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Colloquium: What do we mean by ‘active matter’?

Michael te Vrugt*, Benno Liebchen, and Michael E. Cates

Michael te Vrugt*

Benno Liebchen

Michael E. Cates

  • *Contact author: tevrugtm@uni-mainz.de
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Rev. Mod. Phys. 98, 031001 – Published 13 July, 2026

DOI: https://doi.org/10.1103/wd4f-q7kv

Abstract

As the study of active matter has developed into one of the most rapidly growing subfields of condensed matter physics, more and more kinds of physical systems have been included in this framework. While the word ‘active’ is often thought of as referring to self-propelled particles, it is also applied to a large variety of other systems. Developing novel forms of active matter, as attempted, for example, in the framework of quantum active matter, requires a clear idea of what active matter is. Active matter is defined here based on commonly employed criteria as a system that is driven out of equilibrium by a local energy influx and that generates motion and/or forces whose direction is not determined by external fields. The limits of such a definition and to what extent a unified definition of ‘active matter’ is still possible are also critically discussed. Moreover, we discuss the definition of an ‘active field theory,’ where ‘active’ is used as an attribute of a theoretical model rather than of a physical system. We show that the usage of the term ‘active’ requires agreement on a description that neglects certain degrees of freedom. The meaning of this word is discussed both in general terms and via the specific examples of chemically driven particles, particles with nonreciprocal interactions, intracellular phase separation, and quantum active matter.

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References (290)

  1. Adachi, Kyosuke, Kazuaki Takasan, and Kyogo Kawaguchi, 2022, “Activity-induced phase transition in a quantum many-body system,” Phys. Rev. Res. 4, 013194.
  2. Aditi Simha, R., and Sriram Ramaswamy, 2002, “Hydrodynamic Fluctuations and Instabilities in Ordered Suspensions of Self-Propelled Particles,” Phys. Rev. Lett. 89, 058101.
  3. Albert, David Z., 2000, Time and Chance (Harvard University Press, Cambridge, MA).
  4. Alicea, Jason, Leon Balents, Matthew P. A. Fisher, Arun Paramekanti, and Leo Radzihovsky, 2005, “Transition to zero resistance in a two-dimensional electron gas driven with microwaves,” Phys. Rev. B 71, 235322.
  5. Altman, Ehud, Lukas M. Sieberer, Leiming Chen, Sebastian Diehl, and John Toner, 2015, “Two-Dimensional Superfluidity of Exciton Polaritons Requires Strong Anisotropy,” Phys. Rev. X 5, 011017.
  6. Antonov, Alexander P., Sangyun Lee, Benno Liebchen, Hartmut Löwen, Jannis Melles, Giovanna Morigi, Yehor Tuchkov, and Michael te Vrugt, 2026, “Modeling dissipation in quantum active matter,” Phys. Rev. A 114, 012201.
  7. Antonov, Alexander P., Yuanjian Zheng, Benno Liebchen, and Hartmut Löwen, 2025, “Engineering active motion in quantum matter,” Phys. Rev. Res. 7, 033008.
  8. Aoki, Ichiro, 1982, “A simulation study on the schooling mechanism in fish.” Nippon Suisan Gakkaishi 48, 1081–1088.
  9. Archer, A. J., and R. Evans, 2004, “Dynamical density functional theory and its application to spinodal decomposition,” J. Chem. Phys. 121, 4246–4254.
  10. Archer, Andrew J., and Markus Rauscher, 2004, “Dynamical density functional theory for interacting Brownian particles: Stochastic or deterministic?,” J. Phys. A 37, 9325–9333.
  11. Astrobiology at NASA, 2026a, “About life detection,” https://astrobiology.nasa.gov/research/life-detection/about/ (accessed April 11, 2026).
  12. Avni, Yael, Michel Fruchart, David Martin, Daniel Seara, and Vincenzo Vitelli, 2025, “Nonreciprocal Ising Model,” Phys. Rev. Lett. 134, 117103.
  13. Avron, J. E., 1998, “Odd viscosity,” J. Stat. Phys. 92, 543–557.
  14. Avron, J. E., Ruedi Seiler, and Petr G. Zograf, 1995, “Viscosity of Quantum Hall Fluids,” Phys. Rev. Lett. 75, 697.
  15. Baker-Rasooli, Myrann, Tangui Aladjidi, Tiago D. Ferreira, Alberto Bramati, Mathias Albert, Pierre-Élie Larré, and Quentin Glorieux, 2026, “Swimming against a Superfluid Flow: Self-Propulsion via Vortex-Antivortex Shedding in a Quantum Fluid of Light,” Phys. Rev. Lett. 136, 223401.
  16. Banerjee, Debarghya, Anton Souslov, Alexander G. Abanov, and Vincenzo Vitelli, 2017, “Odd viscosity in chiral active fluids,” Nat. Commun. 8, 1573.
  17. Barberis, Lucas, and Fernando Peruani, 2016, “Large-Scale Patterns in a Minimal Cognitive Flocking Model: Incidental Leaders, Nematic Patterns, and Aggregates,” Phys. Rev. Lett. 117, 248001.
  18. Bauermann, Jonathan, Giacomo Bartolucci, Job Boekhoven, Christoph A. Weber, and Frank Jülicher, 2023, “Formation of liquid shells in active droplet systems,” Phys. Rev. Res. 5, 043246.
  19. Bayrhuber, Horst, Wolfgang Hauber, and Ulrich Kull, 2010, Eds., Linder Biologie—Lehrbuch für die Oberstufe [Lindner Biology—Textbook for High School], 23rd ed. (Westermann, Braunschweig).
