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arxiv: 1906.12041 · v1 · pith:76KKZWLNnew · submitted 2019-06-28 · ❄️ cond-mat.stat-mech · physics.class-ph

Thermalization of non-stochastic Hamiltonian systems

K. S. Glavatskiy , V. L. Kulinskii This is my paper

Pith reviewed 2026-05-25 13:55 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.class-ph
keywords thermalizationHamiltonian systemsadiabatic invariancenon-stochastic dynamicsclassical statistical mechanicsequilibrium relaxation
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The pith

Thermalization occurs in non-stochastic Hamiltonian systems via adiabatic invariance.

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

The paper proves that thermalization can take place in Hamiltonian systems even when their dynamics remain fully deterministic and non-stochastic. It does so by invoking adiabatic invariance to link the slow evolution of certain quantities to the emergence of equilibrium distributions. The argument applies to any classical Hamiltonian, makes no appeal to the thermodynamic limit, and places no restrictions on the form of the interaction potential. This shows that stochasticity is not required for thermalization and therefore need not be assumed when performing coarse-grained analysis of relaxation.

Core claim

We prove thermalization for non-stochastic Hamiltonian systems. This shows that thermalization happens in both stochastic and non-stochastic systems, reducing the need to rely on stochasticity in a coarse-grained analysis. The result is valid for an arbitrary classical Hamiltonian system and does not rely on the thermodynamic limit or the particular form of the interaction potential. It utilizes the property of adiabatic invariance, and reveals a deep relation between the structure of the microscopic Hamiltonian and macroscopic thermodynamics.

What carries the argument

Adiabatic invariance, the property that certain phase-space quantities remain approximately constant under sufficiently slow changes, which directly implies the approach to thermal equilibrium distributions.

If this is right

  • Thermalization does not require stochastic dynamics.
  • The result holds for every classical Hamiltonian without invoking the thermodynamic limit.
  • No special assumptions about the interaction potential are needed.
  • Macroscopic thermodynamics follows from the microscopic Hamiltonian through adiabatic invariance alone.
  • Coarse-grained descriptions can explain relaxation without assuming stochasticity.

Where Pith is reading between the lines

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

  • The argument opens the possibility of deriving equilibrium properties for systems previously excluded from statistical mechanics because they lack stochasticity.
  • It suggests that adiabatic invariants may serve as a more general bridge between deterministic evolution and equilibrium than ergodic or mixing assumptions.
  • The same logic could be examined in time-dependent Hamiltonians where parameters change slowly enough to preserve the invariants.

Load-bearing premise

Adiabatic invariance by itself is enough to establish thermalization for any non-stochastic classical Hamiltonian system without further conditions on the potential or phase-space structure.

What would settle it

A concrete classical Hamiltonian that satisfies adiabatic invariance for its relevant quantities yet whose long-time distribution fails to match the expected thermal form would disprove the claim.

Figures

Figures reproduced from arXiv: 1906.12041 by K. S. Glavatskiy, V. L. Kulinskii.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic representation of the individual traject [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
read the original abstract

Ability of dynamical systems to relax to equilibrium has been investigated since the invention of statistical mechanics, which establishes the connection between dynamics of many-body Hamiltonian systems and phenomenological thermodynamics. The key link in this connection is stochasticity, which translates the deterministic evolution of a dynamical system to its probabilistic exploration of the state space. To-date research focuses on determining the conditions of stochasticity for particular systems. Here we propose an alternative agenda and prove thermalization for non-stochastic Hamiltonian systems. This shows that thermalization happens in both stochastic and non-stochastic systems, reducing the need to rely on stochasticity in a "coarse-grained" analysis. The result is valid for an arbitrary classical Hamiltonian system and does not rely on the thermodynamic limit or the particular form of the interaction potential. It utilizes the property of adiabatic invariance, and reveals a deep relation between the structure of the microscopic Hamiltonian and macroscopic thermodynamics.

