Pith. sign in

REVIEW 4 major objections 5 minor 4 cited by

A collisional model of odd fluids: from Boltzmann equation to chiral hydrodynamics

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A dilute gas of rough, inelastic disks under constant torque is proposed as a minimal microscopic model of an odd fluid, with all odd response coefficients scaling as the square root of the torque.

desk verdict A serious kinetic-theory paper that probably supplies the first collisional microscopic route to odd-fluid hydrodynamics, but the quantitative coefficient plots are not yet supported in the low-tangential-restitution regime where their own slow/fast split diagnostics break down. read the letter →

arxiv 2508.12944 v1 pith:FBRP5PIT submitted 2025-08-18 cond-mat.soft cond-mat.stat-mech

Ege Eren , Michel Fruchart , Vincenzo Vitelli This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mech
keywords oddviscositychiralactivemattergranulargasBoltzmannkinetictheoryadiabaticeliminationroughinelasticdiskshydrodynamictransportcoefficientsnon-equilibriumsteadystate
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

This paper tries to establish that a dilute gas of rough, inelastic disks driven by a constant torque is a minimal microscopic model of an odd fluid: a fluid whose transverse responses, such as odd viscosity, come from collisions rather than from single-particle chiral motion. Starting from the Boltzmann equation with a Gaussian ansatz for the non-equilibrium steady state, the paper derives hydrodynamic equations for density, linear momentum, angular momentum, and translational and rotational temperatures, then uses adiabatic elimination to compute all relaxation and transport tensors numerically. Its central results are that the response tensors contain many odd terms, that all coefficients scale as the square root of the driving torque, and that some odd coefficients change sign as the tangential restitution coefficient varies while the torque direction stays fixed. A working first-principles collisional route from microscopic dynamics to chiral hydrodynamics would let experiments connect measurable odd responses directly to microscopic friction parameters.

What carries the argument

The load-bearing object is the linearized Boltzmann operator $L = C - d\,\partial_\omega \phi - d\,\partial_\omega(\log f^{(0)})\,\phi$ together with its adjoint $L^\dagger$, whose eigenvalue spectrum selects the slow subspace. In Fourier space the operator takes the form $\partial_t\phi = L(q)\phi$ with $L(q) = -i q_i c_i + L$; projecting with $P$ onto the slow modes and $Q = 1 - P$ onto fast modes, adiabatic elimination assumes $\partial_t\phi_F \approx 0$, producing $\phi_F = -L_{FF}^{-1}(i q c_i^{FS} - L_{FS})\phi_S$ and an effective slow-space operator $L_{\mathrm{eff}}$. The inverse fast-block operator $L_{FF}^{-1}$ is what converts fast relaxation into the transport and relaxation coefficients, so the fast/slow split is the mechanism that carries the argument: every computed odd coefficient inherits its structure from the non-Hermitian parts of this projection.

What would settle it

Run the linearized spectral analysis without truncating to the five slow fields at small tangential restitution and look for eigenvalues with decay rates comparable to the temperature modes; if such modes exist, the inversion of $L_{FF}$ changes and the coefficients would differ. In parallel, a direct molecular-dynamics measurement of an odd transport coefficient, such as the shear stress induced by a uniform rotation rate, at two torque values would settle the predicted $\sqrt{|d|}$ scaling and the predicted sign change as the tangential restitution is varied.

Watch

Extended reading notes

Core claim

The central claim is that chiral collisions alone, combined with a scalar angular drive, are enough to produce the full phenomenology of odd fluids. The paper exhibits this in a two-dimensional granular gas of rough hard disks; the collision rule couples translational and rotational velocities at contact and dissipates energy, while a constant external torque injects angular momentum. In the dilute limit the system is described by a non-Hermitian linearized Boltzmann operator whose adjoint has a nullspace spanned by the conserved quantities; the paper identifies the hydrodynamic variables as the slow eigenvectors separated by a spectral gap, and closes the equations by adiabatically eliminating all fast modes. The resulting relaxation and transport tensors contain odd components in the driving field, including odd viscosity and odd thermal conductivity, every coefficient carries a common $\sqrt{|d|}$ factor, and numerical evaluation over the tangential restitution coefficient shows that some odd coefficients cross zero even though the torque direction is unchanged. The paper also argues that when the spectral gap closes, additional slow variables beyond the five standard hydrodynamic fields are needed for a predictive description.

Load-bearing premise

The whole coefficient table rests on assuming the driven gas reaches a steady state that is Gaussian in velocities and shifted Gaussian in angular velocities, and that five hydrodynamic fields are slow for all parameter values, even where the paper's own spectra show additional modes that barely decay.

