Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density

Alessandro Petrini; Lorenzo Caprini; Raphael Maire; Umberto Marini Bettolo Marconi

REVIEW 3 major objections 3 minor 1 cited by

Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density

T0 review · 3 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A transverse 'kick' at every collision is enough to determine the odd transport coefficients of a dilute chiral hard-disk gas.

desk verdict A clean, parameter-free kinetic theory of odd transport in a chiral hard-disk gas, with the torque-density prediction and low-density odd viscosity/diffusivity checks carrying the weight; the odd thermal conductivity claim is weaker than advertised. read the letter →

arxiv 2603.04273 v2 pith:JGSRDZ6B submitted 2026-03-04 cond-mat.stat-mech cond-mat.soft

Raphael Maire , Alessandro Petrini , Umberto Marini Bettolo Marconi , Lorenzo Caprini This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.soft

keywords chiralactivematteroddviscositykinetictheoryBoltzmann-EnskogequationChapman-Enskogexpansiontorquedensityhard-diskgasgranularspinners

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the minimal ingredient for odd transport in a chiral fluid is a transverse 'kick' delivered at every binary collision. From that collision rule alone, it derives closed-form expressions for the torque density, odd viscosity, odd thermal conductivity, and odd self-diffusivity of a dilute two-dimensional hard-disk gas. The predictions have no free parameters beyond the disk geometry, the restitution coefficient, and the kick amplitude, and they match event-driven molecular dynamics at low packing fraction. If correct, the work gives a microscopic derivation of the full set of parity-odd transport coefficients in a chiral fluid.

What carries the argument

The central object is the parity-breaking collision rule v'₁ = v₁ − ((1+α)/2)(v₁₂·σ̂)σ̂ − Δ σ̂⊥, with the transverse impulse Δσ̂⊥ injecting orbital angular momentum. The derivation rests on the Boltzmann–Enskog collision operator closed by molecular chaos, the Gaussian homogeneous solution with steady-state temperature T=mΔ²/(1−α²), and a Chapman-Enskog expansion whose first-order distortion functions are decomposed into longitudinal and transverse tensor sectors D,A and D⊥,A⊥. The linearized collision operator mixes the longitudinal and transverse sectors through off-diagonal matrix elements L∥⊥ and M∥⊥; these mixing elements, which vanish when parity is preserved, are what generate the odd

What would settle it

Measure odd viscosity as α→1 at fixed density (or with an external thermostat holding T finite): the theory gives η_o ∝ (1−α) while the torque density diverges as (1−α²)^−1/2. Observing η_o not tending to zero, or τ not diverging, at the approach to the elastic limit would falsify the central kinetic prediction.

Watch

Extended reading notes

Core claim

Chirality here is not built into the particles' rotation but into the collision rule: when two disks touch, velocities are updated by reversing the normal relative velocity with restitution α and adding a fixed transverse velocity Δ. The Boltzmann–Enskog equation with this rule has a Gaussian steady state with temperature T=mΔ²/(1−α²), and a Chapman–Enskog expansion around it produces a stress tensor containing an antisymmetric, homogeneous torque-density term τ=nφχ m sqrt(4πΔ²/(1−α²)) Δ and a viscous response with odd viscosity η_o=2m(1−α)Δ/[χσ(1+α)P(α)]. The same expansion gives odd thermal conductivity κ_o=8(1−α)Δ/[χσQ(α)] and odd self-diffusivity D_o=πσΔ/[2φχ(1+α)R(α)], with P,Q,R positi

Load-bearing premise

The calculation assumes that, at steady state, the single-particle velocity distribution is a Gaussian with temperature mΔ²/(1−α²) and that the first-order Chapman-Enskog distortion functions are constants; if non-Gaussian or velocity-dependent corrections are not negligible, the closed-form transport coefficients lose quantitative accuracy.