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 / 0 minor

Summary. The manuscript claims to prove that thermalization occurs in arbitrary non-stochastic classical Hamiltonian systems (with no restrictions on the potential or phase-space structure and without invoking the thermodynamic limit) by appealing to the property of adiabatic invariance. This is presented as an alternative to stochasticity-based explanations, showing that thermalization holds for both stochastic and non-stochastic dynamics.

Significance. If the central claim were established with a rigorous derivation, the result would be significant for statistical mechanics, as it would demonstrate thermalization without stochasticity or special assumptions on the Hamiltonian and thereby reduce the explanatory role of coarse-graining to stochastic processes.

major comments (2)
  1. [Abstract] Abstract: the assertion that adiabatic invariance alone implies thermalization (i.e., full phase-space exploration) for any time-independent Hamiltonian is not supported. Standard adiabatic invariance applies only when an external parameter varies slowly compared with the natural frequencies; no mechanism for introducing such slow variation or effective time dependence is supplied for a closed, time-independent system, rendering the mapping from invariance to thermalization undefined for arbitrary Hamiltonians.
  2. [Abstract] Abstract: the claim of a proof is made without any equations, steps, or verification supplied in the available material, so the central assertion that the result holds for arbitrary classical Hamiltonian systems cannot be assessed or reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments. We address each major comment below, clarifying the role of adiabatic invariance in closed systems and noting that the full derivation appears in the manuscript body.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that adiabatic invariance alone implies thermalization (i.e., full phase-space exploration) for any time-independent Hamiltonian is not supported. Standard adiabatic invariance applies only when an external parameter varies slowly compared with the natural frequencies; no mechanism for introducing such slow variation or effective time dependence is supplied for a closed, time-independent system, rendering the mapping from invariance to thermalization undefined for arbitrary Hamiltonians.

    Authors: The manuscript supplies the required mapping by treating the slow evolution of action variables (arising from the separation of timescales inherent in the Hamiltonian structure) as the effective adiabatic parameter, even in the absence of external time dependence. This is derived explicitly in the body of the paper for arbitrary time-independent Hamiltonians. We will revise the abstract to indicate this mechanism more explicitly. revision: partial

  2. Referee: [Abstract] Abstract: the claim of a proof is made without any equations, steps, or verification supplied in the available material, so the central assertion that the result holds for arbitrary classical Hamiltonian systems cannot be assessed or reproduced.

    Authors: The abstract is a concise summary only. The full manuscript contains the complete step-by-step derivation, all equations, and verification that thermalization holds for arbitrary classical Hamiltonian systems without stochasticity, the thermodynamic limit, or restrictions on the potential. We are prepared to supply the relevant sections if the referee had access only to the abstract. revision: no

Circularity Check

0 steps flagged

No circularity: derivation invokes standard adiabatic invariance as external input

full rationale

The paper states that thermalization follows from the property of adiabatic invariance for arbitrary time-independent classical Hamiltonians, without thermodynamic limit or restrictions on the potential. This is presented as a proof resting on a known physical property rather than any self-definition, parameter fitting, or self-citation chain. No equations or steps in the provided abstract reduce the claimed result to its own inputs by construction; the central claim retains independent content relative to the cited invariance property. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Information extracted solely from the abstract; full derivation and assumptions not available for detailed audit.

axioms (1)
  • domain assumption Adiabatic invariance applies to the Hamiltonian systems under consideration
    The result utilizes the property of adiabatic invariance as stated in the abstract.

pith-pipeline@v0.9.0 · 5683 in / 1108 out tokens · 51755 ms · 2026-05-25T13:55:09.230194+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Invariance of a property during slow transformation of the parameters of the dynamic system is called the adiabatic invariance... each of the action variables... remains constant, ˙Ii=0... J depends on the abbreviated actions Ii... Since every Ii is constant during this process, so is J, which is the adiabatic invariant as well.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the deterministic entropy Sd... dSd ≡ DN dJ/J... the entropy of a dynamical system is its adiabatic invariant.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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13 extracted references · 13 canonical work pages

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