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arxiv: 2605.18216 · v1 · pith:ARS74LLWnew · submitted 2026-05-18 · ⚛️ physics.flu-dyn

A discrete Boltzmann model with state-dependent power-law relaxation time for nonequilibrium transport in compressible flows

Demei Li , Zhongyi He , Huilin Lai , Yanbiao Gan , Hailong Liu , Pengfei Lin This is my paper

Pith reviewed 2026-05-20 00:30 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords discrete Boltzmann modelpower-law relaxation timenonequilibrium transportcompressible flowsshock wavesviscous stressheat fluxSod shock tube
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The pith

A discrete Boltzmann model with power-law relaxation time depending on density and temperature recovers nonequilibrium transport across shocks and discontinuities.

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

The paper introduces a discrete Boltzmann model called DTRT-DBM that makes the relaxation time depend on local density and temperature through a power law. This change allows the model to account for how nonequilibrium effects vary across different parts of a flow, something constant relaxation time models miss. Tests on the Sod shock tube problem show that the model correctly reproduces both the overall flow patterns like shocks and rarefactions and the detailed nonequilibrium measures such as viscous stresses and heat fluxes. By varying the power-law exponents, the authors map out how these nonequilibrium quantities behave and find that density and temperature exponents play different roles depending on whether density or temperature changes are stronger.

Core claim

The DTRT-DBM incorporates a relaxation time of the form τ = τ₀ (ρ/ρ₀)^a (T/T₀)^b. This formulation extends the discrete Boltzmann framework to flows with spatially varying nonequilibrium intensity. Validation against the Sod shock tube and analytical solutions for viscous stress and heat flux shows that the model recovers macroscopic wave structures and nonequilibrium quantities across shock waves, rarefaction waves, and contact discontinuities. Phase diagrams of viscous stress and heat flux reveal exponential dependence of extrema on a and b, with asymmetric sensitivity: more to a for density-dominated gradients and more to b for temperature-dominated ones.

What carries the argument

The state-dependent relaxation time τ=τ₀(ρ/ρ₀)^a(T/T₀)^b inside the discrete Boltzmann collision operator, which adjusts the rate at which the distribution relaxes toward equilibrium according to local density and temperature.

Load-bearing premise

A simple power-law dependence on density and temperature alone is enough to represent how nonequilibrium intensity changes from place to place in the flows examined.

What would settle it

Measurements or high-resolution simulations of viscous stress and heat flux in a compressible flow featuring strong chemical reactions or ionization, where the model's predictions would deviate noticeably from the data if the power-law form fails to capture the true state dependence.

Figures

Figures reproduced from arXiv: 2605.18216 by Demei Li, Hailong Liu, Huilin Lai, Pengfei Lin, Yanbiao Gan, Zhongyi He.