  20. Bebon, Robin, Joshua F. Robinson, and Thomas Speck, 2025, “Thermodynamics of Active Matter: Tracking Dissipation across Scales,” Phys. Rev. X 15, 021050.
  21. Bechinger, Clemens, Roberto Di Leonardo, Hartmut Löwen, Charles Reichhardt, Giorgio Volpe, and Giovanni Volpe, 2016, “Active particles in complex and crowded environments,” Rev. Mod. Phys. 88, 045006.
  22. Bénard, Henri, 1901, “Les tourbillons cellulaires dans une nappe liquide.—Méthodes optiques d’observation et d’enregistrement [Cellular vortices (Bénard cells) in a liquid layer—Optical methods of observation and recording],” J. Phys. Theor. Appl. 10, 254–266.
  23. Bergmann, Alexander M., Jonathan Bauermann, Giacomo Bartolucci, Carsten Donau, Michele Stasi, Anna-Lena Holtmannspötter, Frank Jülicher, Christoph A. Weber, and Job Boekhoven, 2023, “Liquid spherical shells are a non-equilibrium steady state of active droplets,” Nat. Commun. 14, 6552.
  24. Berry, Joel, Clifford P. Brangwynne, and Mikko Haataja, 2018, “Physical principles of intracellular organization via active and passive phase transitions,” Rep. Prog. Phys. 81, 046601.
  25. Betti, Enrico, 1872, “Teoria della elasticità [Theory of elasticity],” Nuovo Cimento 7, 69–97.
  26. Bilotto, Pierpaolo, Lorenzo Caprini, and Angelo Vulpiani, 2021, “Excess and loss of entropy production for different levels of coarse graining,” Phys. Rev. E 104, 024140.
  27. Bore, Sigbjørn Løland, Michael Schindler, Khanh-Dang Nguyen Thu Lam, Eric Bertin, and Olivier Dauchot, 2016, “Coupling spin to velocity: Collective motion of Hamiltonian polar particles,” J. Stat. Mech. 033305.
  28. Bowick, Mark J., Nikta Fakhri, M. Cristina Marchetti, and Sriram Ramaswamy, 2022, “Symmetry, Thermodynamics, and Topology in Active Matter,” Phys. Rev. X 12, 010501.
  29. Brems, Maarten A., Tobias Sparmann, Simon M. Fröhlich, Leonie-C. Dany, Jan Rothörl, Fabian Kammerbauer, Elizabeth M. Jefremovas, Oded Farago, Mathias Kläui, and Peter Virnau, 2025, “Realizing Quantitative Quasiparticle Modeling of Skyrmion Dynamics in Arbitrary Potentials,” Phys. Rev. Lett. 134, 046701.
  30. Bricard, Antoine, Jean-Baptiste Caussin, Nicolas Desreumaux, Olivier Dauchot, and Denis Bartolo, 2013, “Emergence of macroscopic directed motion in populations of motile colloids,” Nature (London) 503, 95–98.
  31. Burgardt, Sabrina, et al., 2026, “Quantum-enabled active matter at the atomic scale,” arXiv:2606.24615.
  32. Bürger, Gottfried August, 1786, Wunderbare Reisen zu Wasser und Lande, Feldzüge und lustige Abentheuer des Freyherrn von Münchhausen, wie er dieselben bey der Flasche im Cirkel seiner Freunde selbst zu erzählen pflegt [Marvelous Travels by Sea and Land, Campaigns and Merry Adventures of Baron Munchausen, as He Is Wont to Tell Them Himself over a Bottle in the Circle of His Friends] (Dieterich, Göttingen).
  33. Burla, Federica, Yuval Mulla, Bart E. Vos, Anders Aufderhorst-Roberts, and Gijsje H. Koenderink, 2019, “From mechanical resilience to active material properties in biopolymer networks,” Nat. Rev. Phys. 1, 249–263.
  34. Caflisch, Russel E., Chjan C. Lim, Jonathan H. C. Luke, and Ashok S. Sangani, 1988, “Periodic solutions for three sedimenting spheres,” Phys. Fluids 31, 3175–3179.
  35. Cahn, J. W., and J. E. Hilliard, 1958, “Free energy of a nonuniform system. I. Interfacial free energy,” J. Chem. Phys. 28, 258–267.
  36. Cahn, John W., 1965, “Phase separation by spinodal decomposition in isotropic systems,” J. Chem. Phys. 42, 93–99.
  37. Cameron, Samuel, and Elsen Tjhung, 2025, “Exact results in stochastic processes with division, death, and diffusion: Spatial correlations, marginal entropy production, and macroscopic currents,” Phys. Rev. E 112, 034109.
  38. Caprini, Lorenzo, and Umberto Marini Bettolo Marconi, 2025a, “Bubble phase induced by odd interactions in chiral systems,” J. Chem. Phys. 162, 161101.
  39. Caprini, Lorenzo, and Umberto Marini Bettolo Marconi, 2025b, “Odd active solids: Vortices, velocity oscillations and dissipation-free modes,” New J. Phys. 27, 054401.
  40. Casimir, Hendrik Brugt Gerhard, 1945, “On Onsager’s principle of microscopic reversibility,” Rev. Mod. Phys. 17, 343.
  41. Casiulis, Mathias, Marco Tarzia, Leticia F. Cugliandolo, and Olivier Dauchot, 2020, “Velocity and Speed Correlations in Hamiltonian Flocks,” Phys. Rev. Lett. 124, 198001.