Editorial extensions

If this is right

If the central claim holds, odd viscosity, odd thermal conductivity, and odd self-diffusivity are not fitting parameters but functions of α, Δ, and packing fraction; simulations in the low-density regime confirm the functional forms.
The torque density τ enters the hydrodynamic momentum balance, so chiral collisions alone produce an antisymmetric stress that drives transverse flow and edge-type responses without any external rotation.
Odd viscosity vanishes as α→1, implying that dissipative normal collisions are necessary for odd viscosity in this model; odd self-diffusivity stays finite in the elastic limit.
The kinetic framework provides a benchmark for testing more complex chiral fluids, since any candidate microscopic model must reproduce these coefficients in the dilute hard-disk limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension is to add an external Langevin bath or damping; the paper's own perturbation argument suggests the odd-viscosity scaling at fixed temperature should become measurable and should scale as (1−α), a prediction that could be tested by thermostatting.
The dilute approximation neglects collisional-transfer contributions; at moderate density one expects antisymmetric viscous coefficients η_A and η_R to appear, and the torque density to acquire finite-density corrections beyond the Gaussian closure.
If the collision rule is realized with short-ranged soft repulsion, the qualitative form of the transport coefficients should persist; this is testable in Brownian-dynamics simulations of chiral colloids with odd interactions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuinely useful paper. The collision rule—inelastic normal bounce plus a fixed transverse kick, no spinner degrees of freedom—is minimal, and the paper turns it into closed-form predictions: torque density τ, odd viscosity η_o, odd thermal conductivity κ_o, odd self-diffusivity D_o, all explicit functions of Δ, α, φ, and χ with no fitted constants. That alone is worth having.

What it does well: the homogeneous temperature T=mΔ²/(1−α²) comes out of the same collision balance, and the torque density Eq. (6) is verified against event-driven simulations across densities—solid even where the rest of the theory degrades. Odd viscosity and odd self-diffusivity match in the dilute regime. The derivation is transparent: Boltzmann-Enskog, Gaussian homogeneous state, Chapman-Enskog with constant distortion coefficients, and the integrals are in a SymPy notebook. The authors are also unusually candid about the weak spots.

Where it is soft: the κ_o comparison is the weakest link. Appendix A3 states plainly that the Green–Kubo measurement does not yield a converged value and that systematic discrepancies remain, yet the main text says theory and simulations “agree well” (Fig. 3b). That is an overclaim, though not a hidden one. Second, the Chapman-Enskog closure sets the distortion functions to constants (Sec. IV.F). That is the standard first-Sonine approximation, and at low φ it works, but there is no check against a two-mode truncation, so the odd coefficient ratios could drift. It is a legitimate concern, not a fatal one. Third, low-α data are excluded because the system goes inhomogeneous, so the agreement is mostly tested where α is not too small; the paper discloses this.

None of this sinks the central claim: a minimal collisional model yields explicit, parameter-free odd transport coefficients with real predictive content in the dilute homogeneous regime. The κ_o section should be re-scoped and the closure tested, but the paper points the reader at both issues itself.

Recommendation: send to peer review. It deserves a serious referee. I would bring it to a reading group for the torque-density derivation alone, and I would cite it if I worked on odd transport.

Referee Report

3 major / 3 minor

Summary. The paper introduces a minimal two-dimensional hard-disk gas in which chirality is generated solely by a transverse impulse during collisions. Starting from a Boltzmann–Enskog description and a Gaussian homogeneous steady state, the authors derive hydrodynamic equations and, via a Chapman–Enskog expansion truncated at first order in gradients, obtain closed-form expressions for the homogeneous torque density (Eq. 6), odd viscosity (Eq. 9), odd thermal conductivity (Eq. 11), and odd self-diffusivity (Eq. 13). These formulas depend only on the microscopic parameters Δ, α, φ, and the equilibrium hard-disk pair correlation χ, with no free fitting parameters. The predictions are compared with event-driven molecular dynamics simulations, showing good agreement for the torque density and for the odd viscosity and odd self-diffusivity at low packing fractions. The derivation is supported by a SymPy notebook in the Supplementary Material.

