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arxiv: 2605.03455 · v1 · submitted 2026-05-05 · ❄️ cond-mat.soft · cond-mat.stat-mech

Recognition: unknown

Dynamic properties of a confined quasi-two-dimensional granular fluid driven by a stochastic bath with friction

David Gonz\'alez M\'endez , Rub\'en G\'omez Gonz\'alez , Vicente Garz\'o
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Pith reviewed 2026-05-07 13:29 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mech
keywords quasi-two-dimensional granular fluidEnskog kinetic equationDelta-modeltransport coefficientsviscous dragstochastic bathlinear stabilityDSMC validation
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The pith

External driving from an interstitial gas modifies the transport coefficients of a quasi-two-dimensional granular fluid.

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

The paper develops a kinetic theory for a moderately dense quasi-two-dimensional granular fluid that experiences both inelastic collisions and continuous driving from an interstitial gas. It incorporates viscous drag and stochastic forces into the Delta-model and solves the resulting Enskog equation using the Chapman-Enskog expansion to obtain the Navier-Stokes transport coefficients. These coefficients, including viscosity and thermal conductivity, depend on the strength of the gas coupling in ways that differ from earlier dry granular models. The derived expressions predict a steady temperature and velocity distribution kurtosis that match computer simulations, and they indicate that the uniform state does not develop instabilities.

Core claim

The central claim is that the transport coefficients derived from the Enskog kinetic equation for the driven granular system differ from those of the corresponding dry granular gas because the external bath supplies energy continuously through drag and fluctuations. Specifically, the shear viscosity, bulk viscosity, thermal conductivity, and cooling rate acquire explicit dependence on the friction coefficient and the noise intensity, which reduces the effective cooling and alters the density dependence. This leads to analytical predictions for the steady-state temperature and the fourth velocity moment that are confirmed by DSMC simulations, while a linear stability analysis of the uniform (

What carries the argument

The Enskog kinetic equation for the Delta-model augmented with a viscous drag term and a stochastic Langevin force, expanded to first order in spatial gradients via the Chapman-Enskog procedure.

Load-bearing premise

The modeling of energy injection through the specific combination of modified collision rules, viscous drag, and Langevin noise accurately represents the real physics of the interstitial gas at the densities studied.

What would settle it

Measuring the shear viscosity in a quasi-two-dimensional granular experiment while systematically varying the background gas pressure and comparing to the predicted dependence on friction coefficient would falsify the claim if systematic disagreement appears.

Figures

Figures reproduced from arXiv: 2605.03455 by David Gonz\'alez M\'endez, Rub\'en G\'omez Gonz\'alez, Vicente Garz\'o.