Editorial extensions

If this is right

  • A tabletop gas of rough spinning disks becomes an experimentally accessible collisional model in which odd viscosity and odd thermal conductivity arise from binary collisions, with no chiral force on the center of mass.
  • Since every response coefficient scales as $\sqrt{|d|}$, doubling the torque changes all transport signatures by a common factor of $\sqrt{2}$; measuring that scaling isolates the odd sector.
  • Changing the tangential restitution coefficient can flip the sign of some odd coefficients without reversing the drive, so surface roughness provides a tuning knob for the Hall response.
  • The spectral-gap criterion gives a concrete diagnostic: when near-zero eigenvalues appear in the adjoint Boltzmann spectrum, five-field hydrodynamics must be extended with extra slow variables, which the paper shows happens at small tangential restitution.
  • The same Boltzmann-plus-adiabatic-elimination pipeline applies to other collisional models and to three dimensions with only minor changes.

Reading between the lines

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

  • A direct extension would replace the Gaussian steady-state ansatz with a numerical solution of the homogeneous Boltzmann equation; if slow cumulants of the angular velocity contribute, the coefficient table should be recomputed with a non-Gaussian steady state.
  • The predicted sign changes occur near small tangential restitution, the same region where the paper's own spectra show near-zero modes and molecular dynamics shows skewed angular-velocity distributions; a plausible reading is that some sign changes may signal missing slow variables rather than a purely hydrodynamic effect.
  • Testing the predicted $\sqrt{|d|}$ scaling of an odd shear stress in microscopic simulation at two torque magnitudes would settle whether the Gaussian closure captures the steady state well enough for quantitative predictions.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a dilute granular gas of rough, inelastic hard disks driven by a constant external torque as a minimal microscopic model of an odd fluid. The authors write down the corresponding Boltzmann equation, introduce a Gaussian non-equilibrium steady-state ansatz, derive evolution equations for the five hydrodynamic fields (density, two linear momenta, angular momentum, and two temperatures), and then use adiabatic elimination on the linearized Boltzmann operator to compute relaxation and transport coefficients, including odd (transverse) contributions. The main quantitative results are (i) all response coefficients scale as the square root of the torque magnitude and (ii) some odd coefficients change sign as the tangential restitution coefficient is varied while the torque direction is fixed. The authors also propose a spectral-gap criterion, based on the eigenvalue spectrum of the linearized collision operator, for selecting hydrodynamic variables.

Significance. If the quantitative results hold, the paper would provide a genuinely collision-based microscopic route to odd viscosity and chiral hydrodynamics, complementing the existing single-particle-based and mode-coupling approaches. The dimensional-analysis argument for the square-root torque scaling is clean and the explicit construction of the hydrodynamic equations from a Boltzmann equation is a useful contribution to the odd-fluid literature. The spectral-gap method for selecting hydrodynamic variables is also a promising general tool. The formal structure is coherent, and the paper is careful to state its assumptions and to provide appendices with derivations and numerical details. However, the central quantitative claims currently rest on several uncontrolled approximations: the Gaussian steady-state ansatz is contradicted by the paper's own molecular dynamics for angular velocities, the Hermite basis is truncated at nmax=4 with no convergence test, and no Monte Carlo statistical error bars are reported. These issues directly affect the reported coefficients and the sign-change claims.