Figure 1
Figure 1. Figure 1: Discrete velocity set of the D2V25 model. spatial derivatives are computed with a fifth-order weighted essentially non-oscillatory (WENO) finite-difference scheme to suppress spurious oscillations near steep gradients (Jiang & Shu 1996), while temporal integration is performed with a second-order implicit– explicit Runge–Kutta scheme (Ascher et al. 1997). The accuracy and efficiency of DBM also depend crit… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the DTRT-DBM results with the exact Riemann solution for the Sod shock tube at t = 0.1 with a = 10 and b = 10: (a) density, (b) pressure, (c) velocity, and (d) temperature. 0.30 0.15 0.00 0.15 0.30 0.0 0.5 1.0 0.30 0.15 0.00 0.15 0.30 0.0 0.5 1.0 0.30 0.15 0.00 0.15 0.30 0.0 0.5 1.0 0.30 0.15 0.00 0.15 0.30 0.5 1.0 1.5 DBM Exact r x (b) P x u x x (a) T x (c) (d) [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the DTRT-DBM results with the exact Riemann solution for the Sod shock tube at t = 0.1 with a = −3 and b = −3: (a) density, (b) pressure, (c) velocity, and (d) temperature. Riemann solution near the discontinuities, as shown in figures 3(a,b). More pronounced deviations appear in the velocity and temperature profiles in figures 3(c,d), where both the shock front and the contact discontinuity … view at source ↗
Figure 4
Figure 4. Figure 4: Profile of the local Knudsen numbers calculated from T, ρ and P along the line y = 0.5Ly at t = 0.1 for a = −3 and b = −3. relaxation time is τ = τ0P −3 . In the initial right state, P = 0.1, and hence τ = 103 τ0. In low-pressure regions, the local state causes τ to increase substantially, thereby enhancing nonequilibrium transport and smoothing the flow field. By contrast, in high-pressure regions, τ decr… view at source ↗
Figure 5
Figure 5. Figure 5: Profiles of viscous stress along the centreline y = 0.5Ly at t = 0.04 for different parameter combinations (a = −1, 0, 1.5, 2; b = −2, −1, 1, 3). DTRT-DBM results are compared with the corresponding analytical solutions. 3.3.1. Viscous stress To verify the accuracy of the DTRT-DBM in describing viscous stress, we prescribe the initial conditions as ρ(x, y) = ρL + ρR 2 − ρL − ρR 2 tanh  x − (Nx∆x)/2 Lρ  ,… view at source ↗
Figure 6
Figure 6. Figure 6: Profiles of ∆∗ 2xx along the centreline y = 0.5Ly at t = 0.04 for four representative cases: (a) a = −1, b = −2; (b) a = 0, b = 5; (c) a = 2, b = −1; and (d) a = 2, b = 4. Parameter Combination RTNE Relative Nonequilibrium State in the System a = −1, b = −2 0 Zero Relative Nonequilibrium a = 0, b = 5 0.15 Weak Relative Nonequilibrium a = 2, b = −1 0.5 Moderate Relative Nonequilibrium a = 2, b = 4 1.1 Stron… view at source ↗
Figure 7
Figure 7. Figure 7: Profiles of heat flux along the centreline y = 0.5Ly at t = 0.0075 for different parameter combinations (a = −1, 0, 1.5, 2; b = 0, 1.5, 4, 5). DTRT-DBM results are compared with the corresponding analytical solutions. nonequilibrium state. These results show how different parameter combinations are reflected in the nonequilibrium state of the system. 3.3.2. Heat flux To verify the accuracy of the model in … view at source ↗
Figure 8
Figure 8. Figure 8: Profiles of ∆∗ 3,1x along the centreline y = 0.5Ly at t = 0.0075 for four representative cases: (a) a = −1, b = 1; (b) a = −1, b = 4; (c) a = 2, b = 1.5; and (d) a = 2, b = 5. (a = 2, b = 1.5), their effects on RTNE become competitive: the larger parameter enhances nonequilibrium, whereas the smaller one suppresses it. The system then remains in a weak-to-moderate nonequilibrium state, with the effect of b… view at source ↗
Figure 9
Figure 9. Figure 9: Phase diagram of the extremum of total viscous stress as a function of b for different values of a. Brown solid lines denote linear fits in the first-order-TNE-dominated regime, whereas red dashed lines denote linear fits in the second-order-TNE-dominated regime. 4.1. Viscous stress 4.1.1. Phase diagram of the total viscous stress [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Phase diagrams of the logarithms of the extrema of first- and second-order viscous stresses as functions of b for different values of a: (a) first-order viscous stress; (b) second-order viscous stress. Lines denote linear fits. 4.1.2. Phase diagrams of the first- and second-order viscous stresses [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Variation with a of the linear-fit slopes for the extrema of viscous stresses of different orders, namely k1, k2, k1,1, and k2,1. (ii) k2,1 decreases overall as a increases. When a < 0, the system is dominated by first￾order nonequilibrium and the second-order contribution is nearly zero. Once a becomes positive, the second-order contribution is activated. As a continues to increase, second￾order TNE beco… view at source ↗
Figure 12
Figure 12. Figure 12: Phase diagram of the extremum of total heat flux as a function of a for different values of b. Brown solid lines denote linear fits in the first-order-TNE-dominated regime, whereas red dashed lines denote linear fits in the second-order-TNE-dominated regime. second-order nonequilibrium, the second-order term carries exponents 2a and 2b, so one expects k2 ≈ 2k1 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Phase diagrams of the extrema of first- and second-order heat fluxes as functions of a for different values of b: (a) first-order heat flux; (b) second-order heat flux. Lines denote linear fits. (i) Combined and competing effects of a and b Taking (a, b) = (0, 0) as the reference point, figure 12 shows that the nonequilibrium intensity exhibits distinct behaviour in the four quadrants. In the third quadra… view at source ↗
Figure 14
Figure 14. Figure 14: Variation with b of the linear-fit slopes for the extrema of heat fluxes of different orders, namely k1, k2, k1,1, k2,1, and k2,2. that increasing a accelerates the growth of second-order heat flux and progressively makes it dominant. (iii) Coupling effects and the decrease in the growth rate of total heat flux Together with figures 12 and 13(a,b), these results show that the staged behaviour of total hea… view at source ↗
read the original abstract

