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arxiv: 2603.19605 · v1 · pith:P3DJLTRMnew · submitted 2026-03-20 · ❄️ cond-mat.soft · cond-mat.stat-mech

Discontinuous change of viscosity in a sheared granular gas with velocity-dependent restitution

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

Pith reviewed 2026-05-21 11:26 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech
keywords granular gasshear viscosityvelocity-dependent restitutiondiscontinuous transitionkinetic theoryBagnold scalingrheology
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The pith

Velocity-dependent restitution produces a discontinuous jump in shear viscosity of a granular gas

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

The paper derives the shear viscosity of a dilute gas of hard spheres whose collisions lose more energy above a threshold relative speed. Kinetic theory yields an S-shaped curve of viscosity versus shear rate, which forces a discontinuous switch between a low-shear state and a high-shear state. Both states obey the quadratic Bagnold scaling of stress with shear rate, yet the jump is generated solely by the changing sampling of the two restitution values in the velocity distribution. A reader would care because the mechanism supplies a purely collisional route to abrupt rheological changes that does not invoke friction or dense packing.

Core claim

When the restitution coefficient takes one value for slow collisions and a distinctly lower value for fast collisions, the steady shear viscosity obtained from the Boltzmann equation under uniform shear develops an S-shaped dependence on the imposed shear rate. This S-shape produces a discontinuous change in viscosity at a critical shear rate that separates a low-shear regime from a high-shear regime, with both regimes obeying Bagnold scaling.

What carries the argument

The step-function velocity dependence of the restitution coefficient; it splits the collision integral into two regimes whose relative contributions shift with shear rate and thereby generate the non-monotonic viscosity response.

If this is right

  • The viscosity discontinuity occurs without any change in packing fraction or introduction of friction.
  • Both the low-shear and high-shear regimes exhibit stress proportional to the square of the shear rate.
  • The location of the transition is set by the threshold speed at which restitution switches.
  • The same kinetic mechanism can produce discontinuous rheology in the absence of jamming or frictional contacts.

Where Pith is reading between the lines

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

  • A smooth rather than step-like velocity dependence would replace the discontinuity with a steep but continuous crossover.
  • Event-driven simulations with the identical step-function restitution rule would directly test the predicted jump location.
  • The mechanism may appear in other driven granular flows where dissipation changes sharply with impact speed.

Load-bearing premise

The restitution coefficient must switch in an abrupt, step-like manner at one fixed collision speed.

What would settle it

Run an event-driven simulation of hard spheres with restitution equal to 0.9 below relative speed 1 and 0.5 above it; the measured viscosity versus shear rate must exhibit an abrupt vertical jump at the shear rate where the typical collision speed crosses the threshold.

Figures

Figures reproduced from arXiv: 2603.19605 by Makoto R. Kikuchi, Satoshi Takada, Yuria Kobayashi.

Figure 2
Figure 2. Figure 2: This implies that, within a certain range of shear rates, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Three-dimensional plot of the shear-rate dependence of the viscos￾ity for fixed e2 = 0.99 and varying e1. The solid line connects the points at which ∂η∗ /∂γ˙ ∗ diverges. of the shear rate. The solid line in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Phase diagram in the (e1, e2) plane indicating whether discontinu￾ous shear thickening occurs. The shaded region corresponds to the existence of an S-shaped constitutive relation [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

We investigate the rheology of a sheared granular gas composed of hard spheres with a velocity-dependent restitution coefficient. Using kinetic theory, we derive the shear viscosity and show that it exhibits an S-shaped dependence on the shear rate when the restitution coefficient switches between two values depending on the collision velocity. As a result, a discontinuous change of viscosity emerges between low- and high-shear regimes, both characterized by Bagnold-type scaling. While the phenomenology resembles the Wyart-Cates scenario for dense suspensions, the present transition arises purely from kinetic effects without frictional contacts or jamming.

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 rheology of a sheared granular gas of hard spheres whose restitution coefficient switches discontinuously between two constant values at a threshold collision velocity. Kinetic theory is used to derive the shear viscosity, which is reported to display an S-shaped dependence on shear rate; this produces a discontinuous jump in viscosity between a low-shear and a high-shear Bagnold regime. The transition is presented as arising purely from kinetic effects, without frictional contacts or jamming, and is contrasted with the Wyart-Cates scenario for dense suspensions.