Significance. If the results hold, this is a valuable contribution: it supplies one of the first microscopic kinetic-theory derivations of odd transport coefficients and a torque density in a chiral fluid, with parameter-free closed-form expressions that can serve as benchmarks for simulations and for coarse-grained chiral hydrodynamic theories. The machine-checked symbolic evaluation of the collision integrals is a notable strength, as is the direct simulation validation for several transport coefficients. The main limitations are the uncontrolled truncation of the Chapman–Enskog distortion functions and the incomplete/failed validation of one of the claimed coefficients, which currently prevent the paper from making its central 'good agreement' claim in full.

major comments (3)

[Sec. III, Eq. (11); Appendix A3] The main text states that 'the theory and simulations agree well' for the odd thermal conductivity κ_o, but Appendix A3 explicitly reports that the imposed-gradient method 'does not yield a converged value' and that the Green–Kubo evaluation shows 'systematic discrepancies between the measurements and the theory.' Thus Fig. 3(b) does not validate Eq. (11). This is a load-bearing inconsistency: the abstract and Sec. III claim agreement for κ_o, while the appendix concedes the measurement is not converged. Please either develop a reliable measurement or restate the κ_o claim as a prediction that remains to be tested, with the discrepancies quantified.
[Sec. IV.F, Eq. (43); Eqs. (48)-(50), (C23), (C39)] The assumption that the distortion functions D(c^2), D_⊥(c^2), A(c^2), A_⊥(c^2) are constants is a one-mode Galerkin truncation of the linearized collision operator. The exact solution of Eq. (42) generally contains higher Sonine modes, and all final transport coefficients are computed from matrix elements in this single-mode subspace. Since the odd coefficients are ratios of off-diagonal to diagonal matrix elements (e.g., Eq. (48)), even modest errors in either can shift the predictions substantially. No convergence check against a two- or three-Sonine truncation is reported. Please provide such a check, or an estimate of the truncation error, before claiming parameter-free quantitative agreement.
[Sec. III and Figs. 2-3; Appendix A2] The validation window is narrower than the text implies. Low-α data are omitted because the system develops inhomogeneous 'bubble' phases (Fig. 2 caption, Fig. 4 caption, Sec. IV.B), and for φ=0.1 the odd-viscosity data are missing because accessible shear rates were too large (Fig. 3(a) caption). Additionally, Fig. 5(b) shows that the homogeneous temperature deviates from its Gaussian prediction at moderate φ, which indirectly affects the torque-density prediction (Eq. (6)) through T_Gauss. The paper should state clearly which (α, φ) regions are actually tested and how deviations from molecular chaos are expected to affect the claimed agreement.

minor comments (3)

[Sec. IV.F, Eq. (43)] The symbols D, D_⊥ are used both for the tensor basis functions and for the scalar distortion coefficients (e.g., 'D(c^2)=D_0'). This makes the projection equations (44)-(47) hard to follow. Consider using different letters or explicit arguments throughout.
[Sec. III, Eq. (6) and Fig. 2] The statement that Eq. (6) agrees 'even at high densities' should be qualified by the fact that low-α data at high φ are not shown because of inhomogeneous configurations. Also, the use of the equilibrium pair correlation χ_eq should be noted as an approximation in the main text, not only in the derivation.
[Appendix A3, Eq. (A6)] The Green–Kubo formula for κ_o is applied in a nonequilibrium steady state where standard fluctuation-dissipation relations are not guaranteed. The paper itself notes this limitation; please consider moving this caveat into the main text near Eq. (11), since it directly affects the reliability of the comparison in Fig. 3(b).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: transport coefficients are derived from explicit collision-matrix integrals with no fitted parameters; approximations are stated and simulations serve as ex post checks.

