Pith. sign in

REVIEW 3 major objections 4 minor 60 references

Chiral active fluids: what can we learn from the total momentum?

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read The paper claims that in chiral active fluids, symmetry of the total momentum stress leaves only two measurable odd transport coefficients—odd viscosity and odd pressure—and makes center-of-mass dynamics alone insufficient.

desk verdict A serious theory paper that reduces the observable odd transport coefficients in chiral active fluids from three to two via the symmetry of the total momentum stress; Eq. (16) is robust and new, but the Galilean invariance assumption leading to Eq. (18) is load-bearing and uncertain. read the letter →

arxiv 2411.15812 v1 pith:NRGJPH65 submitted 2024-11-24 cond-mat.soft

Tomer Markovich , Tom C. Lubensky This is my paper · ORCID
classification cond-mat.soft
keywords chiralactivefluidsoddviscositytotalmomentumspinangularcenter-of-massstresspressurehydrodynamicstime-reversalsymmetrybreaking
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

Chiral active fluids contain molecules that spin, so the fluid carries both ordinary center-of-mass momentum and spin angular momentum. The paper argues that the stress a rheometer actually measures is the flux of the total momentum—the sum of these two—and that this total momentum stress must be symmetric. Imposing that symmetry, together with Galilean invariance, forces the odd transport coefficients to satisfy a definite relation, leaving only two experimentally accessible odd coefficients: an odd (Hall) viscosity and an odd pressure. A consequence is that the usual center-of-mass description of such fluids is incomplete, even after the spin relaxes.

What carries the argument

The central object is the total momentum density $g_i=g^c_i+\tfrac12\epsilon_{ij}\nabla_j\ell$, the sum of the center-of-mass momentum and the rotational contribution from the spin angular momentum density $\ell$. The mechanism is the requirement that the stress conjugate to this total momentum—the actual force on boundaries—be symmetrizable, since total linear momentum is conserved except for explicit external forces. Writing the most general symmetry-allowed center-of-mass stress and transforming to total momentum, the symmetry condition produces the relation (16) between odd coefficients, and Galilean invariance fixes $\chi$ via Eq. (17). These constraints are the machinery that eliminates the odd torque and reduces the odd sector to $\tilde\eta_o$ and $\bar\eta_B$.

What would settle it

One could compute, in a molecular-dynamics simulation of chiral active particles with known central and spin-spin forces, the values of $\eta_B$, $\bar\eta_B$, $\eta_C$, and $\ell$ from the stress and moment densities; if $\eta_B-\bar\eta_B-\tfrac12(\eta_C-\ell)$ is systematically nonzero beyond numerical error while Galilean invariance holds, the paper's central relation is wrong. Alternatively, a rheological measurement of the boundary stress and an independent measurement of the center-of-mass stress would show a difference proportional to $\ell_0$ predicted by comparing Eq. (22) and Eq. (18).

Watch

Extended reading notes

Core claim

The central result is Eq. (16): for an isotropic two-dimensional chiral active fluid, symmetry of the total momentum stress requires $\eta_B=\bar\eta_B+\tfrac12(\eta_C-\ell)$, where $\eta_B$ is the odd-pressure coefficient in the center-of-mass stress, $\bar\eta_B$ is the truly odd pressure coupling vorticity to pressure, $\eta_C$ is a spin-spin coupling coefficient, and $\ell$ is the spin angular momentum density. Combined with the Galilean-invariance condition $\chi_{ijkl}=-\frac{\eta_C}{4}(\gamma^o_{ijkl}-2\epsilon_{lk}\delta_{ij})$, this reduces the total momentum stress at hydrodynamic order to Eq. (18), which contains only an odd viscosity with modified coefficient $\tilde\eta_o=\eta_o+\eta_C-\ell$ and an odd pressure $\bar\eta_B$. The paper further shows that the center-of-mass stress after spin relaxation, Eq. (22), is not equivalent to the total momentum stress: it lacks the kinetic $\ell_0$ contribution and, when the active torque is inhomogeneous, contains an antisymmetric term that violates Galilean invariance and total-angular-momentum balance at the center-of-mass level.