  42. Castéra, Jean-Paul, and Paul-Louis Meunier, 1989, “Nonreciprocal devices in integrated optics,” Fiber Integr. Opt. 8, 71–85.
  43. Cates, M. E., and J. Tailleur, 2015, “Motility-induced phase separation,” Annu. Rev. Condens. Matter Phys. 6, 219–244.
  44. Cates, Michael E., 2022, “Active field theories,” in Active Matter and Nonequilibrium Statistical Physics, Lecture Notes of the Les Houches Summer School Vol. 112, edited by Julien Tailleur, Gerhard Gompper, M. Cristina Marchetti, Julia M. Yeomans, and Christophe Salomon (Oxford University Press, Oxford), pp. 180–216.
  45. Cates, Michael Elmhurst, and Cesare Nardini, 2025, “Active phase separation: New phenomenology from non-equilibrium physics,” Rep. Prog. Phys. 88, 056601.
  46. Cavagna, Andrea, and Irene Giardina, 2014, “Bird flocks as condensed matter,” Annu. Rev. Condens. Matter Phys. 5, 183–207.
  47. Chaté, Hugues, Francesco Ginelli, and Raúl Montagne, 2006, “Simple Model for Active Nematics: Quasi-Long-Range Order and Giant Fluctuations,” Phys. Rev. Lett. 96, 180602.
  48. Chen, Y. X., and T. Kolokolnikov, 2014, “A minimal model of predator-swarm interactions,” J. R. Soc. Interface 11, 20131208.
  49. Chow, Yunshyong, and F. Y. Wu, 1987, “Residual entropy and validity of the third law of thermodynamics in discrete spin systems,” Phys. Rev. B 36, 285–288.
  50. Clausen, Sigmund, Geir Helgesen, and Arne T. Skjeltorp, 1998, “Braid description of collective fluctuations in a few-body system,” Phys. Rev. E 58, 4229–4237.
  51. Cook, H. E., 1970, “Brownian motion in spinodal decomposition,” Acta Metall. 18, 297–306.
  52. Coulais, Corentin, Dimitrios Sounas, and Andrea Alu, 2017, “Static non-reciprocity in mechanical metamaterials,” Nature (London) 542, 461–464.
  53. Cross, Mark C., and Pierre C. Hohenberg, 1993, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112.
  54. Dadhichi, Lokrshi Prawar, Jitendra Kethapelli, Rahul Chajwa, Sriram Ramaswamy, and Ananyo Maitra, 2020, “Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks,” Phys. Rev. E 101, 052601.
  55. Dadhichi, Lokrshi Prawar, Ananyo Maitra, and Sriram Ramaswamy, 2018, “Origins and diagnostics of the nonequilibrium character of active systems,” J. Stat. Mech. 123201.
  56. Das, Tanisha, and Shirin Sultana, 2024, “Multifaceted applications of micro/nanorobots in pharmaceutical drug delivery systems: A comprehensive review,” Future J. Pharm. Sci. 10, 2.
  57. Dean, D. S., 1996, “Langevin equation for the density of a system of interacting Langevin processes,” J. Phys. A 29, L613–L617.
  58. de Gennes, Pierre-Gilles, 1992, “Soft matter (Nobel lecture),” Angew. Chem., Int. Ed. Engl. 31, 842–845.
  59. del Ser, Nina, and Vivek Lohani, 2023, “Skyrmion jellyfish in driven chiral magnets,” SciPost Phys. 15, 065.
  60. Derényi, I., P. Tegzes, and T. Vicsek, 1998, “Collective transport in locally asymmetric periodic structures,” Chaos 8, 657–664.
  61. Deseigne, Julien, Olivier Dauchot, and Hugues Chaté, 2010, “Collective Motion of Vibrated Polar Disks,” Phys. Rev. Lett. 105, 098001.
  62. de Souza Silva, Clécio C., Matheus V. Correia, and J. C. Piña Velásquez, 2025, “Emergent Self-Propulsion of Skyrmionic Matter in Synthetic Antiferromagnets,” Phys. Rev. Lett. 135, 086701.
  63. Dhont, Jan K. G., 1996, An Introduction to Dynamics of Colloids, 1st ed., Studies in Interface Science Vol. 2 (Elsevier Science, Amsterdam).
  64. Dieball, Cai, and Aljaž Godec, 2025, “Perspective: Time irreversibility in systems observed at coarse resolution,” J. Chem. Phys. 162, 090901.
  65. Ding, Mingnan, and Michael E. Cates, 2025, “Hamiltonian heat baths, coarse-graining and irreversibility: A microscopic dynamical entropy from classical mechanics,” arXiv:2503.11334.
  66. DiVincenzo, David P., 2000, “The physical implementation of quantum computation,” Fortschr. Phys. 48, 771–783.
  67. Dombrowski, Christopher, Luis Cisneros, Sunita Chatkaew, Raymond E. Goldstein, and John O. Kessler, 2004, “Self-Concentration and Large-Scale Coherence in Bacterial Dynamics,” Phys. Rev. Lett. 93, 098103.
  68. Doostmohammadi, Amin, Jordi Ignés-Mullol, Julia M. Yeomans, and Francesc Sagués, 2018, “Active nematics,” Nat. Commun. 9, 3246.
  69. Dopierala, Dawid, Hugues Chaté, Xia-qing Shi, and Alexandre Solon, 2025, “Inescapable Anisotropy of Nonreciprocal XY Models,” Phys. Rev. Lett. 135, 088302.
