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arxiv: 2605.17881 · v1 · pith:GG2XPFJXnew · submitted 2026-05-18 · ❄️ cond-mat.soft · cond-mat.stat-mech

Mpemba effect in a sheared granular gas with velocity-dependent restitution

Makoto R. Kikuchi , Yuria Kobayashi , Satoshi Takada This is my paper

Pith reviewed 2026-05-20 01:10 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech
keywords Mpemba effectgranular gassheared flowvelocity-dependent restitutionkinetic theoryrelaxation dynamicsshear viscosity
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The pith

A hotter isotropic granular gas relaxes its temperature faster than a cooler sheared steady state when restitution depends on velocity.

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

The paper investigates the Mpemba effect in a dilute sheared granular gas whose restitution coefficient varies with particle speed. Kinetic theory shows that an isotropic initial state with higher temperature can reach the driven steady state quicker than a system that already sits in steady shear. The same pattern appears in the relaxation of shear viscosity. Velocity dependence of restitution introduces an extra intrinsic timescale that produces multiple crossings in the relaxation curves.

Core claim

Despite having a higher initial temperature, a system starting from an isotropic state can relax faster than a system prepared in a sheared steady state, demonstrating a clear Mpemba effect in the temperature evolution. Multiple crossings arise due to an additional intrinsic timescale introduced by the velocity dependence of the restitution coefficient, providing a minimal kinetic mechanism for multiple Mpemba effects in driven granular gases.

What carries the argument

Grad's moment method applied to the Boltzmann equation, which tracks the evolution of temperature and shear stress after an abrupt change in shear rate.

If this is right

  • Temperature relaxation curves cross at least once and can cross multiple times.
  • Shear viscosity also displays a Mpemba effect with crossings in its relaxation curve.
  • The velocity dependence of restitution supplies an extra timescale that enables multiple Mpemba crossings.
  • The mechanism works in the dilute limit under sudden changes of shear rate.

Where Pith is reading between the lines

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

  • Similar velocity-dependent dissipation could produce multiple Mpemba crossings in other driven dissipative gases or suspensions.
  • The extra timescale might be tuned by choosing different restitution laws to control the number of crossings observed.
  • If confirmed, the effect could be used to accelerate cooling or stress relaxation in granular processing by choosing appropriate initial states.

Load-bearing premise

Grad's moment method accurately captures the relaxation dynamics and the velocity distribution under velocity-dependent restitution in the dilute limit.

What would settle it

Molecular dynamics simulation of the same granular gas that shows the isotropic initial condition relaxing no faster than the sheared initial condition.

Figures

Figures reproduced from arXiv: 2605.17881 by Makoto R. Kikuchi, Satoshi Takada, Yuria Kobayashi.

Figure 3
Figure 3. Figure 3: shows the phase diagram of the temperature Mpemba effect in the parameter space spanned by the initial shear rate ˙γini and the initial temperature T (FI) ini . The Mpemba effect occurs only in the region where T (FI) ini & T (FS) ini . When T (FI) ini ≫ T (FS) ini , however, the relaxation of the FI system be￾comes too slow to overtake that of the FS system, and the 000 00 00 00 00 000 00 γ˙ ∗  00 00… view at source ↗
Figure 2
Figure 2. Figure 2: (Coler online) Typical time evolutions of the viscosity for the FS and FI protocols. The sets of parameters are identical to those used in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We investigate the Mpemba effect in a dilute sheared granular gas with a velocity-dependent restitution coefficient. Using kinetic theory based on Grad's moment method, we analyze the relaxation dynamics following a sudden change in the shear rate. We show that, despite having a higher initial temperature, a system starting from an isotropic state can relax faster than a system prepared in a sheared steady state, demonstrating a clear Mpemba effect in the temperature evolution. We further demonstrate the emergence of a viscosity Mpemba effect, characterized by crossings in the relaxation curves of the shear viscosity. Remarkably, multiple crossings arise due to an additional intrinsic timescale introduced by the velocity dependence of the restitution coefficient, providing a minimal kinetic mechanism for multiple Mpemba effects in driven granular gases.

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 investigates the Mpemba effect in a dilute sheared granular gas with velocity-dependent restitution coefficient. Using kinetic theory based on Grad's moment method, the authors analyze relaxation dynamics after a sudden change in shear rate. They claim that an isotropic initial state relaxes faster in temperature than a sheared steady state despite higher initial temperature, demonstrating a Mpemba effect, and report a viscosity Mpemba effect with multiple crossings in relaxation curves arising from an additional intrinsic timescale due to the velocity dependence of the restitution coefficient.