Load-bearing premise

The load-bearing premise is that active bulk interactions respect Galilean invariance, so any violation enters only through the external force and external torque; if that fails, the coefficient relation and the two-odd-coefficient conclusion would have to be rebuilt.

Editorial extensions

If this is right

  • Rheological experiments on chiral active fluids will measure only two odd transport coefficients—the modified odd viscosity $\tilde\eta_o$ and the odd pressure $\bar\eta_B$—because the total momentum stress is what couples to boundaries.
  • The odd torque $\eta_A$, although allowed by structural symmetry, is not accessible in rheology or in center-of-mass stress measurements after spin relaxation; reported simulation values likely reflect short-time or boundary-condition effects.
  • Center-of-mass dynamics alone do not conserve total angular momentum in these fluids; a complete hydrodynamic description must retain total momentum and spin even after the spin relaxes.
  • The coefficient $\eta_B$ in the center-of-mass stress is not an independent odd pressure: symmetry ties it to the spin-spin coupling $\eta_C$ and to the kinetic spin density $\ell$, so central-force and spin-spin interactions cannot be tuned independently in the hydrodynamic odd response.

Reading between the lines

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

  • If this is right, past odd-viscosity measurements and simulations that report a center-of-mass-based coefficient may actually be reporting a combination of central-force and spin-spin effects; the field would need to specify which stress is being measured before comparing numbers.
  • The same total-momentum-symmetry argument should carry over to three-dimensional chiral active fluids, where parity-violating viscosities have more structure; it likely selects a smaller set of rheologically accessible odd coefficients than the full symmetry-allowed set.
  • A testable extension is to measure whether the antisymmetric part of the center-of-mass stress after spin relaxation, which the paper predicts for inhomogeneous active torques, appears in experiments with spatially varying torque; its absence would indicate additional constraints or broken assumptions.
  • The relation between $\eta_B$ and $\eta_C$ suggests a microscopic design rule: changing spin-spin interactions (for example by particle shape) shifts the measurable odd pressure, so rheology could be used to infer the strength of spin exchange in chiral active suspensions.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper develops a phenomenological hydrodynamic description of two-dimensional isotropic chiral active fluids, distinguishing the center-of-mass (CM) momentum from the total momentum that includes molecular spin angular momentum. The central result is that requiring the total momentum stress tensor to be symmetric imposes the relation ηB = ¯ηB + (ηC - ℓ)/2 (Eq. 16), and requiring Galilean invariance fixes the non-Galilean coefficient χ_ijkl through Eq. (17). Together these reduce the odd part of the total momentum stress to an odd viscosity with coefficient (ηo + ηC - ℓ)/4 and an odd pressure ¯ηB (Eq. 18). The paper then eliminates the spin angular momentum in the hydrodynamic limit and derives the resulting CM stress (Eq. 22), concluding that the CM stress is not equivalent to the total momentum stress and that CM dynamics are generally insufficient for chiral active fluids.

Significance. If the derivation is correct, the paper makes a significant and falsifiable claim: rheological measurements of chiral active fluids access only two odd coefficients, the odd viscosity and the odd pressure, rather than the three odd terms allowed by structural symmetry. It also identifies a previously unnoticed coupling between central-force interactions and spin-spin interactions. Strengths of the manuscript include the transparent derivation from conservation laws, the consistency check with the non-interacting Poisson-bracket result of Section IIA, and the explicit rheological prediction. The main weaknesses are the unproven Galilean-invariance assumption underlying Eq. (17) and the lack of a microscopic check of that assumption.

