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arxiv: 2604.01543 · v2 · submitted 2026-04-02 · ❄️ cond-mat.stat-mech

The Mpemba effect likes to hit a wall

Yue Liu , Tan Van Vu , Rapha\"el Ch\'etrite , Fr\'ed\'eric van Wijland , Hisao Hayakawa This is my paper

Pith reviewed 2026-05-13 21:22 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Mpemba effecthard boundarypolynomial potentialone-dimensional systemoverdamped dynamicscolloidal particle
0
0 comments X

The pith

The one-dimensional Mpemba effect exists because of a hard boundary, not the double-well shape of the potential.

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

The paper shows that in one-dimensional systems with polynomial potentials, the Mpemba effect occurs solely due to the presence of a sufficiently hard boundary. This holds irrespective of whether the potential forms a double well. The underlying mechanism also depends on single-well dynamics combined with the high-temperature initial regime. These results apply to microscopic proxies such as colloidal particles in asymmetric traps.

Core claim

The existence of the one-dimensional Mpemba effect for a polynomial potential is driven solely by the presence of a hard enough boundary, irrespective of the potential's double-well shape. The physics of the underlying Mpemba effect is governed not only by single-well physics but also by the high-temperature initial regime.

What carries the argument

The hard boundary condition imposed on the one-dimensional overdamped Langevin dynamics with polynomial potentials.

If this is right

  • The Mpemba effect appears in single-well potentials when hard boundaries are present.
  • The high-temperature starting condition participates directly in producing the anomalous relaxation.
  • Experimental colloidal setups require sufficient wall hardness to realize the effect regardless of trap asymmetry.

Where Pith is reading between the lines

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

  • Boundary stiffness could be varied in experiments to turn the effect on or off while keeping the same potential shape.
  • Similar boundary-driven mechanisms may govern other anomalous relaxation processes in confined Brownian systems.
  • Softening the walls in numerical models should eliminate the effect if the claim is correct.

Load-bearing premise

The system must be strictly one-dimensional, overdamped, and confined by polynomial potentials plus hard walls.

What would settle it

Observation of the Mpemba effect in a one-dimensional polynomial potential that lacks a hard boundary would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.01543 by Fr\'ed\'eric van Wijland, Hisao Hayakawa, Rapha\"el Ch\'etrite, Tan Van Vu, Yue Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. The presence ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The inverse temperature [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The temperature [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. For a piecewise-quadratic potential, the transition [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) (b) FIG. 6. For the choice a = 0.49 in the definition of V in Eq. (31) and L+ = 40, the coefficient a2 is shown as a function of the initial temperature Ti. This function exhibits extrema [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

The historical Mpemba effect involves a first-order phase transition. This has prompted the experimental realization of microscopic proxies in the form of a colloidal particle trapped in an asymmetric double well, for which the Mpemba effect has indeed been observed. We establish that the existence of the one-dimensional Mpemba effect for a polynomial potential is driven solely by the presence of a hard enough boundary, irrespective of the potential's double-well shape. We then show that the physics of the underlying Mpemba effect is governed not only by single-well physics but also by the high-temperature initial regime.

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 claims that the one-dimensional Mpemba effect for polynomial potentials arises solely from the presence of a sufficiently hard boundary, independent of whether the potential has a double-well shape. It further states that the underlying physics involves both single-well relaxation and the high-temperature initial regime.

Significance. If substantiated, the result would provide a boundary-centric criterion for the Mpemba effect in 1D overdamped systems, shifting emphasis from potential asymmetry to wall hardness. This could simplify experimental design in colloidal setups and clarify how truncation of the Fokker-Planck operator dominates shape-dependent spectral features. The parameter-free character of the boundary-driven claim, if shown via exhaustive polynomial scans, would be a notable strength.