  70. Dormann, Dorothee, and Edward Anton Lemke, 2024, “Adding intrinsically disordered proteins to biological ageing clocks,” Nat. Cell Biol. 26, 851–858.
  71. Elgeti, J., R. G. Winkler, and G. Gompper, 2015, “Physics of microswimmers—Single particle motion and collective behavior: A review,” Rep. Prog. Phys. 78, 056601.
  72. Enculescu, M., and H. Stark, 2011, “Active Colloidal Suspensions Exhibit Polar Order under Gravity,” Phys. Rev. Lett. 107, 058301.
  73. Erkamp, Nadia A., Tomas Sneideris, Hannes Ausserwöger, Daoyuan Qian, Seema Qamar, Jonathon Nixon-Abell, Peter St George-Hyslop, Jeremy D. Schmit, David A. Weitz, and Tuomas P. J. Knowles, 2023, “Spatially non-uniform condensates emerge from dynamically arrested phase separation,” Nat. Commun. 14, 684.
  74. Evans, R., 1979, “The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids,” Adv. Phys. 28, 143–200.
  75. Farkas, Z., P. Tegzes, A. Vukics, and T. Vicsek, 1999, “Transitions in the horizontal transport of vertically vibrated granular layers,” Phys. Rev. E 60, 7022.
  76. Feng, Kai, José Carlos Ureña Marcos, Aritra K. Mukhopadhyay, Ran Niu, Qiang Zhao, Jinping Qu, and Benno Liebchen, 2023, “Self-solidifying active droplets showing memory-induced chirality,” Adv. Sci. Lett. 10, e2300866.
  77. Feynman, Richard P., 1961, “There’s plenty of room at the bottom,” in Miniaturization, edited by Horace D. Gilbert (Reinhold, New York), pp. 282–296.
  78. Feynman, Richard P., R. Leighton, and M. Sands, 1964, The Feynman Lectures on Physics, Vol. I (Addison-Wesley, Reading, MA).
  79. Finlayson, B. A., and L. E. Scriven, 1969, “Convective instability by active stress,” Proc. R. Soc. A 310, 183–219.
  80. Fleischer, Richard, 1966, director, Fantastic Voyage (Twentieth Century Fox, Los Angeles).
  81. Fodor, Étienne, Robert L. Jack, and Michael E. Cates, 2022, “Irreversibility and biased ensembles in active matter: Insights from stochastic thermodynamics,” Annu. Rev. Condens. Matter Phys. 13, 215–238.
  82. Fossati, Michele, Colin Scheibner, Michel Fruchart, and Vincenzo Vitelli, 2024, “Odd elasticity and topological waves in active surfaces,” Phys. Rev. E 109, 024608.
  83. Freimuth, R. D., and L. Lam, 1992, “Active walker models for filamentary growth patterns,” in Modeling Complex Phenomena Woodward Conference, edited by L. Lam and V. Naroditsky (Springer, New York), pp. 302–313.
  84. Fruchart, Michel, Colin Scheibner, and Vincenzo Vitelli, 2023, “Odd viscosity and odd elasticity,” Annu. Rev. Condens. Matter Phys. 14, 471–510.
  85. Fukushima, Kunihiko, 1980, “Neocognitron: A self-organizing neural network model for a mechanism of pattern recognition unaffected by shift in position,” Biol. Cybern. 36, 193–202.
  86. Fürth, Reinhold, 1920, “Die Brownsche Bewegung bei Berücksichtigung einer Persistenz der Bewegungsrichtung. Mit Anwendungen auf die Bewegung lebender Infusorien [Brownian motion taking into account a persistence of the direction of motion. With applications to the motion of living infusoria],” Z. Phys. 2, 244–256.
  87. Gaijin Games Kft., 2026, “Active matter,” https://activematter.game/en/ (accessed January 7, 2026).
  88. Ganeshan, Sriram, and Alexander G. Abanov, 2017, “Odd viscosity in two-dimensional incompressible fluids,” Phys. Rev. Fluids 2, 094101.
  89. Ge, Yuqing, Jan Rothörl, Maarten A. Brems, Nico Kerber, Raphael Gruber, Takaaki Dohi, Mathias Kläui, and Peter Virnau, 2023, “Constructing coarse-grained skyrmion potentials from experimental data with iterative Boltzmann inversion,” Commun. Phys. 6, 30.
  90. Ghimenti, Federico, Ludovic Berthier, Grzegorz Szamel, and Frédéric van Wijland, 2023, “Sampling Efficiency of Transverse Forces in Dense Liquids,” Phys. Rev. Lett. 131, 257101.
  91. Ghimenti, Federico, Ludovic Berthier, Grzegorz Szamel, and Frédéric van Wijland, 2024, “Transverse forces and glassy liquids in infinite dimensions,” Phys. Rev. E 109, 064133.
  92. Golestanian, Ramin, 2009, “Anomalous Diffusion of Symmetric and Asymmetric Active Colloids,” Phys. Rev. Lett. 102, 188305.
  93. Gompper, Gerhard, et al., 2020, “The 2020 motile active matter roadmap,” J. Phys. Condens. Matter 32, 193001.
  94. Grabert, Hermann, 1982, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, 1st ed., Springer Tracts in Modern Physics Vol. 95 (Springer-Verlag, Berlin).
  95. Granek, Omer, Yariv Kafri, Mehran Kardar, Sunghan Ro, Julien Tailleur, and Alexandre Solon, 2024, “Colloquium: Inclusions, boundaries, and disorder in scalar active matter,” Rev. Mod. Phys. 96, 031003.