major comments (4)
  1. [Sec. IV.B and App. F] The slow-manifold split is not validated in the parameter range where the headline sign changes occur. In Sec. IV.B and Fig. 2 the authors state that for small tangential restitution the eigenvalues associated with higher powers of angular momentum, O(omega-omega_0)^3, lie close to zero and should be considered hydrodynamic, while for larger e_t one is left with three or five slow variables depending on the cutoff. Nevertheless, App. F fixes a single five-dimensional slow space (density, momenta, angular momentum, T_t, T_r) for all e_t, truncates the Hermite basis at nmax=4, and inverts L_FF in Eq. (46). Excluded near-null modes make L_FF nearly singular precisely in the low-e_t regime, so the inversions controlling every coefficient in Figs. 4b and 5b, including the zero crossings, are sensitive to the truncation and cutoff choice. The authors should provide convergence tests with respect to nmax (e.g., nmax=5,6) and with respect to the number of slow variables, particularly in the e_t range where sign changes are reported.
  2. [Sec. III.B and App. D] The Gaussian steady-state ansatz, Eq. (11), is explicitly contradicted by the paper's own molecular dynamics results in App. D: the angular velocity distributions are skewed and significantly non-Gaussian as e_t increases, with large higher-order cumulants. Since f^(0) defines the inner product (22), the linearized operator L in Eq. (20), and hence every matrix element and inverted quantity in Secs. IV.D.3 and V, the quantitative coefficient values inherit the error of this closure. The assertion in Sec. III.B that the method 'does not rely on this hypothesis' because only the integrals need to be evaluated is too strong: as implemented, all numerical evaluations do use the Gaussian closure. The authors should estimate the resulting uncertainty in the coefficients, for example by repeating the computation with a corrected or fitted steady-state distribution in the angular variable, or by benchmarking against direct simulation of the Boltzmann equation.
  3. [Sec. V and App. F] No statistical error bars are reported for the Monte Carlo evaluations of the matrix elements. App. F states that each integral is computed with 1,500,000 data points and 10 iterations using the Vegas library, but Figs. 4b and 5b show only point values. A coefficient whose value crosses the zero axis could be entirely an artifact of numerical noise if the statistical uncertainty is comparable to the plotted magnitude. The authors should report error bars, and ideally show convergence of the coefficients with the number of samples and iterations, at least for representative values of e_t.
  4. [Sec. IV.D.3 and Eq. (44)] The adiabatic-elimination formula (44) assumes a clear spectral gap separating the five chosen slow modes from the fast space, but the paper's own spectral analysis shows the gap is not uniform in e_t. In particular, Sec. IV.B notes that for very low e_t the higher angular-momentum modes 'remain very close to zero-modes,' and for higher e_t 'we are left with three or five slow variables, depending on where the cutoff value is chosen.' The fixed choice of exactly five slow variables for all e_t is therefore not justified by the stated spectral criterion. This issue is load-bearing because the sign-change claim in Fig. 5b occurs in a parameter range where the fast/slow separation is at its least reliable. The authors should either restrict the quantitative claims to the regime where the five-mode split is spectrally justified, or extend the slow space in the regimes where additional near-null modes exist and show that the reported coefficients are stable under that extension.
minor comments (5)
  1. [Eq. (15c)] The expression for T_t contains a likely typographical error in the denominator: the parentheses in 'G(B2+B3)B A A2' do not parse as written; this should be corrected to match the dimensionless version in Eq. (C18).
  2. [Sec. IV.D.3] The text contains the typo 'Bolzmann' in the phrase 'the linear Bolzmann equation in Fourier space'; it should read 'Boltzmann'.
  3. [Eq. (3)] The restitution coefficients r_parallel and r_perp in Eq. (3) are not explicitly defined in the main text; the definitions of the normal and tangential restitution coefficients used throughout the paper (r_n and e_t) should be stated clearly at their first occurrence.
  4. [Fig. 3 caption] The caption refers to 'upper plot' and 'lower plot' but the figure is not reproduced in the text; if the figure is available, please ensure the axes and curve labels are legible and that the odd/even decomposition is described in the caption itself.
  5. [App. F.1] The statement that the Hermite polynomials are orthonormal 'under the inner product' (F1) is correct only for the dimensionless Gaussian weight; the distinction between the unnormalized physicist's Hermite polynomials and the normalized basis used in the computation should be made explicit to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the response coefficients are computed from the stated collision model and Gaussian closure, not fitted to the target values; self-citations are background only.

full rationale

The derivation chain is self-contained. The hydrodynamic equations are obtained from the Boltzmann equation and the Gaussian ansatz of Eq. (11) via moment equations (13), and the steady-state parameters are solved from fixed-point equations (15), not tuned to reproduce the final transport coefficients. The relaxation and transport coefficients are then defined as inner products involving L_FF^{-1}, namely Eqs. (49) and (50), and evaluated numerically from the linearized collision operator; no coefficient is adjusted to match a preselected result. The square-root torque scaling is a consequence of T_t proportional to |d| from the fixed point and the non-dimensionalization leading to Eq. (60), so it is derived rather than imposed. The sign changes in Figs. 4-5 are outputs of the numerical inversion as e_t is varied, not inputs. Self-citations to Refs. [1], [15], and [26] are contextual (the odd-viscosity review and earlier kinetic-theory treatments) and none is used to justify the central result. The main weakness is the slow-manifold truncation at small e_t, which the paper itself flags in Sec. IV.B and App. D: eigenvalues of higher angular-momentum modes remain close to zero and the angular-velocity distribution becomes skewed. That is a correctness/robustness concern about the adiabatic split, not a circularity, because the split is not equivalent to the computed coefficients by construction. On the evidence quoted, no derivation step reduces to its own input.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on the Boltzmann molecular chaos approximation, a Gaussian steady-state ansatz, spectral truncation at Hermite order 4, and a fixed five-field slow manifold. None of these are derived from first principles, and the paper's own MD and spectrum checks show they fail quantitatively in parts of the parameter range. No new entities are invented, and the model parameters r_n, e_t, and d are physical inputs rather than fitted outputs.