full rationale

The derivation chain is self-contained. The model is specified by the collision rule (1), and all transport predictions follow from explicit kinetic-theory calculations. The steady-state temperature is obtained from the collision-energy balance, Eq. (24), and the transport coefficients are expressed through matrix elements of the linearized collision operator (Eqs. 48-50, C18-C25, C40, C43) as closed functions of the microscopic parameters Δ, α, and φ (with χ taken from the equilibrium hard-disk equation of state, not fitted). No simulation value enters the evaluation of D0, D⊥0, A0, A⊥0, B0, or B⊥0; the Chapman-Enskog ansatz and constant-coefficient truncation are explicit approximations, and the simulations are used as external checks. The admitted limitations — the single-mode truncation, the dilute-limit neglect of collisional transfer, molecular-chaos breakdown at moderate density, and the non-converged κ_o Green-Kubo measurement — weaken validation but do not make any prediction identical to its input by construction. Self-citations appear only for context (e.g., Refs. 25, 42, 43, 45) and are not load-bearing for the derived formulas. No circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to simulation data; Δ, α, and φ are model control parameters, and χ is taken from equilibrium hard-disk theory. The central formulas rest on four modeling approximations: molecular chaos, Gaussian closure with derived temperature, dilute collision locality, and first-order Chapman-Enskog with constant distortion coefficients. These are the assumptions a reader must grant before accepting the closed-form transport coefficients.

assumptions (5)

domain assumption Molecular chaos closure f2 = χ f1 f1 (Sec. IV A, Eq. 17)
Neglects velocity correlations at collision; the paper's own Fig. 5(b) shows this fails at moderate densities.
domain assumption Gaussian ansatz for f^(0) with T = mΔ²/(1−α²) (Sec. IV B-C, Eqs. 21 and 24)
Used to evaluate all kinetic integrals; Sonine corrections are small but non-zero and are neglected.
domain assumption Dilute-limit approximation: collisions occur at the same point and Enskog collisional transfer is neglected (Sec. IV D, Eq. 31)
Limits validity to φ ≪ 1 and excludes collisional transfer from viscous stress, though the homogeneous torque density is retained separately.
domain assumption Chapman-Enskog truncation at first order with constant distortion coefficients D0, D⊥0, A0, A⊥0 (Sec. IV F, Eq. 43)
Assumes hydrodynamic fields vary slowly and that velocity dependence of the perturbed distribution is negligible; standard but approximate.
domain assumption Pair correlation at contact is approximated by the equilibrium hard-disk value χ ≈ χ_eq (Sec. III, Eq. 6; Sec. IV C)
Equilibrium equation-of-state input used for a nonequilibrium collision model; no nonequilibrium correction is included.

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Cite this review

Pith. "Pith review of Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density." pith.science (2026). https://pith.science/paper/JGSRDZ6B

@misc{pith2026260304273,
  author       = {Pith},
  title        = {Pith review of: Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JGSRDZ6B}},
  note         = {Machine review of arXiv:2603.04273}
}

read the original abstract

Parity-odd transport is a central signature of chiral fluids, yet analytical predictions are sparse. Here, we introduce a minimal two-dimensional hard-disk gas in which chirality arises solely from a collision-induced transverse impulse. Motivated by granular spinners, collisions are dissipative and inject orbital angular momentum through a fixed tangential ``kick'' at contact. Starting from a Boltzmann-Enskog description, we derive nonlinear hydrodynamic equations for density, momentum, and temperature, and show that chirality generates an antisymmetric homogeneous stress corresponding to a nonzero torque density. In the dilute limit, a Chapman-Enskog expansion yields analytical predictions for transport coefficients, including odd viscosity, odd thermal conductivity, and odd self-diffusivity, in good agreement with numerical simulations. This minimal kinetic model can serve as a foundation for systematic coarse-graining of chiral fluids and as a tractable benchmark for gaining insight into odd transport across a broader class of chiral systems.

Figures

Figures reproduced from arXiv: 2603.04273 by the authors.

