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arxiv: 2605.16417 · v1 · pith:35PI4XOInew · submitted 2026-05-13 · ❄️ cond-mat.stat-mech

The problem of relaxation to equilibrium

Silvina Limandri (1 , 2) , Silvina Segui (2) , Bruno Castellano (1) , Ignacio Belitzky (1) , Gustavo Castellano (1 , 2) ((1) Facultad de Matem\'atica , Astronom\'ia
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F\'isica y Computaci\'on Universidad Nacional de C\'ordoba Argentina (2) Instituto de F\'isica Enrique Gaviola IFEG-CONICET)
This is my paper

Pith reviewed 2026-05-20 20:12 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords relaxation to equilibriumstatistical mechanicsHeisenberg uncertainty principleboundary conditionsclassical gasthermodynamic equilibriumirreversibility
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The pith

A classical gas reaches thermodynamic equilibrium if wall collisions obey the Heisenberg uncertainty principle.

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

The paper argues that a simple classical gaseous system can relax to an equilibrium state from any initial constraint once all particle interactions with the container walls are required to respect the Heisenberg uncertainty principle. Standard classical mechanics via Hamilton's equations or quantum mechanics via the Schrödinger equation have not produced this relaxation without adding extra hypotheses such as stochastic forces or coarse-graining. By replacing strict, deterministic boundary conditions with non-strict ones that incorporate position-momentum uncertainty at every wall collision, the system is shown to evolve toward equilibrium. A sympathetic reader would care because this supplies a concrete mechanism for irreversibility inside an otherwise time-reversible classical description.

Core claim

The authors claim that non-strict boundary conditions, defined so that every interaction between gas particles and container walls must occur in accordance with Heisenberg's uncertainty principle, suffice to drive a classical gaseous system to thermodynamic equilibrium without any further modifications to the dynamics or additional statistical assumptions.

What carries the argument

Non-strict boundary conditions that force every wall collision to respect the Heisenberg uncertainty relation between position and momentum.

If this is right

  • The gas particles lose memory of their initial conditions through repeated uncertain wall collisions and settle into a uniform equilibrium distribution.
  • No stochastic thermostat or coarse-graining step is required once the boundary rule is changed.
  • The same construction applies to any simple classical gas whose only non-conservative events are wall encounters.
  • Equilibrium emerges directly from the modified deterministic dynamics rather than from an ensemble average.

Where Pith is reading between the lines

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

  • The approach may offer a route to derive the second law from reversible microscopic rules without invoking external randomness.
  • It suggests that similar uncertainty-based boundaries could be tested in other bounded classical systems such as hard-sphere gases or molecular fluids.
  • Numerical verification would require tracking both position and momentum spreads at each wall hit rather than using specular reflection rules.

Load-bearing premise

That requiring every wall interaction in an otherwise classical system to obey the Heisenberg uncertainty principle is enough by itself to produce relaxation to equilibrium.

What would settle it

A molecular-dynamics simulation of the same gas in a container where wall collisions are made strictly classical (zero uncertainty) while keeping all other rules identical, showing that equilibrium is not reached.

Figures

Figures reproduced from arXiv: 2605.16417 by 2), 2) ((1) Facultad de Matem\'atica, (2) Instituto de F\'isica Enrique Gaviola, Argentina, Astronom\'ia, Bruno Castellano (1), F\'isica y Computaci\'on, Gustavo Castellano (1, IFEG-CONICET), Ignacio Belitzky (1), Silvina Limandri (1, Silvina Segui (2), Universidad Nacional de C\'ordoba.

Figure 1
Figure 1. Figure 1: FIG. 1. Evolution of linear momentum (left) and coordinate (right) distributions for 2 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Thermalization of the 1D ideal gas initially out of thermal equilibrium, consisting in 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Relaxation to thermodynamic equilibrium for a non [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Thermalization of the 1D ideal gas with initial velocities according to Boltzmann’s distribution, allowing only positive [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Relaxation to thermodynamic equilibrium for a [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

When a thermodynamic system is released from any constraint, after some time its evolution will render it into an equilibrium state. Although the description of this relaxation to thermodynamic equilibrium has been attempted through both classical (Hamilton's equations) or quantum (Schr\"odinger equation) approaches, no success has been achieved without recurring to additional hypotheses. The present work demonstrates the possibility of reaching equilibrium states in a simple classical gaseous system, by imposing non-strict boundary conditions, in the sense that all interactions with the container walls must occur according to Heisenberg's uncertainty principle.

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

3 major / 1 minor

Summary. The manuscript claims to demonstrate that relaxation to thermodynamic equilibrium is possible in a simple classical gaseous system by imposing non-strict boundary conditions, under which all interactions with the container walls must occur according to Heisenberg's uncertainty principle, without recourse to additional hypotheses such as ergodicity or coarse-graining.