  96. Grosberg, Alexander Y., and Jean-François Joanny, 2018, “Dissipation in a system driven by two different thermostats,” Polym. Sci., Ser. C 60, 118–121.
  97. Grosberg, A. Yu., and J.-F. Joanny, 2015, “Nonequilibrium statistical mechanics of mixtures of particles in contact with different thermostats,” Phys. Rev. E 92, 032118.
  98. Guillet, Stéphane, Alexis Poncet, Marine Le Blay, William T. M. Irvine, Vincenzo Vitelli, and Denis Bartolo, 2025, “Melting of nonreciprocal solids: How dislocations propel and fission in flowing crystals,” Proc. Natl. Acad. Sci. U.S.A. 122, e2412993122.
  99. Hallatschek, Oskar, Sujit S. Datta, Knut Drescher, Jörn Dunkel, Jens Elgeti, Bartek Waclaw, and Ned S. Wingreen, 2023, “Proliferating active matter,” Nat. Rev. Phys. 5, 407–419.
  100. Hanai, Ryo, Daiki Ootsuki, and Rina Tazai, 2025, “Photoinduced non-reciprocal magnetism,” Nat. Commun. 16, 8195.
  101. Hardt, Dennis, Reza Doostani, Sebastian Diehl, Nina del Ser, and Achim Rosch, 2025, “Propelling ferrimagnetic domain walls by dynamical frustration,” Nat. Commun. 16, 3817.
  102. Herrera, Pedro, and Mario Sandoval, 2024, “Structure of the active Fokker-Planck equation,” Phys. Rev. E 109, 014140.
  103. Hickey, David J., Ramin Golestanian, and Andrej Vilfan, 2023, “Nonreciprocal interactions give rise to fast cilium synchronization in finite systems,” Proc. Natl. Acad. Sci. U.S.A. 120, e2307279120.
  104. Hinch, E. J., 1988, “Sedimentation of small particles,” in Disorder and Mixing: Convection, Diffusion and Reaction in Random Materials and Processes, edited by J.-P. E. Guyon and Y. Pomeau Nadal (Springer, Dordrecht), pp. 153–162.
  105. Hohenberg, P. C., and B. I. Halperin, 1977, “Theory of dynamic critical phenomena,” Rev. Mod. Phys. 49, 435–479.
  106. Hu, Mengyi, Xuemei Ge, Xuan Chen, Wenwei Mao, Xiuping Qian, and Wei-En Yuan, 2020, “Micro/nanorobot: A promising targeted drug delivery system,” Pharmaceutics 12, 665.
  107. Huang, Zhi-Feng, Andreas M. Menzel, and Hartmut Löwen, 2020, “Dynamical Crystallites of Active Chiral Particles,” Phys. Rev. Lett. 125, 218002.
  108. Huang, Zhi-Feng, Michael te Vrugt, Raphael Wittkowski, and Hartmut Löwen, 2024, “Active pattern formation emergent from single-species nonreciprocity,” arXiv:2404.10093.
  109. Huang, Zhi-Feng, Michael te Vrugt, Raphael Wittkowski, and Hartmut Löwen, 2025, “Anomalous grain dynamics and grain locomotion of odd crystals,” Proc. Natl. Acad. Sci. U.S.A. 122, e2511350122.
  110. Hyman, Anthony A., Christoph A. Weber, and Frank Jülicher, 2014, “Liquid-liquid phase separation in biology,” Annu. Rev. Cell Dev. Biol. 30, 39–58.
  111. Illien, Pierre, 2025, “The Dean-Kawasaki equation and stochastic density functional theory,” Rep. Prog. Phys. 88, 086601.
  112. Joanny, Jean-François, and Jacques Prost, 2009, “Active gels as a description of the actin-myosin cytoskeleton,” HFSP J. 3, 94–104.
  113. Jülicher, Frank, Armand Ajdari, and Jacques Prost, 1997, “Modeling molecular motors,” Rev. Mod. Phys. 69, 1269.
  114. Jülicher, Frank, and Christoph A. Weber, 2024, “Droplet physics and intracellular phase separation,” Annu. Rev. Condens. Matter Phys. 15, 237–261.
  115. Kabrisky, Matthew, 1966, A Proposed Model for Visual Information Processing in the Human Brain (University of Illinois Press, Urbana, IL).
  116. Kalz, Erik, Hidde Derk Vuijk, Iman Abdoli, Jens-Uwe Sommer, Hartmut Löwen, and Abhinav Sharma, 2022, “Collisions Enhance Self-Diffusion in Odd-Diffusive Systems,” Phys. Rev. Lett. 129, 090601.
  117. Kardar, M., G. Parisi, and Y.-C. Zhang, 1986, “Dynamic Scaling of Growing Interfaces,” Phys. Rev. Lett. 56, 889–892.
  118. Kawasaki, Kyozi, 1994, “Stochastic model of slow dynamics in supercooled liquids and dense colloidal suspensions,” Physica (Amsterdam) 208A, 35–64.
  119. Khasseh, Reyhaneh, Sascha Wald, Roderich Moessner, Christoph A. Weber, and Markus Heyl, 2025, “Active Quantum Flocks,” Phys. Rev. Lett. 135, 248302.
  120. Kistler, Kathryn E., Tatjana Trcek, Thomas R. Hurd, Ruoyu Chen, Feng-Xia Liang, Joseph Sall, Masato Kato, and Ruth Lehmann, 2018, “Phase transitioned nuclear Oskar promotes cell division of Drosophila primordial germ cells,” eLife 7, e37949.