major comments (3)
  1. [Sec. IIIA, Eq. (17)] The central simplification to Eq. (18) and the two-odd-coefficient claim depend on the Galilean-invariance condition χ_ijkl = -ηC/4(γo_ijkl - 2ε_lk δ_ij). The manuscript itself states in Sec. IIIA that it is not clear to what extent Galilean invariance can be imposed on active materials, and it simply assumes all violations are contained in f and τ_ex. This is load-bearing: if a momentum-conserving active torque or velocity-dependent interaction produces a non-Galilean χ, then the total momentum stress retains an extra term v_k ∇_l[χ_ijkl + (ηC/4)(γo_ijkl - 2ε_lk δ_ij)], changing both boundary tractions and the CM-stress reduction leading to Eq. (22). The authors should either derive Eq. (17) from a microscopic model—for example, by extending the Poisson-bracket calculation of Sec. IIA to include spin-spin interactions—or explicitly restrict the main claim to systems where Eq. (17) can be justified.
  2. [Sec. IIIA, Eq. (16)] The symmetry constraint on the total momentum stress is applied to the particular local representation of the stress given by Eq. (14). However, the total momentum stress is not unique: adding the divergence of a third-rank tensor with suitable antisymmetry does not change the momentum balance and can symmetrize any stress. The paper should clarify whether Eq. (16) is a physical constraint on measurable boundary tractions or a condition on the chosen local representation, and justify the choice. Without this clarification, the relation ηB = ¯ηB + (ηC - ℓ)/2 may be a gauge artifact rather than a robust consequence of total momentum conservation.
  3. [Sec. IV, Eq. (22)] The elimination of the spin angular momentum leading to Eq. (22) is presented only through the brief statement of Eq. (21). The intermediate algebra that shows how the coefficients ηA and ηC enter and eventually cancel (or drop out) is not shown. Since Eq. (22) is used to support the claim that the CM stress after relaxation differs from the total momentum stress, the derivation should be given explicitly or at least the order-by-order cancellation should be stated. This is particularly important because Eq. (20) still contains ηA through the combination τ_ex - ˙ℓ, so its disappearance in Eq. (22) is not immediately obvious.
minor comments (4)
  1. [Abstract and Sec. I] The abstract contains a typo: 'constraints the ammount' should be 'constrains the amount'; similarly, in Sec. I 'transnational' should be 'translational'.
  2. [Appendix A, Eq. (A1)] The advection term ∇j(vc_j gc_j) appears to have a repeated index typo; it should be ∇j(vc_j gc_i). Please check the index structure.
  3. [Sec. IV, Eq. (21)] The notation Γ_T is used in Eq. (21) and Appendix C without a prior definition at the point of use; define Γ_T = Γ + Γ_Ω explicitly before Eq. (21).
  4. [Eq. (22)] In Eq. (22), the relation between τ_ex and τ is not repeated; recalling τ_ex = τ - Γ_Ω Ω near the equation would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central constraint (Eq. 16) is derived from total-momentum symmetrization, not assumed; the acknowledged Galilean-invariance assumption (Eq. 17) is a physical premise, not a self-referential reduction.

full rationale

The paper's central result, Eq. (16), comes from an algebraic symmetrization of the total-momentum stress (Appendix B), starting from the structural CM stress (Eq. 11), the SAM dynamics (Eq. 13), and the resulting total-momentum stress (Eq. 14). The constraint ηB = ¯ηB + 1/2(ηC − ℓ) is imposed by requiring the total-momentum stress to be symmetric, which is a conservation-law input rather than the desired conclusion. Eq. (17) likewise fixes the non-Galilean coefficient χijkl by imposing Galilean invariance on the total momentum; this is an assumption that is explicitly flagged as uncertain in Sec. IIIA ('It is not clear to what extent Galilean invariance can actually be imposed on active materials'), so it is a correctness risk, not a circular step. The only same-author inputs are the non-interacting Poisson-bracket results from Ref. [22], which are used as a consistency check and to identify the kinetic contribution ℓ/2 in ¯ηB; they do not generate Eq. (16) or Eq. (18). No fitted parameter is relabeled as a prediction, no uniqueness theorem from the authors is used to forbid alternatives, and no known result is merely renamed. The derivation is self-contained for the symmetrization constraint, though its physical reach depends on the unverified Galilean-invariance premise.