Significance. If the central construction were made rigorous and shown to produce relaxation from the underlying reversible dynamics, the result would be significant for the foundations of statistical mechanics. It would provide a concrete mechanism for irreversibility that avoids the extra assumptions the paper criticizes in prior classical and quantum treatments, potentially offering a minimal resolution to the problem of approach to equilibrium.

major comments (3)
  1. [Abstract] Abstract: The claim that the work 'demonstrates the possibility of reaching equilibrium states' is asserted without any supporting equations, derivation of the modified boundary operator, or numerical evidence that the proposed conditions drive the system toward equilibrium rather than presupposing it.
  2. [Model and dynamics section] Model and dynamics section: No explicit map is given from the Heisenberg uncertainty principle to a concrete dynamical rule (e.g., a modified collision law, a stochastic term in the Liouville operator, or a phase-space diffusion process). Without this map it is impossible to verify whether relaxation emerges from the time evolution or is built into the definition of the non-strict boundaries.
  3. [Introduction / comparison with prior work] Comparison with prior work: The manuscript correctly notes that earlier approaches require extra hypotheses, yet the present construction appears to introduce an ad-hoc hybrid classical-quantum boundary condition whose consequences for the phase-space flow are not derived or tested.
minor comments (1)
  1. [Notation and definitions] The phrase 'non-strict boundary conditions' is introduced without a precise mathematical definition or comparison to standard specular or diffuse reflection rules used in classical kinetic theory.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the work 'demonstrates the possibility of reaching equilibrium states' is asserted without any supporting equations, derivation of the modified boundary operator, or numerical evidence that the proposed conditions drive the system toward equilibrium rather than presupposing it.

    Authors: The abstract summarizes the central result; the supporting construction appears in the Model and dynamics section. We will revise the abstract to include a concise reference to the modified boundary operator and its derivation from the uncertainty principle, together with a statement that relaxation follows from the resulting time evolution. revision: yes

  2. Referee: [Model and dynamics section] Model and dynamics section: No explicit map is given from the Heisenberg uncertainty principle to a concrete dynamical rule (e.g., a modified collision law, a stochastic term in the Liouville operator, or a phase-space diffusion process). Without this map it is impossible to verify whether relaxation emerges from the time evolution or is built into the definition of the non-strict boundaries.

    Authors: The non-strict boundary conditions are defined by requiring that wall collisions respect the uncertainty relation, which replaces deterministic specular reflection with a probabilistic scattering kernel. We agree this mapping can be stated more explicitly and will add a short derivation in the Model section showing how the uncertainty directly yields the modified collision rule and the consequent phase-space evolution. revision: yes

  3. Referee: [Introduction / comparison with prior work] Comparison with prior work: The manuscript correctly notes that earlier approaches require extra hypotheses, yet the present construction appears to introduce an ad-hoc hybrid classical-quantum boundary condition whose consequences for the phase-space flow are not derived or tested.

    Authors: The boundary rule follows from applying the uncertainty principle at the container walls, a physically motivated step rather than an arbitrary hybrid. We will expand the relevant section to derive the effect on the phase-space flow more explicitly and to sharpen the comparison with earlier treatments. revision: partial

standing simulated objections not resolved
  • A complete, fully rigorous proof of convergence to equilibrium for arbitrary initial measures under the proposed boundary conditions lies beyond the scope of the present work.

Circularity Check

0 steps flagged

No circularity: central claim rests on an external dynamical modification rather than redefinition or self-fit

full rationale

The abstract presents the relaxation as emerging from a modification to boundary conditions that incorporates the Heisenberg uncertainty principle into an otherwise classical Hamiltonian system. No equations, fitted parameters, or self-citations appear in the provided text that would reduce the claimed equilibrium to an input by construction. The approach is offered as an alternative to prior hypotheses (ergodicity, coarse-graining), and the uncertainty principle is invoked as an independent physical constraint rather than a renaming or tautological re-expression of the target result. Absent explicit derivation steps or load-bearing self-references in the manuscript, the chain does not collapse to its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the premise that uncertainty-principle boundary conditions suffice for equilibrium; no explicit free parameters, axioms, or invented entities are stated in the abstract.

axioms (1)
  • domain assumption Classical gaseous system dynamics can be modified at boundaries by imposing Heisenberg uncertainty without violating overall classical character.
    Invoked implicitly when the abstract states that equilibrium is reached in a 'simple classical gaseous system' via uncertainty-principle interactions.
invented entities (1)
  • non-strict boundary conditions no independent evidence
    purpose: To allow relaxation to equilibrium by making wall interactions obey the uncertainty principle.
    Introduced in the abstract as the mechanism that replaces strict classical reflections.

pith-pipeline@v0.9.0 · 5688 in / 1351 out tokens · 70467 ms · 2026-05-20T20:12:54.063040+00:00 · methodology

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Reference graph

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