  121. Kolmakov, G. V., and Igor S. Aranson, 2021, “Superfluid swimmers,” Phys. Rev. Res. 3, 013188.
  122. Kopp, Robin A., and Sabine H. L. Klapp, 2023a, “Persistent motion of a Brownian particle subject to repulsive feedback with time delay,” Phys. Rev. E 107, 024611.
  123. Kopp, Robin A., and Sabine H. L. Klapp, 2023b, “Spontaneous velocity alignment of Brownian particles with feedback-induced propulsion,” Europhys. Lett. 143, 17002.
  124. Kroy, Klaus, and Frank Cichos, 2023, “Hot Brownian motion,” in Diffusive Spreading in Nature, Technology and Society, edited by A. Bunde, J. Caro, C. Chmelik, J. Kärger, and G. Vogl (Springer, Cham, Switzerland), pp. 133–151.
  125. Kümmel, Felix, Borge ten Hagen, Raphael Wittkowski, Daisuke Takagi, Ivo Buttinoni, Ralf Eichhorn, Giovanni Volpe, Hartmut Löwen, and Clemens Bechinger, 2014, “Kümmel et al. Reply:,” Phys. Rev. Lett. 113, 029802.
  126. Kundu, Aritra, 2025, “Run and tumble dynamics of biased quantum trajectories in a monitored qubit,” arXiv:2512.20519.
  127. Laertius, Diogenes, 2018, Lives of the Eminent Philosophers, translated by and edited by James Miller (Oxford University Press, New York).
  128. Langer, Amelie, Abhinav Sharma, Ralf Metzler, and Erik Kalz, 2024, “Dance of odd-diffusive particles: A Fourier approach,” Phys. Rev. Res. 6, 043036.
  129. Lee, Jin Gyun, Ritu R. Raj, Nicole B. Day, and C. Wyatt Shields IV, 2023, “Microrobots for biomedicine: unsolved challenges and opportunities for translation,” ACS Nano 17, 14196–14204.
  130. Lee, Sangyun, Jae Sung Lee, and Jong-Min Park, 2025, “Speed limits for discrete-time Markov chains,” Phys. Rev. E 112, 044132.
  131. Lee, Sangyun, Yehor Tuchkov, Alexander P. Antonov, Benno Liebchen, Hartmut Löwen, Giovanna Morigi, and Michael te Vrugt, 2026, “Anomalous mean-squared displacement in quantum active matter from a Wigner phase-space framework,” arXiv:2604.22600.
  132. Lemaitre, J., A. Gervois, H. Peerhossaini, D. Bideau, and J. P. Troadec, 1990, “An air table designed to study two-dimensional disc packings: Preliminary tests and first results,” J. Phys. D 23, 1396.
  133. Lima, F. W. S., and D. Stauffer, 2006, “Ising model simulation in directed lattices and networks,” Physica (Amsterdam) 359A, 423–429.
  134. Lipowski, Adam, António Luis Ferreira, Dorota Lipowska, and Krzysztof Gontarek, 2015, “Phase transitions in Ising models on directed networks,” Phys. Rev. E 92, 052811.
  135. Lish, Samantha R., Lukas Hupe, Ramin Golestanian, and Philip Bittihn, 2024, “Isovolumetric dividing active matter,” arXiv:2409.20481.
  136. Liška,Vojtěch, Tereza Zemánková, Petr Jákl, Martin Šiler, Stephen H. Simpson, Pavel Zemánek, and Oto Brzobohatý, 2024, “PT-like phase transition and limit cycle oscillations in non-reciprocally coupled optomechanical oscillators levitated in vacuum,” Nat. Phys. 20, 1622–1628.
  137. Liu, Maoji, Zhanglin Hou, Hiroyuki Kitahata, Linli He, and Shigeyuki Komura, 2023, “Non-reciprocal phase separations with non-conserved order parameters,” J. Phys. Soc. Jpn. 92, 093001.
  138. Liu, Zeng Tao, Yan Shi, Yongfeng Zhao, Hugues Chaté, Xia-qing Shi, and Tian Hui Zhang, 2021, “Activity waves and freestanding vortices in populations of subcritical Quincke rollers,” Proc. Natl. Acad. Sci. U.S.A. 118, e2104724118.
  139. Liverpool, Tanniemola B., 2006, “Active gels: Where polymer physics meets cytoskeletal dynamics,” Phil. Trans. R. Soc. A 364, 3335–3355.
  140. Loewe, Benjamin, Anton Souslov, and Paul M. Goldbart, 2018, “Flocking from a quantum analogy: Spin-orbit coupling in an active fluid,” New J. Phys. 20, 013020.
  141. Loos, Sarah A. M., and Sabine H. L. Klapp, 2019, “Heat flow due to time-delayed feedback,” Sci. Rep. 9, 2491.
  142. Loos, Sarah A. M., Sabine H. L. Klapp, and Thomas Martynec, 2023, “Long-Range Order and Directional Defect Propagation in the Nonreciprocal XY Model with Vision Cone Interactions,” Phys. Rev. Lett. 130, 198301.
  143. López, H. M., J. Gachelin, C. Douarche, H. Auradou, and E. Clément, 2015, “Turning Bacteria Suspensions into Superfluids,” Phys. Rev. Lett. 115, 028301.
  144. Lotka, Alfred James, 1925, Elements of Physical Biology (Williams and Wilkins, Philadelphia).
  145. Lou, Xin, Qing Yang, Yu Ding, Peng Liu, Ke Chen, Xin Zhou, Fangfu Ye, Rudolf Podgornik, and Mingcheng Yang, 2022, “Odd viscosity-induced Hall-like transport of an active chiral fluid,” Proc. Natl. Acad. Sci. U.S.A. 119, e2201279119.