Significance. If the moment closure remains accurate, the work supplies a minimal kinetic mechanism for multiple Mpemba effects in driven granular gases by linking an extra relaxation timescale to the velocity dependence of restitution. This extends prior studies on constant-restitution cases and could inform understanding of anomalous relaxation in non-equilibrium systems with speed-dependent interactions.

major comments (2)
  1. [Kinetic theory analysis] Kinetic theory analysis section: the central claims rest on the closed moment equations obtained via Grad's moment method, yet the manuscript provides no comparison with direct simulation Monte Carlo solutions of the Boltzmann equation or extension to higher-order moments. With velocity-dependent restitution the collision integrals acquire explicit speed dependence that couples higher moments more strongly; without such a check the reported ordering of isotropic versus sheared trajectories and the multiple crossings may be artifacts of the truncation rather than robust features of the underlying kinetic equation.
  2. [Results] Results on temperature and viscosity evolution: the paper does not report error estimates, sensitivity to the parameters that define the velocity dependence of the restitution coefficient, or tests that the crossings survive when those parameters are varied within the dilute limit; this weakens the assertion that the additional intrinsic timescale is a generic minimal mechanism.
minor comments (2)
  1. [Abstract and Introduction] The abstract states that the approach relies on an established method but the main text should include a brief derivation or explicit statement of the moment closure ansatz and the form of the velocity-dependent restitution coefficient for reproducibility.
  2. [Figures] Figure captions and axis labels should explicitly indicate the units or dimensionless groups used for temperature and shear rate to aid clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below, indicating where revisions will be incorporated to improve clarity and robustness while maintaining the core analytical approach.

read point-by-point responses

  1. Referee: [Kinetic theory analysis] Kinetic theory analysis section: the central claims rest on the closed moment equations obtained via Grad's moment method, yet the manuscript provides no comparison with direct simulation Monte Carlo solutions of the Boltzmann equation or extension to higher-order moments. With velocity-dependent restitution the collision integrals acquire explicit speed dependence that couples higher moments more strongly; without such a check the reported ordering of isotropic versus sheared trajectories and the multiple crossings may be artifacts of the truncation rather than robust features of the underlying kinetic equation.

    Authors: We appreciate the referee's concern regarding the accuracy of the Grad closure for velocity-dependent restitution. Grad's moment method is a standard tool in kinetic theory for granular gases and has been shown to capture the essential relaxation dynamics, including non-Gaussian effects, in prior studies with both constant and velocity-dependent restitution. The additional intrinsic timescale arises explicitly from the velocity dependence in the collision integrals already at this level, leading to the observed multiple crossings. While a direct DSMC validation or higher-moment extension would provide further confirmation, such numerical work lies outside the scope of the present analytical investigation and would require a separate study. In the revised manuscript we will expand the discussion of the closure's validity, referencing its performance in related constant-restitution cases, and clarify why the qualitative features are expected to be robust rather than truncation artifacts. revision: partial

  2. Referee: [Results] Results on temperature and viscosity evolution: the paper does not report error estimates, sensitivity to the parameters that define the velocity dependence of the restitution coefficient, or tests that the crossings survive when those parameters are varied within the dilute limit; this weakens the assertion that the additional intrinsic timescale is a generic minimal mechanism.

    Authors: We agree that explicit sensitivity tests would strengthen the claim of a generic mechanism. In the revised manuscript we will add a new subsection (or appendix) showing the temperature and viscosity relaxation curves for several values of the parameters that control the velocity dependence of the restitution coefficient, while remaining within the dilute regime. These additional results confirm that the multiple crossings persist and that the extra timescale remains operative. Because the moment equations are deterministic, conventional statistical error bars do not apply; we will instead discuss the systematic approximation error associated with the Grad closure in the methods section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from standard moment closure to numerical integration of resulting ODEs

full rationale

The paper starts from the Boltzmann equation for a dilute granular gas, applies Grad's moment method to obtain a closed set of ODEs for the relevant moments (including temperature and shear stress), and integrates those ODEs forward in time for two different initial conditions to exhibit the Mpemba crossings. The velocity dependence of the restitution coefficient enters the collision integrals as an explicit functional form and thereby supplies an extra intrinsic timescale; this is an input assumption, not a quantity fitted to or derived from the target relaxation curves. No equation is shown to be identical to its own input by algebraic rearrangement, no parameter is fitted to a subset of the reported trajectories and then relabeled as a prediction, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The reported crossings are therefore genuine outputs of the truncated dynamical system rather than tautological restatements of its construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of Grad's moment method to this driven system and on the specific functional form chosen for velocity-dependent restitution that supplies the extra timescale; no independent evidence for the closure accuracy is given in the abstract.

free parameters (1)
  • parameters defining velocity dependence of restitution coefficient
    The model requires a functional form or coefficients for how restitution varies with relative velocity; these are not specified but are necessary to generate the additional timescale.
axioms (1)
  • domain assumption Grad's moment method supplies a sufficient closure for the velocity distribution function during relaxation after a shear-rate jump.
    Directly invoked to analyze the dynamics in the abstract.

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Works this paper leans on

51 extracted references · 51 canonical work pages · 2 internal anchors

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