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

No numerical fitting is performed. The paper introduces no new particles or fields; spin angular momentum density and odd transport coefficients are standard. The central claim is a relation among phenomenological coefficients, so the ledger lists the modeling assumptions rather than fitted parameters.

assumptions (5)
  • domain assumption Total momentum decomposes as g_i = g^c_i + (1/2) ε_ij ∇_j ℓ.
    Sec II Eq (3). Assumes a hydrodynamic description of an isotropic 2D fluid of complex molecules; taken from Refs [21,22,52].
  • domain assumption The total momentum stress can be written symmetric when total linear momentum is conserved, up to the external torque term (1/2) ε_ij τ_ex.
    Sec IIIA and Appendix B. This is the load-bearing physical constraint behind Eq (16).
  • domain assumption Galilean invariance holds for all internal interactions; only f and τ_ex may break it.
    Invoked to set χ_ijkl in Eq (17). The authors note in Sec IIIA that active materials may break Galilean invariance, but relegate such effects to f and τ_ex.
  • domain assumption The hydrodynamic gradient expansion is truncated at first order in ∇v.
    Sec III Eqs (9)-(11). Standard for long-wavelength hydrodynamics, but excludes higher-order odd terms.
  • domain assumption Spin angular momentum relaxes quickly absent active torque and relaxes to ℓ0 = τ/ΓT in the hydrodynamic limit.
    Sec IV and Appendix C. Used to eliminate the spin angular momentum; Eqs (21)-(22) depend on this timescale separation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Chiral active fluids: what can we learn from the total momentum?." pith.science (2026). https://pith.science/paper/NRGJPH65

@misc{pith2026241115812,
  author       = {Pith},
  title        = {Pith review of: Chiral active fluids: what can we learn from the total momentum?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NRGJPH65}},
  note         = {Machine review of arXiv:2411.15812}
}
read the original abstract

Chiral active materials are those that break both time-reversal symmetry and parity microscopically, which results in average rotation of the material's complex molecules around their center-of-mass (CM). These materials are far from equilibrium due to their local non-vanishing spin angular momentum. In this paper we show that, unlike passive fluids, the non vanishing spin angular momentum brings about a difference between the CM momentum and the total momentum, which accounts for the momentum of all atoms that compose the complex rotating molecules. This is in stark contrast to equilibrium fluids where the CM stress and the total momentum are essentially equivalent. In fact, we find that generally the CM dynamics are insufficient to describe the dynamics of a chiral active material. The total momentum, other than being experimentally accessible in simple rheological experiments, also imposes another constraint -- its stress must be allowed to be written in a symmetric way. We find that the latter imposes a relation between possible central-force interactions and spin-spin interactions, and constraints the ammount of {\it odd} viscosities in the system to the well-known odd (Hall) viscosity and the odd pressure.

Figures

Figures reproduced from arXiv: 2411.15812 by the authors.

Figure 1
Figure 1. Schematic of our model system. Each molecule with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 50 canonical work pages

  1. [1]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Rev. Mod. Phys.85, 1143 (2013)

    2013

  2. [2]

    Markovich, É

    T. Markovich, É. Fodor, E. Tjhung, and M. E. Cates, Thermodynamics of active field theories: Energetic cost of coupling to reservoirs, Physical Review X11, 021057 (2021)

    work page 2021

  3. [3]

    Fodor, R

    É. Fodor, R. L. Jack, and M. E. Cates, Irreversibility and biased ensembles in active matter: Insights from stochas- tic thermodynamics, Annual Review of Condensed Mat- ter Physics13, 215 (2022)

    work page 2022

  4. [4]

    Hatwalne, S

    Y. Hatwalne, S. Ramaswamy, M. Rao, and R. A. Simha, Rheology of active-particle suspensions, Phys. Rev. Lett. 92, 118101 (2004)

    2004

  5. [5]

    Prost, F

    J. Prost, F. Jülicher, and J. F. Joanny, Active gel physics, Nat. Phys.11, 111 (2015)

    work page 2015

  6. [6]