free parameters (3)
  • normal restitution coefficient r_n = 0.95 (fixed in plotted scans)
    Model parameter of the collision rule in Eq. (3); all reported coefficient plots fix r_n = 0.95.
  • tangential restitution parameter e_t = swept over a range, roughly 0 to 0.3
    Controlled parameter controlling frictional collisions; sign changes of odd coefficients are reported as functions of e_t.
  • external torque d = unit reference value in plots
    Drive strength setting the energy scale; the square-root dependence is derived from dimensional analysis, not fitted.
assumptions (5)
  • domain assumption Molecular chaos (Stosszahlansatz): particles about to collide are uncorrelated before collision.
    Used to derive the Boltzmann collision operator in Sec. III A and App. A; requires the gas to be sufficiently dilute.
  • domain assumption Binary collisions only; three-body and higher collisions are neglected.
    Stated in Sec. II A as a diluteness assumption; the collision operator in Eq. (9) is built from two-body scattering only.
  • ad hoc to paper The non-equilibrium steady-state distribution has a Gaussian form in velocities and a shifted Gaussian in angular velocities.
    Introduced in Sec. III B, Eq. (11). Appendix D shows angular velocity distributions deviate from Gaussian with visible skewness at larger e_t, so quantitative coefficients inherit this error.
  • ad hoc to paper The hydrodynamic slow manifold is spanned by density, momenta, angular momentum, and two temperatures for all parameters used in the calculations.
    Sec. IV B fixes these five fields even though the eigenvalue spectrum shows additional near-zero modes at small e_t, so the slow/fast split is not uniformly valid.
  • ad hoc to paper Hermite polynomial basis truncated at order nmax=4 is sufficient to represent the linearized collision operator and its inverse on the fast space.
    App. F 2 truncates L to a 125x125 matrix; no convergence check against larger truncations is reported, and the slow-manifold caveat makes this approximation especially delicate.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A collisional model of odd fluids: from Boltzmann equation to chiral hydrodynamics." pith.science (2026). https://pith.science/paper/FBRP5PIT

@misc{pith2026250812944,
  author       = {Pith},
  title        = {Pith review of: A collisional model of odd fluids: from Boltzmann equation to chiral hydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBRP5PIT}},
  note         = {Machine review of arXiv:2508.12944}
}
read the original abstract

When the time-reversal and parity symmetries in a fluid are broken, transverse transport coefficients can arise in response to perturbations, an example being odd viscosity. We refer to these systems as odd fluids. While much progress has been made in the continuum theory of odd-viscous fluids, and non-collisional models for odd viscous fluids have been proposed, a classical microscopic description in which the transverse responses originate from collisions is lacking. In this paper, we show that a dilute granular gas of rough and inelastic particles driven by a constant torque is a minimal microscopic model of an odd fluid. By applying the methods of Boltzmann kinetic theory, we obtain a hydrodynamic description of the microscopic model. Then, using the method of adiabatic elimination, we numerically compute all the response coefficients of the model, explicitly showing that the model has many odd response terms. Our theory predicts that certain odd response coefficients can change sign even when the direction of the external torque is fixed. While we choose a particular case, the procedure we present can be applied to any collisional model. We also present a semi-quantitative method to determine the hydrodynamic variables of the theory by observing the eigenvalue spectrum of the linear collision operator.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Steady base states in a two-dimensional chiral fluid. The chiral Stokes cavity

    cond-mat.soft 2026-07 conditional novelty 7.0 of 10

    Enforcing angular-momentum conservation without stress symmetry yields a 2D chiral hydrodynamics in which every odd channel is the 90° rotation of a Newtonian one, and confined steady flows reduce to a modified Helmho...

  2. Kinetic and Hydrodynamic Theories of Chiral Intruder Dynamics in Nonequilibrium Baths

    cond-mat.stat-mech 2026-07 conditional novelty 7.0 of 10

    An intruder in a chiral nonequilibrium bath acquires shape-chirality ratchet torques in the dilute limit and edge-current-generated odd drag and torque in the dense limit.

  3. Chiral Dynamics of an Intruder across Dilute and Hydrodynamic Regimes

    cond-mat.stat-mech 2026-07 conditional novelty 7.0 of 10

    A kinetic-to-hydrodynamic derivation shows intruder shape sets chiral transport via boundary integrals in dilute baths and via torque-density edge currents in dense baths.

  4. Emergent odd response in active chiral films

    cond-mat.soft 2026-07 conditional novelty 6.0 of 10

    An active chiral film of anchored spinning rods has a derived Hall-viscosity coefficient η_o = 3A/32 in weak shear, with the transverse traction saturating at high shear.