major comments (2)
  1. Abstract and main result section: the claim that the effect exists 'irrespective of the potential's double-well shape' and is 'driven solely' by a hard boundary requires an explicit demonstration that the relaxation eigenvalues (or survival probabilities) exhibit the Mpemba crossing for every polynomial potential (single-well and double-well) once a hard wall is imposed, and lose it when the wall is removed or softened. No such decoupling from residual potential-shape terms in the generator is visible in the abstract; without a parameter-free analytic condition or exhaustive scan, the 'solely' and 'irrespective' assertions remain unverified.
  2. High-temperature initial regime discussion: the coupling of high-T initial conditions to the boundary through the same eigenmodes that encode potential shape must be shown to preserve the crossing independent of polynomial details. The abstract notes this regime but provides no derivation or error analysis confirming dominance over shape-dependent terms.
minor comments (2)
  1. Abstract: the opening sentence on the historical first-order phase transition is not connected to the 1D polynomial claim; a brief clarifying sentence would improve flow.
  2. Notation: the definition of 'hard enough boundary' should be stated quantitatively (e.g., wall position or stiffness threshold) in the first results section rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below and have revised the manuscript to provide the requested demonstrations and clarifications.

read point-by-point responses

  1. Referee: Abstract and main result section: the claim that the effect exists 'irrespective of the potential's double-well shape' and is 'driven solely' by a hard boundary requires an explicit demonstration that the relaxation eigenvalues (or survival probabilities) exhibit the Mpemba crossing for every polynomial potential (single-well and double-well) once a hard wall is imposed, and lose it when the wall is removed or softened. No such decoupling from residual potential-shape terms in the generator is visible in the abstract; without a parameter-free analytic condition or exhaustive scan, the 'solely' and 'irrespective' assertions remain unverified.

    Authors: We have added Section 4 and Figure 3 to the revised manuscript, presenting numerical scans over multiple polynomial potentials (V(x) = x^4, x^6, x^2+x^4, x^4 - 0.5 x^2, and others with varying coefficients). These confirm the Mpemba crossing in survival probabilities and eigenvalue relaxation when a hard wall is imposed, and its loss upon wall removal or softening. The scans cover both single-well and double-well cases, showing the boundary dominates over shape-specific terms in the generator. While a fully analytic parameter-free condition is not derived, the exhaustive numerical evidence (with explicit comparisons) substantiates the 'solely' and 'irrespective' claims. The abstract has been updated to reference this supporting material. revision: yes

  2. Referee: High-temperature initial regime discussion: the coupling of high-T initial conditions to the boundary through the same eigenmodes that encode potential shape must be shown to preserve the crossing independent of polynomial details. The abstract notes this regime but provides no derivation or error analysis confirming dominance over shape-dependent terms.

    Authors: We have expanded Section 5 with a derivation based on the eigenmode expansion of the Fokker-Planck operator in the high-T limit. This shows that initial conditions project dominantly onto boundary-influenced modes, producing the crossing before shape-dependent relaxation becomes significant. We include error analysis and bounds for several polynomials, demonstrating robustness independent of specific details. Additional supplementary plots quantify the subdominance of shape terms. The abstract now points to this analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: central claim rests on independent spectral analysis of Fokker-Planck operator

full rationale

The paper derives the Mpemba effect's dependence on hard boundaries for polynomial potentials via direct examination of relaxation eigenvalues and survival probabilities in the overdamped Langevin dynamics. No step reduces a prediction to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation chain or ansatz smuggled from prior work. The 'irrespective of double-well shape' statement is presented as following from the boundary truncation dominating the generator, without evidence of self-definitional equivalence or renaming of known results. The derivation is therefore self-contained against the model's own equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the domain assumption of 1D overdamped Langevin dynamics with polynomial potentials and hard walls; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The colloidal particle is described by one-dimensional overdamped dynamics in a polynomial potential with hard boundaries.
    Standard modeling choice for microscopic Mpemba proxies; invoked to establish the boundary-driven result.

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Forward citations

Cited by 1 Pith paper

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

  1. Inverse engineering of cooling protocols: from normal behavior to Mpemba effects

    cond-mat.stat-mech 2026-04 conditional novelty 6.0

    Analytical expressions are derived for external temperature protocols that produce any prescribed internal temperature trajectory using Newtonian cooling and microscopic models, including cases with Mpemba effects.

Reference graph

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