Figure 1
Figure 1. Figure 1: Schematic illustration of the confined quasi–two view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the (scaled) steady temperature θs(α)/θs(1) versus the coefficient of restitution α for d = 2, ϕ = 0.1, ∆∗ b = 1, and T ∗ b = 0.8. Here, θs(1) refers to the value of the steady temperature for elastic collisions. The solid line is the theoretical result obtained from Eq. (62) while the dashed line corresponds to the theoretical result obtained by numerically solving Eqs. (63) and (64). Symbols are … view at source ↗
Figure 3
Figure 3. Figure 3: (A) Plot of the steady fourth–cumulant a2,s(α) ver￾sus the coefficient of restitution α for d = 2, ∆∗ b = 1, T ∗ b = 0.8, and two different volume fractions: ϕ = 0.1 and ϕ = 0.2. The solid lines are the theoretical results obtained from Eq. (63) while the dashed lines correspond to the theoretical results obtained by numerically solving Eqs. (63) and (64). Sym￾bols are the DSMC results. Here, we have assum… view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the derivatives Υθ, Υλ, and Υχ as a function of the coefficient of restitution α for d = 2, ϕ = 0.2, ∆∗ b = 0.5, and T ∗ b = 1. property over the whole parameter space. In what fol￾lows, the derivatives are evaluated at the HSS for each set of parameters considered. Finally, we also observe that these derivatives exhibit a complex dependence on the coefficient of restitution. B. First–order approxi… view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the (scaled) shear viscosity η(α)/η(1) versus the coefficient of restitution α for d = 2, ∆∗ b = 0.5, and T ∗ b = 1. Three different values of ϕ have been considered: ϕ = 0.05 (solid), ϕ = 0.1 (dashed) and ϕ = 0.2 (dotted). Here, η(1) refers to the shear viscosity coefficient for elastic collisions. 0.0 0.2 0.4 0.6 0.8 1.0 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the (scaled) bulk viscosity ηb(α)/ηb(1) ver￾sus the coefficient of restitution α for d = 2, ∆∗ b = 0.5, and T ∗ b = 1. Three different values of ϕ have been considered: ϕ = 0.05 (solid), ϕ = 0.1 (dashed) and ϕ = 0.2 (dotted). Here, ηb(1) refers to the bulk viscosity coefficient for elastic collisions. VI. SOME ILLUSTRATIVE SYSTEMS IN THE TWO–DIMENSIONAL CASE The results obtained in Sec. V provide t… view at source ↗
Figure 7
Figure 7. Figure 7: Plot of the (reduced) thermal conductivity view at source ↗
Figure 8
Figure 8. Figure 8: Plot of the (reduced) heat diffusive coefficient view at source ↗
Figure 9
Figure 9. Figure 9: Plot of the (reduced) first–order contribution to view at source ↗
Figure 10
Figure 10. Figure 10: (A): Plot of the eigenvalue M (0) 33 versus the coeffi￾cient of restitution α for d = 2, ∆∗ b = 1, T ∗ b = 0.9, and three different values of ϕ: ϕ = 0.05 (solid), ϕ = 0.1 (dashed) and ϕ = 0.2 (dotted). (B): Plot of the eigenvalue M (0) 33 versus the coefficient of restitution α for d = 2, ϕ = 0.25, T ∗ b = 0.9, and three different values of ∆∗ b: ∆∗ b = 0.5 (solid), ∆∗ b = 1 (dashed) and ∆∗ b = 1.5 (dotte… view at source ↗
read the original abstract

This paper investigates the dynamic properties of a confined quasi-two-dimensional granular fluid at moderate densities, modeled within the framework of the Enskog kinetic equation. The system is described using the so-called $\Delta$-model, which incorporates energy injection through modified collision rules, and is further extended to account for the influence of an interstitial gas via a viscous drag force and a stochastic Langevin-like term. By applying the Chapman-Enskog method, the Navier-Stokes transport coefficients and the cooling rate are derived analytically considering the leading terms in a Sonine polynomial expansion. The study focuses on steady-state conditions and examines how the combined effects of inelastic collisions and external driving influence transport properties such as the viscosity and the thermal conductivity. Theoretical predictions for the steady temperature and the kurtosis are validated against direct simulation Monte Carlo (DSMC) results, showing excellent agreement. The findings reveal that the external driving significantly alters the transport coefficients compared to dry (no gas phase) granular systems, challenging previous assumptions that neglected these effects. Additionally, a linear stability analysis demonstrates that the homogeneous steady state is stable across the explored parameter space.

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

1 major / 2 minor

Summary. The manuscript derives the Navier-Stokes transport coefficients and cooling rate for a quasi-two-dimensional granular fluid at moderate densities using the Enskog kinetic equation. The system incorporates the Δ-model for inelastic collisions with energy injection, augmented by viscous drag and a stochastic Langevin term to model interstitial gas effects. Employing the Chapman-Enskog expansion with the leading Sonine polynomial approximation, analytic expressions are obtained for viscosity, thermal conductivity, and related quantities under steady-state conditions. DSMC simulations are used to validate predictions for the homogeneous steady-state temperature and kurtosis, with reported good agreement. The authors conclude that external driving alters the transport coefficients relative to dry granular systems and that the homogeneous steady state is linearly stable.