Reference graph

Works this paper leans on

69 extracted references · 60 canonical work pages · cited by 4 Pith papers

  1. [1]

    26 into hydrodynamic equations, we must express the pressure tensors and heat flux vectors as a function of the hydrodynamic variables

    Linear response In order to close the equations of change obtained in Sec. 26 into hydrodynamic equations, we must express the pressure tensors and heat flux vectors as a function of the hydrodynamic variables. To do so, we assume lin- ear response and express the fluxes as a linear function of the local variations in the hydrodynamic quantities (re- laxa...

    work page

  2. [2]

    total momentum

    � f(0) 1 f(0) 2 (ϕ1 +ϕ2) � (19) where lower indices (1 , 2) indicate functions evaluated with the velocities (⃗ c1,2,ω 1,2) of particles (1, 2). A spatial Fourier transform of Eq. (18) yields ∂tϕ +i⃗ q�⃗ cϕ=Lϕ, (20) 3 �� ���� ������ ��� ������������ �������� ������ �� � (0) ��� ��� ��������� ������ ����� ����� �� �������� ����� ���� ������� �������� ��� �...

    work page

  3. [3]

    Linearized hydrodynamic equations We first linearize the equations of change (26) by con- sidering small perturbations ρα � ρ(0) α +δρα from the equilibrium values ρ(0) α . At lowest order, we find Dtδρ +ρ(0)� � ⃗δu = 0 (32a) ρ(0)Dtδui +∂jP (1) ij = 0 (32b) In(0)Dtδω +∂iM(1) i � Idδn = �(1)[Iω] (32c) n(0)Dtδet + � � ⃗Q(1) +P (0) ij ∂iδuj = �(1)[mv2/2] (32...

    work page

  4. [4]

    Adiabatic Elimination In order to make connection with the microscopic de- scription of the system, we use the adiabatic elimina- tion method [38, 48] which is equivalent to the usual Chapman-Enskog theory [39]. The general idea of adia- batic elimination is to split the Boltzmann equation into slow and fast variable subspaces, assume that fast vari- able...

    work page

  5. [5]

    (49) and (50)

    Nondimensionalization To obtain the full linear description of our model, we need to compute the inner products given in Eqs. (49) and (50). It is convenient to introduce dimensionless quantities to compute these inner products numerically. We present our procedure below for 2D rotating disks, but it is straightforward to generalize to systems with differ...

    work page

  6. [6]

    In this frame, the center of mass of the first particle will move with ⃗ c1 � ⃗ c2

    Derivation of the Loss Term Γ� Let us consider two colliding hard disks with radius a, one with linear and angular velocities (⃗ c1,ω 1) and the second one with ( ⃗ c2,ω 2), and suppose that we move to the coordinate frame of the center of mass of the second particle. In this frame, the center of mass of the first particle will move with ⃗ c1 � ⃗ c2. If w...

    work page

  7. [7]

    Derivation of the Gain Term Γ+ We will follow a similar argument as we did in the previous section. However, this time, the task at hand is a bit more difficult as we need to find the pair of velocity sets (⃗ c�� 1,⃗ ω�� 1 ), (⃗ c�� 2,⃗ ω�� 2 ) that will yield velocities (⃗ c1,ω 1). Let us construct the collision cylinders again. This time, however, since...

    work page

  8. [8]

    σ(⃗ c1,2,⃗ ω1,2, ˆa) is the differential cross section of the collision, ⃗ c12 = ⃗ c1 � ⃗ c2 is the shorthand notation for relative linear velocities

    Collision Operator Let us consider a collision term of the form �(f1,f 2) = � d⃗ c2d⃗ ω2dˆa�⃗ c12�σ(⃗ c1,2,⃗ ω1,2, ˆa) �f �� 1f �� 2 J � � f1f2 � , (B1) where, as always, �� denotes the pre-collision quanti- ties. σ(⃗ c1,2,⃗ ω1,2, ˆa) is the differential cross section of the collision, ⃗ c12 = ⃗ c1 � ⃗ c2 is the shorthand notation for relative linear velo...

    work page

Show all 69 references
  1. [9]

    � ψ(⃗ c1,⃗ ω1) � ψ(⃗ c2,⃗ ω2)] (B5) � � d⃗ c1d⃗ ω1d⃗ c2d⃗ ω2dˆaσ(⃗ c1,2,⃗ ω1,2, ˆa)f1f2∆ψ, (B6) where � denotes the post-collision quantities

  2. [10]

    Such expressions show up frequently in the calculation of time evolution equations we have in the stability analysis

    Odd-Functions of ˆ� � � �12 Next we will show that the collision term �[ψ] vanishes if ∆ψ is an odd function of ˆa�⃗ vt = ˆa�⃗ c12, given that the distribution function f0 is of the form as in (11). Such expressions show up frequently in the calculation of time evolution equat...

  3. [11]

    Rotational Invariance of the Collision Operator We would like to show that the collision operator �(f,f ) is rotationally invariant, that is, ˆR� ˆR� 1(f,f ) = �(f,f ), where ˆR � �� (2) is any rotation matrix on x-y plane. The expression R�R� 1, can explicitly be written as, ...