**Figure 2.** FIG. 2. Comparison between numerical measurements and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Odd transport coefficients as functions of the restitution coefficient [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Comparison of the steady-state velocity probability distribution [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Homogeneous temperature [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Even transport coefficients as functions of the restitution coefficient [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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Forward citations

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Reference graph

Works this paper leans on

16 extracted references · 1 linked inside Pith · cited by 1 Pith paper

[1]

collides

Overall, the theory and simulations agree well, as seen in Fig. 3(b), with the low-density discrepancies most likely due to systematic measurement errors as will be explained later. Before deriving these results, we briefly comment on transport coefficients that do not enter the hydrodynamic equations [Eqs. (4)]. Since the fluid velocity is conserved and ...
[2]

Stress and heat current in event-driven molecular dynamics The stress and heat current are generically given by a kinetic part plus an impulsive collisional contribution. For hard-disks with instantaneous collisions, their micro- scopic expressions read 127 Π(t) =− m L2 NX α=1 cα(t)⊗c α(t) + σ L2 X α<β ˆσαβ(t)⊗I αβ(t)δ t−t coll αβ , J(t) = + m 2L2 NX α=1 ...
[3]

This introduces a time-dependent shear offsetδx(t) = ˙γtL y between adjacent replicas

Viscosities The viscosities are measured under a controlled sim- ple shear of rate ˙γimposed via Lees-Edwards boundary conditions129 in a box of sizeL x ×L y where the peri- odic images above and below the simulation cell slide along ˆxwith velocities±˙γL y/2 (equivalently, a relative velocity ˙γLy between the top and bottom images). This introduces a tim...
[4]

However, a gradient generated by fixingTat two slabs (e.g., via a M¨ uller- Plathe scheme132) is screened by bulk relaxation due to energy non-conservation during collision

Conductivities It is possible to measure the thermal conductivities by imposing a thermal gradient. However, a gradient generated by fixingTat two slabs (e.g., via a M¨ uller- Plathe scheme132) is screened by bulk relaxation due to energy non-conservation during collision. Therefore, it is easier to impose the thermal gradient by directly fixing the tempe...

2000
[5]

29: D= lim t→∞ D x(t)−x(0) vx(0) + y(t)−y(0) vy(0) E 2 , Do = lim t→∞ D x(t)−x(0) vy(0)− y(t)−y(0) vx(0) E 2

Self-diffusivities The ordinary and odd self-diffusivities are obtained from the velocity-displacement correlations introduced in Ref. 29: D= lim t→∞ D x(t)−x(0) vx(0) + y(t)−y(0) vy(0) E 2 , Do = lim t→∞ D x(t)−x(0) vy(0)− y(t)−y(0) vx(0) E 2 . (A7) Unlike stress or heat current autocorrelations in hard- particle systems, these estimators are free of imp...
[6]

= 0, (B8) and, forp= 2, m∆2 m∆2 +T(2α 2 + 7) +T 2(α2 −1)(2α 2 + 7) − a2 16 T 2(1 +α) 30α2(α−1) + 177α−209 − m∆2 m∆2 −T(6α 2 + 29) +O(a 2
[7]

(B9) Solving these two equations determinesT(α, m∆ 2) and a2(α, m∆2) at this order, leading to the theoretical prediction given in Fig

= 0. (B9) Solving these two equations determinesT(α, m∆ 2) and a2(α, m∆2) at this order, leading to the theoretical prediction given in Fig. 5. The solution is density- independent because the sole density dependence enters throughχ, which reduces to a constant prefactor inJfor homogeneous states. Other closures are possible (e.g., usingp= 1 andp= 3 to ge...
[8]

Derivation of the hydrodynamics equations To derive the hydrodynamics equations forn,uand T[Eqs. (4)], we start from the Boltzmann-Enskog equa- tion (14): ∂tf(r,v, t) +v·∇f(r,v, t) =J(r,v|f, f).(C1) For any test functionA(v), multiplying byAand inte- grating overvgives ∂t Z Af dv+∇· Z Avf dv= Z AJdv.(C2) Density—WithA= 1, mass conservation by collisions i...