    Tjhung, D

    E. Tjhung, D. Marenduzzo, and M. E. Cates, Sponta- neous symmetry breaking in active droplets provides a generic route to motility, Proc. Natl. Acad. Sci. U.S.A. 31, 12381 (2013)

    work page 2013

  7. [7]

    C. P. Broedersz and F. C. MacKintosh, Molecular mo- tors stiffen non-affine semiflexible polymer networks, Soft Matter 7, 3186 (2011)

    work page 2011

  8. [8]

    M. E. Cates and J. Tailleur, Motility-induced phase sepa- ration, Annu. Rev. Condens. Matter Phys.6, 219 (2015)

    2015

Show all 60 references
  1. [9]

    Markovich, E

    T. Markovich, E. Tjhung, and M. E. Cates, Shear- induced first-order transition in polar liquid crystals, Phys. Rev. Lett.122, 088004 (2019)

    2019

  2. [10]

    S. J. Kole, G. P. Alexander, S. Ramaswamy, and A. Maitra, Layered chiral active matter: Beyond odd elasticity, Phys. Rev. Lett.126, 248001 (2021)

    2021

  3. [11]

    S. Chen, T. Markovich, and F. C. MacKintosh, Motor- free contractility in active gels, Phys. Rev. Lett. 125, 208101 (2020)

    2020

  4. [12]

    A. P. Solon, J.-B. Caussin, D. Bartolo, H. Chaté, and J. Tailleur, Pattern formation in flocking models: A hy- drodynamic description, Phys. Rev. E92, 062111 (2015)

    2015

  5. [13]

    Bricard, J

    A. Bricard, J. B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Emergence of macroscopic directed mo- tion in populations of motile colloids, Nature 503, 95 (2013)

    2013

  6. [14]

    Shitara, O

    A.Bricard, J.B.Caussin, D.Das, C.Savoie, V.Chikkadi, K. Shitara, O. Chepizhko, F. Peruani, D. Saintillan, and D. Bartolo, Emergent vortices in populations of colloidal rollers, Nat. Comm.6, 7470 (2015)

    2015

  7. [15]

    Buttinoni, J

    I. Buttinoni, J. Bialké, K. Kümmel, J. Löwen, C. Bechinger, and T. Spec, Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles, Phys. Rev. Lett.110, 238301 (2013)

    2013

  8. [16]

    Brahmachari, T

    S. Brahmachari, T. Markovich, F. C. MacKintosh, and J. N. Onuchic, Temporally correlated active forces drive segregation and enhanced dynamics in chromosome poly- mers, PRX Life2, 033003 (2024)

    2024

  9. [17]

    Toner, Y

    J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics318, 170 (2005), special Issue

    2005

  10. [18]

    Fruchart, C

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

    2023

  11. [19]

    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, 201102 (2020)

    2020

  12. [20]

    X. Lou, Q. Yang, Y. Ding, P. Liu, K. Chen, X. Zhou, F.Ye, R.Podgornik,andM.Yang,Oddviscosity-induced hall-like transport of an active chiral fluid, Proceedings of the National Academy of Sciences119, e2201279119 (2022)

    2022

  13. [21]

    T.MarkovichandT.C.Lubensky,Oddviscosityinactive matter: Microscopic origin and 3d effects, Phys. Rev. Lett. 127, 048001 (2021). 8

    2021

  14. [22]

    Markovich and T

    T. Markovich and T. C. Lubensky, Nonreciprocity and odd viscosity in chiral active fluids, Proceedings of the National Academy of Sciences121, e2219385121 (2024)

    2024

  15. [23]

    Markovich, E

    T. Markovich, E. Tjhung, and M. E. Cates, Chiral ac- tive matter: microscopic ‘torque dipoles’ have more than one hydrodynamic description, New J. Phys.21, 112001 (2019)

    2019

  16. [24]