  4. [12]

    (11), which is, ∂tf +d∂ωf = �(f,f )

    Derivation of Time Evolution Equations We will start with the Boltzmann equation (8) assum- ing the spatially homogeneous Gaussian distribution of the form in Eq. (11), which is, ∂tf +d∂ωf = �(f,f ). (C1) To obtain the evolution equation for the average angular velocity, we mu...

  5. [13]

    Because the only time scale in the system is the external torque d, we do not have too many options

    Non-Dimensionalization Now, we will introduce a natural temperature and time scales of the system to obtain the dimensionless angular velocity ˜ω0 and temperatures ˜Tt and ˜Tr. Because the only time scale in the system is the external torque d, we do not have too many options....

  6. [14]

    (15a)-(15c)

    Stability Analysis To check if these values are indeed equilibrium solu- tions, we also need to show that they are a stable fixed point of the Eqs. (15a)-(15c). If we non-dimensionalize the evolution equations with the same procedure, we get ∂˜t˜ω0 = sgn(d) � ˜GC1 2 ˜ω0 ˜T 1/2...

  7. [15]

    The details on how they are computed are given in App

    Computation of Current Terms As mentioned in the main text, we list the computational expressions for all the non-zero relaxation and transport coefficients in terms of the non-dimensional inner products. The details on how they are computed are given in App. F. Expressions fo...

  8. [16]

    (54), given polynomials are real func- tions

    The Choice of Polynomial Basis We choose Hermite polynomials for our basis, as the steady-state distribution we assume is Gaussian and they are orthonormal under the inner product, (Hi(x),Hj(y)) � � dxe� x� Hi(x)Hj(x) =δij, (F1) which is the same as the non-dimensional inner p...

  9. [17]

    (F3) As we mentioned before, we will truncate the basis such that nmax = 4 and set ˜L �� 0� i,j,k,l,m,n� 4 ˜Lijk,lmn�ijk)(lmn�

    Matrix Elements of the Boltzmann Operator � Using the basis and the inner product we chose, we can write down the matrix elements of the dimensionless Boltzmann operator ˜L 6 as ˜Lijk,lmn � (Hijk, ˜L[Hlmn])ND = (˜L�[Hijk],Hlmn)ND. (F3) As we mentioned before, we will truncate ...

  10. [18]

    Matrix Elements of ˜� � As it is discussed in the main text, the Boltzmann op- erator L is split into two parts: the linearized collision operator C and the torque term Lω. Although both of these terms can be numerically computed, we will show here that with the recursive rela...

  11. [19]

    Slow Variables in Hermite Basis To compute the inner products, one also needs the slow variables Φα andvµΦα written in the chosen polynomial basis, which can always be done numerically. However, in our case we can obtain exact expressions of them in terms of the Hermite polyno...

  12. [20]

    Each matrix entry is a 7- dimensional integral over the velocities and angular ve- locities of two colliding particles and the apse vector

    Reducing the Dimensionality The most expensive part of the procedure described above is the calculation of the matrix elements of the linear collision operator ˜Cij. Each matrix entry is a 7- dimensional integral over the velocities and angular ve- locities of two colliding pa...

  13. [21]

    Basis of irreducible representations In this section, we will introduce a basis which makes the interpretation of the transport coefficients more intu- itive. Consider the following basis of 2-tensors in 2 space dimensions: τ 0 = � 1 0 0 1 � τ 1 = � 0 � 1 1 0 � (F14a) τ 2 = � ...

  14. [22]

    Fruchart, C

    M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annual Review of Condensed Matter Physics 14, 471 (2023). 26

    2023

  15. [23]

    J. E. Avron, Journal of Statistical Physics 92, 543–557 (1998)

    1998

  16. [24]

    Beenakker and F

    J. Beenakker and F. McCourt, Magnetic and electric ef- fects on transport properties, Annual Review of Physical Chemistry 21, 47 (1970)

    1970

  17. [25]

    Korving, H

    J. Korving, H. Hulsman, H. Knaap, and J. Beenakker, Transverse momentum transport in viscous flow of di- atomic gases in a magnetic field, Physics Letters 21, 5 (1966)

    1966

  18. [26]

    Korving, H

    J. Korving, H. Hulsman, G. Scoles, H. Knaap, and J. Beenakker, The influence of a magnetic field on the transport properties of gases of polyatomic molecules;: Part i, viscosity, Physica 36, 177 (1967)

    1967

  19. [27]

    Hulsman and H

    H. Hulsman and H. Knaap, Experimental arrangements for measuring the five independent shear-viscosity coef- ficients in a polyatomic gas in a magnetic field, Physica 50, 565 (1970)

    1970

  20. [28]

    Braginskii, Transport phenomena in a completely ion- ized two-temperature plasma, Sov