  146. Luo, M., Y. Feng, T. Wang, and J. Guan, 2018, “Micro-/nanorobots at work in active drug delivery,” Adv. Funct. Mater. 28, 1706100.
  147. Maggi, Claudio, Umberto Marini Bettolo Marconi, Nicoletta Gnan, and Roberto Di Leonardo, 2015, “Multidimensional stationary probability distribution for interacting active particles,” Sci. Rep. 5, 10742.
  148. Maitra, Ananyo, 2025, “Activity unmasks chirality in liquid-crystalline matter,” Annu. Rev. Condens. Matter Phys. 16, 275–295.
  149. Malament, David B., 2004, “On the time reversal invariance of classical electromagnetic theory,” Stud. Hist. Philos. Mod. Phys. 35, 295–315.
  150. Manna, Raj Kumar, Abhrajit Laskar, Oleg E. Shklyaev, and Anna C. Balazs, 2022, “Harnessing the power of chemically active sheets in solution,” Nat. Rev. Phys. 4, 125–137.
  151. Marchetti, M. C., J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, Madan Rao, and R. Aditi Simha, 2013, “Hydrodynamics of soft active matter,” Rev. Mod. Phys. 85, 1143–1189.
  152. Marini Bettolo Marconi, Umberto, and Lorenzo Caprini, 2025, “Spontaneous generation of angular momentum in chiral active crystals,” Soft Matter 21, 2586–2606.
  153. Marini Bettolo Marconi, Umberto, and Pedro Tarazona, 1999, “Dynamic density functional theory of fluids,” J. Chem. Phys. 110, 8032–8044.
  154. Mariscal, Carlos, 2026, “Life,” in The Stanford Encyclopedia of Philosophy (Spring 2026 Edition), edited by Edward N. Zalta and Uri Nodelman (Stanford University, Stanford), https://plato.stanford.edu/archives/spr2026/entries/life/.
  155. Markovich, Tomer, and Tom C. Lubensky, 2021, “Odd Viscosity in Active Matter: Microscopic Origin and 3D Effects,” Phys. Rev. Lett. 127, 048001.
  156. Martin, David, Jérémy O’Byrne, Michael E. Cates, Étienne Fodor, Cesare Nardini, Julien Tailleur, and Frédéric van Wijland, 2021, “Statistical mechanics of active Ornstein-Uhlenbeck particles,” Phys. Rev. E 103, 032607.
  157. Maxwell, J. Clerk, 1864, “On the calculation of the equilibrium and stiffness of frames,” London Edinburgh Dublin Philos. Mag. J. Sci. 27, 294–299.
  158. McLeish, Tom, 2020, “Arrest in coarsening in biological liquid-liquid phase separation—or—‘It ain’t necessarily so …,’ ” Journal Club for Condensed Matter Physics, 10.36471/JCCM_March_2020_01.
  159. Meng, Fanlong, Daiki Matsunaga, Benoît Mahault, and Ramin Golestanian, 2021, “Magnetic Microswimmers Exhibit Bose-Einstein-like Condensation,” Phys. Rev. Lett. 126, 078001.
  160. Menzel, A. M., and H. Löwen, 2013, “Traveling and Resting Crystals in Active Systems,” Phys. Rev. Lett. 110, 055702.
  161. Menzel, Andreas M., 2015, “Tuned, driven, and active soft matter,” Phys. Rep. 554, 1–45.
  162. Menzel, Andreas M., Arnab Saha, Christian Hoell, and Hartmut Löwen, 2016, “Dynamical density functional theory for microswimmers,” J. Chem. Phys. 144, 024115.
  163. Meredith, Caleb H., Pepijn G. Moerman, Jan Groenewold, Yu-Jen Chiu, Willem K. Kegel, Alfons van Blaaderen, and Lauren D. Zarzar, 2020, “Predator-prey interactions between droplets driven by non-reciprocal oil exchange,” Nat. Chem. 12, 1136–1142.
  164. Michelin, Sébastien, 2023, “Self-propulsion of chemically active droplets,” Annu. Rev. Fluid Mech. 55, 77–101.
  165. Mishra, Shradha, and Sriram Ramaswamy, 2006, “Active Nematics are Intrinsically Phase Separated,” Phys. Rev. Lett. 97, 090602.
  166. Mitchell, Emma, and Elsen Tjhung, 2022, “Macroscopic current generated by local division and apoptosis in a minimal model of tissue dynamics,” Soft Matter 18, 1082–1088.
  167. Modi, Nisha, Raghavendra Nimiwal, Jane Liao, Yitian Li, Kyle J. M. Bishop, and Allie C. Obermeyer, 2026, “Transient pH changes drive vacuole formation in enzyme-polymer condensates,” Nat. Chem. Eng. 3, 47–56.
  168. Moncho-Jordá, Arturo, and Joachim Dzubiella, 2020, “Controlling the Microstructure and Phase Behavior of Confined Soft Colloids by Active Interaction Switching,” Phys. Rev. Lett. 125, 078001.
  169. Munakata, Toyonori, Shinpei Iwama, and Makoto Kimizuka, 2009, “Linear stochastic system with delay: Energy balance and entropy production,” Phys. Rev. E 79, 031104.
  170. Mundaca-Uribe, Rodolfo, Nelly Askarinam, Ronnie H. Fang, Liangfang Zhang, and Joseph Wang, 2024, “Towards multifunctional robotic pills,” Nat. Biomed. Eng. 8, 1334–1346.
