose equilibrium geometry and dissipation channels are known.
  • Two dissipation channels in the van der Waals gas produce a distinct slow and fast mode whose separation grows away from criticality.

Where Pith is reading between the lines

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

  • The measure might be used to predict how changing the relative strengths of the two dissipation channels alters the relaxation spectrum in other fluids.
  • The linear scaling near criticality offers a simple experimental signature that could be checked in light-scattering or pressure-jump experiments on supercritical gases.
  • If the construction generalizes, it would link geometric descriptions of equilibrium, driven steady states, and relaxation within a single framework.

Load-bearing premise

The Rayleigh quotient applied directly to the thermodynamic geometry accurately reproduces the relaxation rates without further approximations in the dissipation terms of the van der Waals model.

What would settle it

Measure the temperature dependence of the slowest relaxation rate in a near-critical fluid and check whether it vanishes linearly with (T − T_c).

Figures

Figures reproduced from arXiv: 2604.15000 by Hao Wang, Li Zhao, Shuai Deng, Yu-Han Ma.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of a damped piston-cylinder system enclosing a van der Waals fluid. (b)-(d) Temporal evolution under [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Relaxation of the gas volume for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

While the geometry of equilibrium states and driven non-equilibrium processes is clearly understood, a geometric description for relaxation towards equilibrium is still lacking. Here, we propose a thermo-geometric measure based on the Rayleigh quotient, reformulating relaxation as a fundamental competition between entropic stiffness and frictional dissipation. Taking a van der Waals gas with two dissipation channels as an example, we explicitly demonstrate its relaxation landscape. Particularly, we find that upon approaching the critical temperature $T_c$, the slow-mode relaxation rate vanishes linearly as $\lambda_s \propto (T-T_c)/T_c$, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.

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

2 major / 2 minor

Summary. The manuscript proposes a thermo-geometric measure for relaxation to equilibrium based on the Rayleigh quotient, reformulating the process as a competition between entropic stiffness and frictional dissipation. Using a van der Waals gas with two dissipation channels as an example, the relaxation landscape is constructed and the slow-mode relaxation rate is reported to vanish linearly as λ_s ∝ (T - T_c)/T_c near the critical temperature, indicating critical slowing down. This is presented as completing the thermodynamic geometry framework for characterizing relaxation dynamics.

Significance. If the derivation is shown to be free of circularity in the choice of dissipation channels, the work would usefully extend thermodynamic geometry to non-equilibrium relaxation, providing a geometric account of critical slowing down in a mean-field model. The explicit van der Waals calculation is a concrete strength that allows direct checking of the claimed linear scaling.

major comments (2)
  1. [Abstract and Rayleigh quotient derivation] Abstract and the section deriving the Rayleigh quotient: the central claim that λ_s vanishes linearly as λ_s ∝ (T−T_c)/T_c is stated without the explicit steps connecting the thermodynamic metric, the friction matrix, and the minimal Rayleigh quotient to this scaling. The derivation must be supplied to confirm the result follows from the geometry rather than from the specific parametrization of the two dissipation channels.
  2. [Van der Waals gas example] Van der Waals gas example section: the friction matrix is assumed diagonal in the basis of the thermodynamic metric. The manuscript must derive this form from the underlying stochastic dynamics (e.g., collision rules) rather than selecting channels that recover the known mean-field critical slowing down; any temperature dependence or off-diagonal coupling would generically alter the scaling of the minimal Rayleigh quotient near T_c.
minor comments (2)
  1. [Notation] The symbol λ_s and the precise definition of the Rayleigh quotient should be introduced with an equation number at first appearance rather than only in the abstract.
  2. [Figures] The figure of the relaxation landscape would be clearer if the location of T_c and the slow-mode eigenvector were explicitly labeled.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The suggestions have prompted us to clarify the derivations and strengthen the justification of our modeling choices. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and Rayleigh quotient derivation] Abstract and the section deriving the Rayleigh quotient: the central claim that λ_s vanishes linearly as λ_s ∝ (T−T_c)/T_c is stated without the explicit steps connecting the thermodynamic metric, the friction matrix, and the minimal Rayleigh quotient to this scaling. The derivation must be supplied to confirm the result follows from the geometry rather than from the specific parametrization of the two dissipation channels.

    Authors: We agree that the original manuscript did not provide sufficiently explicit intermediate steps linking the thermodynamic metric, the friction matrix, and the minimal value of the Rayleigh quotient to the reported linear scaling of λ_s. In the revised version we have inserted a dedicated subsection that derives the result step by step: the Rayleigh quotient is defined as the ratio of the quadratic dissipation form (involving the friction matrix) to the quadratic entropic form (involving the thermodynamic metric); its minimum eigenvalue λ_s is obtained by solving the associated generalized eigenvalue problem; near T_c the metric components diverge as the isothermal compressibility while the friction coefficients remain finite, yielding the leading-order linear vanishing λ_s ∝ (T − T_c)/T_c after a straightforward expansion. This derivation is independent of the particular numerical values chosen for the two channels and follows directly from the geometric construction. revision: yes

  2. Referee: [Van der Waals gas example] Van der Waals gas example section: the friction matrix is assumed diagonal in the basis of the thermodynamic metric. The manuscript must derive this form from the underlying stochastic dynamics (e.g., collision rules) rather than selecting channels that recover the known mean-field critical slowing down; any temperature dependence or off-diagonal coupling would generically alter the scaling of the minimal Rayleigh quotient near T_c.

    Authors: We acknowledge that the diagonal form of the friction matrix was introduced as a modeling assumption motivated by the physical separation of the two dissipation channels (volume relaxation and thermal relaxation). In the revision we have added an explicit discussion of this choice, including a symmetry argument that off-diagonal elements are forbidden by time-reversal properties of the underlying Langevin dynamics and a brief sensitivity analysis demonstrating that weak temperature dependence or small off-diagonal couplings do not change the leading linear scaling of λ_s. A complete microscopic derivation from collision rules would require a separate kinetic-theory calculation that lies beyond the geometric scope of the present work. revision: partial

standing simulated objections not resolved
  • A first-principles derivation of the friction matrix directly from microscopic collision rules for the van der Waals gas.

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent geometric reformulation

full rationale

The paper proposes a new Rayleigh-quotient-based thermo-geometric measure that reformulates relaxation as competition between entropic stiffness and frictional dissipation, then applies it to the van der Waals model with two explicit dissipation channels to obtain the relaxation landscape and the linear vanishing of the slow-mode rate near Tc. No equations or self-citations are available in the provided text that reduce the central result to a fitted input, a self-defined quantity, or a load-bearing prior result by the same authors. The scaling result is presented as a demonstration within the new framework rather than a tautological restatement of the model's dissipation parameters, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only: the framework assumes standard thermodynamic relations and the validity of the Rayleigh quotient for this context; no explicit free parameters or invented entities are named, but the two dissipation channels in the van der Waals example likely introduce model-specific choices.

axioms (1)
  • domain assumption Thermodynamic geometry applies to non-equilibrium relaxation processes via the Rayleigh quotient
    Invoked in the proposal to reformulate relaxation dynamics

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Forward citations

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Reference graph

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