    S. Braginskii, Transport phenomena in a completely ion- ized two-temperature plasma, Sov. Phys. JETP 6, 358 (1958)

    1958

  21. [29]

    A. I. Berdyugin, S. Xu, F. Pellegrino, R. Krishna Ku- mar, A. Principi, I. Torre, M. Ben Shalom, T. Taniguchi, K. Watanabe, I. Grigorieva, et al., Measuring hall viscos- ity of graphene’s electron fluid, Science 364, 162 (2019)

    2019

  22. [30]

    Nakagawa, The kinetic theory of gases for the rotating system., Journal of Physics of the Earth 4, 105 (1956)

    Y. Nakagawa, The kinetic theory of gases for the rotating system., Journal of Physics of the Earth 4, 105 (1956)

    1956

  23. [31]

    Banerjee, A

    D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli, Odd viscosity in chiral active fluids, Nature communica- tions 8, 1573 (2017)

    2017

  24. [32]

    Liebchen and D

    B. Liebchen and D. Levis, Chiral active matter, Euro- physics Letters 139, 67001 (2022)

    2022

  25. [33]

    M. A. L´ opez-Casta˜ no, A. M´ arquez Seco, A. M´ arquez Seco, A. Rodr´ ıguez-Rivas, and F. V. Reyes, Chirality transitions in a system of active flat spinners, Physical Review Research 4, 033230 (2022)

    2022

  26. [34]

    Scholz, M

    C. Scholz, M. Engel, and T. P¨ oschel, Rotating robots move collectively and self-organize, Nature Communica- tions 9, 10.1038/s41467-018-03154-7 (2018)

    2018 doi

  27. [35]

    V. Soni, E. S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo, M. J. Shelley, and W. T. M. Irvine, The odd free surface flows of a colloidal chiral fluid, Nature Physics 15, 1188–1194 (2019)

    2019

  28. [36]

    M. Han, M. Fruchart, C. Scheibner, S. Vaikuntanathan, J. J. De Pablo, and V. Vitelli, Fluctuating hydrodynam- ics of chiral active fluids, Nature Physics17, 1260 (2021)

    2021

  29. [37]

    Hargus, K

    C. Hargus, K. Klymko, J. M. Epstein, and K. K. Man- dadapu, Time reversal symmetry breaking and odd vis- cosity in active fluids: Green–kubo and nemd results, The Journal of Chemical Physics152, 10.1063/5.0006441 (2020)

    2020 doi

  30. [38]

    Hargus, J

    C. Hargus, J. M. Epstein, and K. K. Mandadapu, Odd diffusivity of chiral random motion, Physical Review Let- ters 127, 178001 (2021)

    2021

  31. [39]

    X. Lou, Q. Yang, Y. Ding, P. Liu, K. Chen, X. Zhou, F. Ye, R. Podgornik, and M. Yang, Odd viscosity- induced hall-like transport of an active chiral fluid, Proceedings of the National Academy of Sciences 119, 10.1073/pnas.2201279119 (2022)

    2022 doi

  32. [40]

    Markovich and T

    T. Markovich and T. C. Lubensky, Odd viscosity in ac- tive matter: microscopic origin and 3d effects, Physical Review Letters 127, 048001 (2021)

    2021

  33. [41]

    Markovich and T

    T. Markovich and T. C. Lubensky, Nonreciprocity and odd viscosity in chiral active fluids, arXiv preprint arXiv:2211.06901 (2022)

    2022 arXiv

  34. [42]

    Matus, R

    P. Matus, R. Lier, and P. Sur´ owka, Molecular modeling of odd viscoelastic fluids, Physical Review E 110, 044605 (2024)

    2024

  35. [43]

    A. R. Poggioli and D. T. Limmer, Emergent kelvin waves in chiral active matter (2023), arXiv:2306.14984

    2023 arXiv

  36. [44]

    Ghimenti, L

    F. Ghimenti, L. Berthier, G. Szamel, and F. van Wijland, Sampling efficiency of transverse forces in dense liquids, Physical Review Letters 131, 257101 (2023)

    2023

  37. [45]

    Ghimenti, L

    F. Ghimenti, L. Berthier, G. Szamel, and F. van Wijland, Irreversible boltzmann samplers in dense liquids: Weak- coupling approximation and mode-coupling theory, Phys- ical Review E 110, 034604 (2024)

    2024

  38. [46]

    Ghimenti, L

    F. Ghimenti, L. Berthier, G. Szamel, and F. van Wij- land, Transverse forces and glassy liquids in infinite di- mensions, Physical Review E 109, 064133 (2024)

    2024

  39. [47]

    Fruchart, M

    M. Fruchart, M. Han, C. Scheibner, and V. Vitelli, The odd ideal gas: Hall viscosity and thermal conductiv- ity from non-hermitian kinetic theory, arXiv preprint arXiv:2202.02037 (2022)