Significance. If the derivations hold, the work is significant for providing closed-form expressions for transport in driven granular fluids that include gas-phase effects, which are often neglected in confined systems. The analytic Chapman-Enskog treatment combined with DSMC validation for the homogeneous state offers a concrete advance over purely dry granular models, and the stability analysis supports the physical relevance of the homogeneous reference state. These results could inform modeling of real quasi-2D granular flows where friction and stochastic forcing from surrounding gas are present.

major comments (1)
  1. [Results section] Results section (DSMC validation): DSMC simulations confirm only the steady temperature and kurtosis in the homogeneous state, as stated in the abstract. No direct comparisons are reported for the derived transport coefficients (shear viscosity, thermal conductivity, or cooling rate) against DSMC under driven non-equilibrium conditions such as shear or heat flux. Because the central claim is that external driving 'significantly alters' these coefficients relative to dry systems, the absence of such tests leaves the quantitative accuracy of the first-Sonine expressions unverified within the model's dynamics.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'showing excellent agreement' lacks quantitative detail (e.g., relative errors, specific values of restitution coefficient, packing fraction, or friction strength used in the DSMC runs). Adding these would strengthen the claim without altering the derivation.
  2. [Model section] Model section: The precise definition of the Δ-model collision rules and the coupling between the viscous drag coefficient and the Langevin noise strength should be stated explicitly with equations, as these choices directly enter the Enskog collision operator and the resulting transport expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and positive assessment of the significance of our work. We provide a point-by-point response to the major comment below.

read point-by-point responses

  1. Referee: [Results section] Results section (DSMC validation): DSMC simulations confirm only the steady temperature and kurtosis in the homogeneous state, as stated in the abstract. No direct comparisons are reported for the derived transport coefficients (shear viscosity, thermal conductivity, or cooling rate) against DSMC under driven non-equilibrium conditions such as shear or heat flux. Because the central claim is that external driving 'significantly alters' these coefficients relative to dry systems, the absence of such tests leaves the quantitative accuracy of the first-Sonine expressions unverified within the model's dynamics.

    Authors: We agree that our DSMC simulations are restricted to the homogeneous steady state, providing validation only for the steady temperature and kurtosis. The Navier-Stokes transport coefficients are derived analytically using the Chapman-Enskog expansion to first order in gradients, with the first Sonine polynomial approximation, around the homogeneous reference state. The claim that external driving alters the coefficients is based on comparing our analytic expressions (which include the stochastic bath and friction terms) to the corresponding expressions for the dry granular gas (without those terms). While we acknowledge that direct DSMC measurements of viscosity or thermal conductivity in driven inhomogeneous states would offer stronger numerical support, implementing such tests requires careful setup of boundary conditions or external fields compatible with the stochastic driving, which goes beyond the current scope of the manuscript. The good agreement for the homogeneous state confirms the accuracy of the collision model and the moment approximations used. We will revise the manuscript to include a brief discussion of the validation scope in the results or conclusions section. revision: partial

Circularity Check

0 steps flagged

Standard first-principles Chapman-Enskog derivation from driven Enskog equation; no reduction to inputs

full rationale

The derivation begins from the Enskog kinetic equation modified by the Δ-model collision rules plus explicit viscous drag and stochastic Langevin terms. The Navier-Stokes transport coefficients and cooling rate are obtained via the standard Chapman-Enskog expansion to first Sonine order, yielding explicit analytic expressions in terms of the model parameters (restitution coefficient, drag strength, noise intensity, density). Steady-state temperature and kurtosis are then computed from these expressions and compared to independent DSMC simulations; the transport coefficients themselves are not fitted to data nor renamed from known results. No self-citation is invoked as a load-bearing uniqueness theorem, and the central claim (altered coefficients relative to dry granular gases) follows directly from the explicit formulas rather than by construction. The approach is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Enskog kinetic equation for dense granular gases, the Δ-model for energy injection, and the leading-order Sonine approximation; these are standard or domain-specific assumptions rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Enskog kinetic equation applies to the quasi-2D confined granular fluid at moderate densities
    Invoked as the starting point for the Chapman-Enskog expansion in the abstract.
  • domain assumption Leading terms in Sonine polynomial expansion suffice for transport coefficients
    Explicitly stated as the approximation used to derive Navier-Stokes coefficients.

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