Thermodynamic nonequilibrium effects play a central role in momentum and energy transport in compressible flows. In conventional BGK kinetic models, the relaxation time $\tau$ is taken as a constant, which neglects the dependence of the relaxation process on local macroscopic states. To overcome this limitation, we develop a discrete Boltzmann model with a density- and temperature-dependent power-law relaxation time, termed DTRT-DBM, in which $\tau=\tau_0(\rho/\rho_0)^a(T/T_0)^b$. This formulation extends the discrete Boltzmann framework to flows with spatially varying nonequilibrium intensity. The model is validated by the Sod shock tube and by analytical solutions for viscous stress and heat flux, demonstrating accurate recovery of both macroscopic wave structures and nonequilibrium quantities across shock waves, rarefaction waves, and contact discontinuities. On this basis, phase diagrams of viscous stress and heat flux are constructed to examine how these quantities depend on the power-law exponents $a$ and $b$. The extrema of these quantities depend exponentially on the model parameters and exhibit regime-dependent behaviour. The roles of $a$ and $b$ are not symmetric: the nonequilibrium response is more sensitive to $a$ when density gradients dominate, but more sensitive to $b$ when temperature gradients dominate. Within the parameter range and flow configurations examined here, higher-order viscous stress increases the growth rate of the total viscous-stress extremum, whereas higher-order heat flux reduces the growth rate of the total heat-flux extremum. These results show that the proposed model can capture different higher-order nonequilibrium responses in compressible flows and provides a framework for the modelling and analysis of multiscale nonequilibrium processes.

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 / 3 minor

Summary. The manuscript develops a discrete Boltzmann model (DTRT-DBM) that replaces the constant relaxation time of standard BGK-type models with a power-law form τ = τ₀(ρ/ρ₀)^a (T/T₀)^b. The model is tested on the Sod shock-tube problem and against analytical expressions for viscous stress and heat flux; the authors report accurate recovery of both macroscopic wave structures and nonequilibrium moments across shocks, rarefactions, and contacts. Phase diagrams are constructed to map the dependence of extremal viscous stress and heat flux on the exponents a and b, revealing asymmetric sensitivity (a more influential under density gradients, b under temperature gradients) and differing effects of higher-order moments.

Significance. If the variable-τ formulation is shown to be consistent with the discrete velocity discretization and the cited analytical benchmarks, the work would supply a compact, tunable extension of discrete Boltzmann methods for flows with spatially varying nonequilibrium intensity. The phase-diagram analysis supplies concrete, falsifiable predictions about how nonequilibrium extrema scale with a and b.