Show all 16 references

[9]

Kinetic integrals We repeatedly encounter matrix elements of the lin- earized collision operator of the form ⟨ϕ f (0) −1 ,L[ψ]⟩ ≡ Z ϕ(c1)L[ψ](c1)dc1 =χσ Z Θ(−c12 · ˆσ12)|c12 · ˆσ12|δϕ ×ψ(c 2)f (0)(c1)f (0)(c2)dc1dc2d ˆσ12 , (C7) 16 for anyϕandψ, and where: δϕ=ϕ(c ′
[10]

These integrals simplify in center-of-massC= (c 1 + c2)/2 and relativec 12 =c 1 −c 2

+ϕ(c ′ 2)−ϕ(c 1)−ϕ(c 2),(C8) is the change of any quantityϕ(c 1) +ϕ(c 2) at collision. These integrals simplify in center-of-massC= (c 1 + c2)/2 and relativec 12 =c 1 −c 2. Using the collision rule (1): c1 =C+ 1 2 c12 , c2 =C− 1 2 c12 , c′ 1 =C+ 1 2 c12 − 1 +α 2 (c12 · ˆσ12) ˆ...
[11]

(C15) Using Eq

Viscosities For the viscosities, we need to compute: L∥∥ =L ⊥⊥ ≡ D f (0) −1 Dij,L[D ij] E , L∥⊥ =−L ⊥∥ ≡ D f (0) −1 Dij,L[D ⊥ ij] E . (C15) Using Eq. (C7), this gives L∥∥ =χσ Z Θ(−c12 · ˆσ)(−c12 · ˆσ)f (0)(c1)f (0)(c2) ×D ij(c2)δ(Dij)dc1dc2d ˆσ, L∥⊥ =χσ Z Θ(−c12 · ˆσ)(−c12 · ˆ...
[12]

After inserting Eq

+ (c⊥ 2 ·c ′ 2)(c2 ·c ′ 2), (C17) withc ⊥ =ε·c. After inserting Eq. (C9) and carrying out the Gaussian integrations, we obtain (see SM) L∥∥ =−χσn 2T 2 r πT m m∆2 T + (1 +α)(7−3α) , L∥⊥ = 2πχσn 2T 2∆(1−α). (C18) From Eq. (53), we have: ηs =−4n 2T 3 L∥∥ L2 ∥∥ +L 2 ∥⊥ , ηo = 4n2T...
[13]

Projecting the thermal part of Eq

Thermal conductivities Similarly to the viscosity calculation, the thermal sec- tor requires the matrix elements Mαβ ≡ D f (0) −1 A(α) i ,L[A (β) i ] E , α, β∈ {∥,⊥}, (C21) withA (∥) i ≡A i andA (⊥) i ≡A ⊥ i . Projecting the thermal part of Eq. (42) onto{A,A ⊥}gives 1 T 4nT 3/...
[14]

+c ′2 2 (c2 ·c ′ 2) −c 2 1(c2 ·c 1)−c 2 2(c2 ·c 2) i , A⊥ i (c2)δ(Ai) m/2 = mc2 2 2 −2T h c′2 1 (c⊥ 2 ·c ′
[15]

(C24) Upon integration, we obtain (see SM) M∥∥ =− √πT 5/2χn2σ 2m3/2 T(1 +α)(19−15α)−3m∆ 2 , M∥⊥ =πχσn 2T 3∆(1−α)/m

+c ′2 2 (c⊥ 2 ·c ′ 2) −c 2 1(c⊥ 2 ·c 1)−c 2 2(c⊥ 2 ·c 2) i . (C24) Upon integration, we obtain (see SM) M∥∥ =− √πT 5/2χn2σ 2m3/2 T(1 +α)(19−15α)−3m∆ 2 , M∥⊥ =πχσn 2T 3∆(1−α)/m . (C25) To relateMto the conductivities, we use the heat cur- rent J= m 2 Z c2cf(c)dc =µ m 2T Z c2cf ...
[16]

Chirality in microswimmer motion: From circle swimmers to active turbulence,

Self-diffusion coefficient We compute the regular self-diffusivityDand its odd counterpartD o from the Boltzmann-Lorentz equation for a tracer following Ref. 51: ∂tfs(r,v, t) +v·∇f s(r,v, t) =J dilute r,v|f s, fB , (C31) wheref B =f (0) is a fixedhomogeneousGaussian and onlyf ...