    Fürthauer, M

    S. Fürthauer, M. Strempel, S. W. Grill, and F. Jülicher, Active chiral fluids, The European physical journal E35, 1 (2012)

    2012

  17. [25]

    S. R. Naganathan, S. Fürthauer, M. Nishikawa, F. Jülicher, and S. W. Grill, Active torque generation by the actomyosin cell cortex drives left–right symmetry breaking, Elife3, e04165 (2014)

    2014

  18. [26]

    Drescher, K

    K. Drescher, K. C. Leptos, I. Tuval, T. Ishikawa, T. J. Pedley, and R. E. Goldstein, Dancing volvox: hydrody- namic bound states of swimming algae, Phys. Rev. Lett. 102, 168101 (2009)

    2009

  19. [27]

    Mater.2, 806 (2003)

    Y.TabeandH.Yokoyama,Coherentcollectiveprecession of molecular rotors with chiral propellers, Nat. Mater.2, 806 (2003)

    2003

  20. [28]

    A. P. Petroff, X.-L. Wu, and A. Libchaber, Fast-moving bacteria self-organize into active two-dimensional crys- talsofrotatingcells,Phys.Rev.Lett. 114,158102(2015)

    2015

  21. [29]

    Snezhko, Complex collective dynamics of active torque-driven colloids at interfaces, Curr

    A. Snezhko, Complex collective dynamics of active torque-driven colloids at interfaces, Curr. Opin. Colloid Interface Sci.21, 65 (2016)

    2016

  22. [30]

    Maggi, F

    C. Maggi, F. Saglimbeni, M. Dipalo, F. De Angelis, and R. Di Leonardo, Micromotors with asymmetric shape that efficiently convert light into work by thermocapil- lary effects, Nat. Commun.6, 1 (2015)

    2015

  23. [31]

    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, Nat. Phys. 15, 1188 (2019)

    2019

  24. [32]

    J. C. Tsai, F. Ye, J. Rodriguez, J. P. Gollub, and T. C. Lubensky, A chiral granular gas, Phys. Rev. Lett.94, 214301 (2005)

    2005

  25. [33]

    B. C. van Zuiden, J. Paulose, W. T. M. Irvine, D. Bar- tolo, and V. Vitelli, Spatiotemporal order and emergent edge currents in active spinner materials, Proc. Natl Acad. Sci. USA113, 12919 (2016)

    2016

  26. [34]

    Banerjee, A

    D. Banerjee, A. Souslov, A. G. Abanov, and V. Vitelli, Odd viscosity in chiral active fluids, Nat. Comm.8, 1573 (2017)

    2017

  27. [35]

    Ganeshan and A

    S. Ganeshan and A. G. Abanov, Odd viscosity in two- dimensional incompressible fluids, Phys. Rev. Fluids2, 094101 (2017)

    2017

  28. [36]

    G. M. Monteiro, A. G. Abanov, and S. Ganeshan, Hamil- tonian structure of 2d fluid dynamics with broken parity, (2021), arXiv:2105.01655 [physics.flu-dyn]

    2021 arXiv

  29. [37]

    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

  30. [38]

    Onsager, Reciprocal relations in irreversible processes ii, Phys

    L. Onsager, Reciprocal relations in irreversible processes ii, Phys. Rev.38, 2265 (1931)

    1931

  31. [39]

    S. R. De Groot and P. Mazur,Non-equilibrium thermo- dynamics (Courier Corporation, 2013)

    2013

  32. [40]

    J. E. Avron, Odd viscosity, J. Stat. Phys.92, 543 (1998)

    1998

  33. [41]

    Kagan and L

    Y. Kagan and L. Maksimov, Kinetic theory of gases tak- ingintoaccountrotationaldegreesoffreedominanexter- nal field, Sov. Phys. JETP24 (1967), [J. Expt. Theoret. Phys.(U.S.S.R.) 51 1893, (1966)]

    1967

  34. [42]

    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. JETP6, 358 (1958), [J. Expt. Theo. Phys. 33, 459 (1957)]

    1958

  35. [43]