major comments (2)
  1. [Model formulation and Chapman-Enskog recovery] § on model formulation and Chapman-Enskog recovery: the power-law τ(ρ,T) necessarily produces additional terms proportional to ∇τ (and therefore to ∇ρ and ∇T) in the first-order stress and heat-flux expressions. These terms are absent from the constant-τ analytical solutions used for validation. The manuscript must either derive the modified transport coefficients explicitly or demonstrate that the discrete velocity set and collision operator cancel the extra contributions for arbitrary a,b; otherwise the reported agreement with analytical viscous stress/heat flux is limited to the hydrodynamic limit or to special (a,b) pairs.
  2. [Validation against analytical viscous stress and heat flux] Validation against analytical viscous stress and heat flux (Sod-shock-tube section): the strongest claim—that the model recovers nonequilibrium quantities across discontinuities—rests on direct comparison with analytical expressions that assume constant τ. Without an accompanying Chapman-Enskog verification that includes the ∇τ contributions, the quantitative match cannot be taken as general evidence for the power-law form.
minor comments (3)
  1. Specify the numerical values and ranges of a and b examined in the phase diagrams, together with any physical constraints (e.g., positivity of τ, realizability of moments).
  2. Add quantitative error measures (L2 norms, maximum deviations) for the Sod-shock-tube profiles rather than qualitative statements of agreement.
  3. Clarify whether the discrete velocity set is the same as in prior DBM papers or modified to accommodate the variable-τ operator.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. The points raised regarding the Chapman-Enskog recovery and validation are well-taken, and we have prepared detailed responses and revisions to address them.

read point-by-point responses

  1. Referee: [Model formulation and Chapman-Enskog recovery] the power-law τ(ρ,T) necessarily produces additional terms proportional to ∇τ (and therefore to ∇ρ and ∇T) in the first-order stress and heat-flux expressions. These terms are absent from the constant-τ analytical solutions used for validation. The manuscript must either derive the modified transport coefficients explicitly or demonstrate that the discrete velocity set and collision operator cancel the extra contributions for arbitrary a,b; otherwise the reported agreement with analytical viscous stress/heat flux is limited to the hydrodynamic limit or to special (a,b) pairs.

    Authors: We agree that the variable relaxation time introduces additional terms involving ∇τ in the Chapman-Enskog expansion. In the revised manuscript, we will derive the modified transport coefficients explicitly, accounting for these contributions. We will also examine whether the specific discrete velocity discretization leads to any cancellations for the chosen power-law form. This will provide a more rigorous foundation for the validation against analytical expressions. revision: yes

  2. Referee: [Validation against analytical viscous stress and heat flux] the strongest claim—that the model recovers nonequilibrium quantities across discontinuities—rests on direct comparison with analytical expressions that assume constant τ. Without an accompanying Chapman-Enskog verification that includes the ∇τ contributions, the quantitative match cannot be taken as general evidence for the power-law form.

    Authors: We acknowledge the limitation in the original validation. We will include a detailed Chapman-Enskog analysis that incorporates the ∇τ terms in the revised manuscript. The comparisons in the Sod shock-tube section will be updated to use the corrected analytical expressions for viscous stress and heat flux, thereby providing stronger evidence for the power-law relaxation time model. revision: yes

Circularity Check

0 steps flagged

No significant circularity; modeling choice validated against external benchmarks

full rationale

The paper introduces the state-dependent relaxation time τ=τ₀(ρ/ρ₀)^a(T/T₀)^b explicitly as a modeling extension to conventional constant-τ BGK models, rather than deriving it from prior equations or fitting it to the target nonequilibrium quantities. Validation proceeds via direct comparison to the independent Sod shock-tube problem and to analytical expressions for viscous stress and heat flux; these benchmarks lie outside the model construction and are not obtained by re-expressing the same inputs. No load-bearing step reduces a claimed prediction to a self-citation chain, a fitted parameter renamed as output, or a definitional equivalence. The derivation chain is therefore self-contained against external references.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central modeling step introduces two free exponents a and b whose values are not derived from first principles but chosen to explore sensitivity; the underlying discrete Boltzmann framework and BGK-style collision operator are taken from prior literature.

free parameters (2)
  • a
    Power-law exponent controlling density dependence of relaxation time; explored via phase diagrams but not derived.
  • b
    Power-law exponent controlling temperature dependence of relaxation time; explored via phase diagrams but not derived.
axioms (1)
  • domain assumption BGK-type single-relaxation-time collision operator remains valid when relaxation time is made state-dependent.
    Standard assumption of the discrete Boltzmann framework invoked to justify the model extension.

pith-pipeline@v0.9.0 · 5858 in / 1284 out tokens · 37370 ms · 2026-05-20T00:30:25.453346+00:00 · methodology

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