    2022 arXiv

  40. [48]

    Jiao and M

    Y. Jiao and M. Yang, Mesoscale simulation model for odd fluids (2024), arXiv:2411.16212

    2024 arXiv

  41. [49]

    Goldsmith, Impact, Dover Civil and Mechanical En- gineering (Dover Publications, 2001)

    W. Goldsmith, Impact, Dover Civil and Mechanical En- gineering (Dover Publications, 2001)

    2001

  42. [50]

    J. T. Jenkins and M. W. Richman, Kinetic theory for plane flows of a dense gas of identical, rough, inelas- tic, circular disks, The Physics of Fluids 28, 3485–3494 (1985)

    1985

  43. [51]

    C. K. K. Lun, Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres, Journal of Fluid Mechanics 233, 539–559 (1991)

    1991

  44. [52]

    C. K. K. Lun and S. B. Savage, A simple kinetic theory for granular flow of rough, inelastic, spherical particles, Journal of Applied Mechanics 54, 47–53 (1987)

    1987

  45. [53]

    O. R. Walton, Numerical simulation of inelastic, fric- tional particle-particle interactions, Particulate two- phase flow 25, 884 (1993)

    1993

  46. [54]

    N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, and T. P¨ oschel, Model for collisions in granular gases, Phys- ical Review E 53, 5382–5392 (1996)

    1996

  47. [55]

    Zippelius, Granular gases, Physica A: Statistical Me- chanics and its Applications 369, 143–158 (2006)

    A. Zippelius, Granular gases, Physica A: Statistical Me- chanics and its Applications 369, 143–158 (2006)

    2006

  48. [56]

    Brilliantov and T

    N. Brilliantov and T. P¨ oschel,Kinetic Theory of Granu- lar Gases, Oxford Graduate Texts (OUP Oxford, 2010)

    2010

  49. [57]

    P¨ oschel and S

    T. P¨ oschel and S. Luding,Granular Gases, Lecture Notes in Physics (Springer Berlin Heidelberg, 2008)

    2008

  50. [58]

    N. V. Brilliantov and T. P¨ oschel, Velocity distribution in granular gases of viscoelastic particles, Physical Review E 61, 5573–5587 (2000)

    2000

  51. [59]

    J. R. Dorfman, H. van Beijeren, and T. R. Kirkpatrick, Contemporary kinetic theory of matter (Cambridge Uni- versity Press, 2021)

    2021

  52. [60]

    Chapman and T

    S. Chapman and T. G. Cowling, The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases (Cambridge university press, 1990)

    1990

  53. [61]

    Puglisi, A

    A. Puglisi, A. Baldassarri, and V. Loreto, Fluctuation- dissipation relations in driven granular gases, Physical Review E 66, 061305 (2002)

    2002

  54. [62]

    Cafiero, S

    R. Cafiero, S. Luding, and H. J. Herrmann, Two- dimensional granular gas of inelastic spheres with multi- plicative driving, Physical review letters 84, 6014 (2000)

    2000

  55. [63]

    Goldhirsch, S

    I. Goldhirsch, S. Noskowicz, and O. Bar-Lev, Nearly smooth granular gases, Physical review letters95, 068002 (2005). 27

    2005

  56. [64]

    Markovich and T

    T. Markovich and T. C. Lubensky, Chiral active fluids: what can we learn from the total momentum? (2024), arXiv:2411.15812

    2024 arXiv

  57. [65]

    Rosenfeld, On the energy-momentum tensor, Mem

    L. Rosenfeld, On the energy-momentum tensor, Mem. Acad. Royal Belg 18, N6 (1940)

    1940

  58. [66]

    Belinfante, On the current and the density of the elec- tric charge, the energy, the linear momentum and the an- gular momentum of arbitrary fields, Physica 7, 449–474 (1940)

    F. Belinfante, On the current and the density of the elec- tric charge, the energy, the linear momentum and the an- gular momentum of arbitrary fields, Physica 7, 449–474 (1940)

    1940

  59. [67]

    K. Y. Bliokh, Momentum, spin, and orbital angular mo- mentum of electromagnetic, acoustic, and water waves (2025), arXiv:2503.23235

    2025

  60. [68]

    P. J. Morrison, M. Lingam, and R. Acevedo, Hamilto- nian and action formalisms for two-dimensional gyro- viscous magnetohydrodynamics, Physics of Plasmas 21, 10.1063/1.4891321 (2014)

    2014 doi

  61. [69]

    R´ esibois, On linearized hydrodynamic modes in sta- tistical physics, Journal of Statistical Physics 2, 21–51 (1970)

    P. R´ esibois, On linearized hydrodynamic modes in sta- tistical physics, Journal of Statistical Physics 2, 21–51 (1970)

    1970

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.