Significance. If the discontinuity survives the averaging inherent in the collision integral, the result would constitute a purely kinetic mechanism for discontinuous rheology in dilute granular gases. This would be of interest for granular rheology and could motivate further study of velocity-dependent restitution in driven granular systems.

major comments (2)
  1. [§3 (kinetic theory derivation) and the paragraph following Eq. (viscosity expression)] The central claim that a step-function restitution produces an actual discontinuity in viscosity requires explicit demonstration that the velocity-distribution averaging in the Boltzmann/Enskog collision integral does not smooth the jump. Near the critical shear rate the pair-velocity distribution has width ~sqrt(T), so a finite fraction of collisions lie on each side of the threshold; the manuscript must show in the moment equations or closure that the effective restitution (or the resulting viscosity) nevertheless remains discontinuous.
  2. [§3 and §4 (results)] The abstract states that kinetic theory yields the S-shaped curve, yet no explicit distribution function, closure approximations, or verification that the assumed step-function restitution is compatible with the steady-state assumptions of the kinetic theory are provided. These details are load-bearing for the discontinuity claim and must be supplied.
minor comments (2)
  1. [Introduction] Define the precise functional form of the velocity-dependent restitution (including the threshold velocity and the two constant values) already in the introduction or model section rather than deferring to the methods.
  2. [Figure captions] Clarify the normalization and units of the shear rate and viscosity in all figures so that the Bagnold scaling regimes are immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript concerning the rheology of a sheared granular gas with velocity-dependent restitution. We address the major comments point by point below, providing clarifications and indicating where revisions will be made to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§3 (kinetic theory derivation) and the paragraph following Eq. (viscosity expression)] The central claim that a step-function restitution produces an actual discontinuity in viscosity requires explicit demonstration that the velocity-distribution averaging in the Boltzmann/Enskog collision integral does not smooth the jump. Near the critical shear rate the pair-velocity distribution has width ~sqrt(T), so a finite fraction of collisions lie on each side of the threshold; the manuscript must show in the moment equations or closure that the effective restitution (or the resulting viscosity) nevertheless remains discontinuous.

    Authors: We appreciate the referee pointing out the need for explicit demonstration regarding the preservation of the discontinuity under averaging. In our kinetic theory approach, the collision integral is evaluated using the Enskog-Boltzmann equation with the step-function restitution coefficient. The key is that the threshold is on the relative velocity, and in the steady-state solution for the shear flow, the moment equations for the stress tensor components lead to a viscosity that depends on the fraction of collisions above and below the threshold. Because the restitution switches discontinuously, the effective cooling rate and the resulting viscosity exhibit a jump when the shear rate makes the typical velocity cross the threshold. We have verified this by solving the algebraic equations for the moments, where the switch occurs sharply. To make this clearer, we will add an explicit calculation showing the discontinuity in the effective restitution coefficient as a function of shear rate in the revised version of §3. revision: yes

  2. Referee: [§3 and §4 (results)] The abstract states that kinetic theory yields the S-shaped curve, yet no explicit distribution function, closure approximations, or verification that the assumed step-function restitution is compatible with the steady-state assumptions of the kinetic theory are provided. These details are load-bearing for the discontinuity claim and must be supplied.

    Authors: We agree that additional details on the distribution function and closures would enhance the manuscript. The derivation assumes a local Maxwellian velocity distribution for the granular gas in the driven steady state, which is a standard closure in kinetic theory for granular gases under shear. The step-function restitution is compatible because the steady state is defined by balance between viscous heating and collisional cooling, and the discontinuity arises from the abrupt change in cooling rate at the critical point. We will expand the description in §3 to include the explicit form of the velocity distribution used and the moment closure approximations. Additionally, we will verify in §4 that the steady-state conditions hold across the transition. These additions will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard kinetic theory to an externally prescribed step-function restitution

full rationale

The paper states that the restitution coefficient is a given velocity-dependent input (switching between two discrete values at a threshold speed) and then derives the shear viscosity via kinetic theory (Boltzmann/Enskog level) for a sheared granular gas. The S-shaped viscosity curve and resulting discontinuity between two Bagnold regimes are presented as consequences of this model. No parameters are fitted to the target discontinuity, no self-citation chain is invoked to justify the central result, and the derivation does not reduce to renaming or self-definition. The skeptic concern about averaging over the velocity distribution is a question of correctness or model validity, not circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The model rests on a step-function restitution law whose two values and switching threshold are introduced to produce the S-shape; kinetic theory is invoked as the calculational framework without further justification in the abstract.

free parameters (3)
  • restitution switch threshold velocity
    Velocity at which the restitution coefficient jumps between its two values; required to define the velocity dependence.
  • low-velocity restitution coefficient
    One of the two discrete restitution values used in the step-function model.
  • high-velocity restitution coefficient
    Second discrete restitution value in the step-function model.
axioms (1)
  • domain assumption Kinetic theory (Boltzmann or Enskog level) remains valid for the steady sheared state with the chosen restitution law.
    The shear viscosity is derived using kinetic theory; the abstract gives no indication that higher-order correlations or non-Gaussian effects are included.

pith-pipeline@v0.9.0 · 5625 in / 1421 out tokens · 37786 ms · 2026-05-21T11:26:00.301904+00:00 · methodology

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