    10 (Perg- amon Press, Oxford, 1981)

    L.D.Landau, E.M.Lifshitz,andL.P.Pitaevskij, Course of Theoretical Physics: Physical Kinetics, Vol. 10 (Perg- amon Press, Oxford, 1981)

    1981

  36. [44]

    Hoyos, Hall viscosity, topological states and effective theories, Int

    C. Hoyos, Hall viscosity, topological states and effective theories, Int. J. Mod. Phys. B28, 1430007 (2014)

    2014

  37. [45]

    Read, Non-abelian adiabatic statistics and hall viscos- ity in quantum hall states andpx +ipy paired superfluids, Phys

    N. Read, Non-abelian adiabatic statistics and hall viscos- ity in quantum hall states andpx +ipy paired superfluids, Phys. Rev. B79, 045308 (2009)

    2009

  38. [46]

    N.ReadandE.H.Rezayi,Hallviscosity, orbitalspin, and geometry: Paired superfluids and quantum hall systems, Phys. Rev. B84, 085316 (2011)

    2011

  39. [47]

    Bradlyn, M

    B. Bradlyn, M. Goldstein, and N. Read, Kubo formulas for viscosity: Hall viscosity, ward identities, and the rela- tion with conductivity, Phys. Rev. B86, 245309 (2012)

    2012

  40. [48]

    L. V. Delacrétaz and A. Gromov, Transport signatures of the hall viscosity, Phys. Rev. Lett.119, 226602 (2017)

    2017

  41. [49]

    Hosaka, S

    Y. Hosaka, S. Komura, and D. Andelman, Nonreciprocal response of a two-dimensional fluid with odd viscosity, Physical Review E103, 042610 (2021)

    2021

  42. [50]

    Khain, C

    T. Khain, C. Scheibner, M. Fruchart, and V. Vitelli, Stokes flows in three-dimensional fluids with odd and parity-violating viscosities, Journal of Fluid Mechanics 934, A23 (2022)

    2022

  43. [51]

    More odd terms are allowed by symmetry in 3D materials with cylindrical symmetry [50]

  44. [52]

    P. C. Martin, O. Parodi, and P. S. Pershan, Unified hy- drodynamic theory for crystals, liquid crystals, and nor- mal fluids, Phys. Rev. A6, 2401 (1972)

    1972

  45. [53]

    K. Y. Bliokh, H. Punzmann, H. Xia, F. Nori, and M. Shats, Field theory spin and momentum in water waves, Science Advances8, eabm1295 (2022)

    2022

  46. [54]

    Lifshitz, Course of Theoretical Physics: Theory of Elas- ticity, Vol

    L.D.Landau, L.P.Pitaevskij, A.M.Kosevich,andE.M. Lifshitz, Course of Theoretical Physics: Theory of Elas- ticity, Vol. 7, 3rd ed. (Butterworth-Heinemann, Oxford, 1986)

    1986

  47. [55]

    The total momentum stress tensor can have antisymmet- ric terms that can be written as a divergence of a third- rank tensor, but these terms can be symmetrized [23, 24, 54]

  48. [56]

    Stark and T

    H. Stark and T. C. Lubensky, Poisson-bracket approach to the dynamics of nematic liquid crystals, Phys. Rev. E 67, 061709 (2003)

    2003

  49. [57]

    Stark and T

    H. Stark and T. C. Lubensky, Poisson bracket approach to the dynamics of nematic liquid crystals: The role of spinangularmomentum,Phys.Rev.E 72,051714(2005)

    2005

  50. [58]

    P. M. Chaikin and L. C. Lubensky,Principles of con- densed matter physics (Cambridge University Press, 1995)

    1995

  51. [59]

    Maitra, P

    A. Maitra, P. Srivastava, M. C. Marchetti, S. Ra- maswamy, and M. Lenz, Swimmer suspensions on sub- strates: Anomalous stability and long-range order, Phys. Rev. Lett.124, 028002 (2020)

    2020

  52. [60]

    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

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.