2016 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

collides

Overall, the theory and simulations agree well, as seen in Fig. 3(b), with the low-density discrepancies most likely due to systematic measurement errors as will be explained later. Before deriving these results, we briefly comment on transport coefficients that do not enter the hydrodynamic equations [Eqs. (4)]. Since the fluid velocity is conserved and ...

[2] [2]

Stress and heat current in event-driven molecular dynamics The stress and heat current are generically given by a kinetic part plus an impulsive collisional contribution. For hard-disks with instantaneous collisions, their micro- scopic expressions read 127 Π(t) =− m L2 NX α=1 cα(t)⊗c α(t) + σ L2 X α<β ˆσαβ(t)⊗I αβ(t)δ t−t coll αβ , J(t) = + m 2L2 NX α=1 ...

[3] [3]

This introduces a time-dependent shear offsetδx(t) = ˙γtL y between adjacent replicas

Viscosities The viscosities are measured under a controlled sim- ple shear of rate ˙γimposed via Lees-Edwards boundary conditions129 in a box of sizeL x ×L y where the peri- odic images above and below the simulation cell slide along ˆxwith velocities±˙γL y/2 (equivalently, a relative velocity ˙γLy between the top and bottom images). This introduces a tim...

[4] [4]

However, a gradient generated by fixingTat two slabs (e.g., via a M¨ uller- Plathe scheme132) is screened by bulk relaxation due to energy non-conservation during collision

Conductivities It is possible to measure the thermal conductivities by imposing a thermal gradient. However, a gradient generated by fixingTat two slabs (e.g., via a M¨ uller- Plathe scheme132) is screened by bulk relaxation due to energy non-conservation during collision. Therefore, it is easier to impose the thermal gradient by directly fixing the tempe...

2000

[5] [5]

29: D= lim t→∞ D x(t)−x(0) vx(0) + y(t)−y(0) vy(0) E 2 , Do = lim t→∞ D x(t)−x(0) vy(0)− y(t)−y(0) vx(0) E 2

Self-diffusivities The ordinary and odd self-diffusivities are obtained from the velocity-displacement correlations introduced in Ref. 29: D= lim t→∞ D x(t)−x(0) vx(0) + y(t)−y(0) vy(0) E 2 , Do = lim t→∞ D x(t)−x(0) vy(0)− y(t)−y(0) vx(0) E 2 . (A7) Unlike stress or heat current autocorrelations in hard- particle systems, these estimators are free of imp...

[6] [6]

= 0, (B8) and, forp= 2, m∆2 m∆2 +T(2α 2 + 7) +T 2(α2 −1)(2α 2 + 7) − a2 16 T 2(1 +α) 30α2(α−1) + 177α−209 − m∆2 m∆2 −T(6α 2 + 29) +O(a 2

[7] [7]

(B9) Solving these two equations determinesT(α, m∆ 2) and a2(α, m∆2) at this order, leading to the theoretical prediction given in Fig

= 0. (B9) Solving these two equations determinesT(α, m∆ 2) and a2(α, m∆2) at this order, leading to the theoretical prediction given in Fig. 5. The solution is density- independent because the sole density dependence enters throughχ, which reduces to a constant prefactor inJfor homogeneous states. Other closures are possible (e.g., usingp= 1 andp= 3 to ge...

[8] [8]

Derivation of the hydrodynamics equations To derive the hydrodynamics equations forn,uand T[Eqs. (4)], we start from the Boltzmann-Enskog equa- tion (14): ∂tf(r,v, t) +v·∇f(r,v, t) =J(r,v|f, f).(C1) For any test functionA(v), multiplying byAand inte- grating overvgives ∂t Z Af dv+∇· Z Avf dv= Z AJdv.(C2) Density—WithA= 1, mass conservation by collisions i...