  171. Nadal, F., and E. Lauga, 2014, “Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies,” Phys. Fluids 26, 082001.
  172. Nadolny, Tobias, Christoph Bruder, and Matteo Brunelli, 2025, “Nonreciprocal Synchronization of Active Quantum Spins,” Phys. Rev. X 15, 011010.
  173. Nadolny, Tobias, Matteo Brunelli, and Christoph Bruder, 2025, “Nonreciprocal Interactions Induce Frequency Shifts in Superradiant Lasers,” Phys. Rev. Lett. 134, 193603.
  174. Naess, S. Nalum, A. Elgsaeter, G. Helgesen, and K. D. Knudsen, 2009, “Carbon nanocones: Wall structure and morphology,” Sci. Technol. Adv. Mater. 10, 065002.
  175. Narayan, Vijay, Sriram Ramaswamy, and Narayanan Menon, 2007, “Long-lived giant number fluctuations in a swarming granular nematic,” Science 317, 105–108.
  176. Nardini, Cesare, Étienne Fodor, Elsen Tjhung, Frédéric van Wijland, Julien Tailleur, and Michael E. Cates, 2017, “Entropy Production in Field Theories without Time-Reversal Symmetry: Quantifying the Non-Equilibrium Character of Active Matter,” Phys. Rev. X 7, 021007.
  177. Nartallo-Kaluarachchi, Ramón, Morten L. Kringelbach, Gustavo Deco, Renaud Lambiotte, and Alain Goriely, 2026, “Nonequilibrium physics of brain dynamics,” Phys. Rep. 1152, 1–43.
  178. Needleman, Daniel, and Zvonimir Dogic, 2017, “Active matter at the interface between materials science and cell biology,” Nat. Rev. Mater. 2, 17048.
  179. Negi, Rajendra Singh, Roland G. Winkler, and Gerhard Gompper, 2022, “Emergent collective behavior of active Brownian particles with visual perception,” Soft Matter 18, 6167–6178.
  180. Nelson, Bradley J., Ioannis K. Kaliakatsos, and Jake J. Abbott, 2010, “Microrobots for minimally invasive medicine,” Annu. Rev. Biomed. Eng. 12, 55–85.
  181. O’Byrne, Jérémy, Yariv Kafri, Julien Tailleur, and Frédéric van Wijland, 2022, “Time irreversibility in active matter, from micro to macro,” Nat. Rev. Phys. 4, 167–183.
  182. Onsager, Lars, 1931a, “Reciprocal relations in irreversible processes. I,” Phys. Rev. 37, 405–426.
  183. Onsager, Lars, 1931b, “Reciprocal relations in irreversible processes. II,” Phys. Rev. 38, 2265–2279.
  184. Ophaus, Lukas, Svetlana V. Gurevich, and Uwe Thiele, 2018, “Resting and traveling localized states in an active phase-field-crystal model,” Phys. Rev. E 98, 022608.
  185. Ouazan-Reboul, Vincent, Jaime Agudo-Canalejo, and Ramin Golestanian, 2023, “Self-organization of primitive metabolic cycles due to non-reciprocal interactions,” Nat. Commun. 14, 4496.
  186. Partridge, Brian L., 1982, “The structure and function of fish schools,” Sci. Am. 246, No. 6, 114–123.
  187. Paxton, Walter F., Kevin C. Kistler, Christine C. Olmeda, Ayusman Sen, Sarah K. St. Angelo, Yanyan Cao, Thomas E. Mallouk, Paul E. Lammert, and Vincent H. Crespi, 2004, “Catalytic nanomotors: Autonomous movement of striped nanorods,” J. Am. Chem. Soc. 126, 13424–13431.
  188. Pedley, T. J., and John O. Kessler, 1992, “Hydrodynamic phenomena in suspensions of swimming microorganisms,” Annu. Rev. Fluid Mech. 24, 313–358.
  189. Penner, Alexander-Georg, Ludmila Viotti, Rosario Fazio, Liliana Arrachea, and Felix von Oppen, 2025, “Heat-to-motion conversion for quantum active matter,” Phys. Rev. B 112, L180303.
  190. Penocchio, Emanuele, Geyao Gu, Alex Albaugh, and Todd R. Gingrich, 2025, “Power strokes in molecular motors: Predictive, irrelevant, or somewhere in between?,” J. Am. Chem. Soc. 147, 1063–1073.
  191. Pollack, Yoav G., Philip Bittihn, and Ramin Golestanian, 2022, “A competitive advantage through fast dead matter elimination in confined cellular aggregates,” New J. Phys. 24, 073003.
  192. Poncet, Alexis, and Denis Bartolo, 2022, “When Soft Crystals Defy Newton’s Third Law: Nonreciprocal Mechanics and Dislocation Motility,” Phys. Rev. Lett. 128, 048002.
  193. Popli, Pankaj, Ananyo Maitra, and Sriram Ramaswamy, 2025, “Ordering and Defect Cloaking in Nonreciprocal Lattice XY Models,” Phys. Rev. Lett. 135, 088303.
  194. Przibram, Karl, 1913, “Über die ungeordnete Bewegung niederer Tiere [On the disordered movement of lower animals],” Pfluegers Arch. Gesamte Physiol. Menschen Tiere 153, 401–405.
  195. Przibram, Karl, 1917, “Über die ungeordnete Bewegung niederer Tiere. II [On the disordered movement of lower animals. II],” Arch. Entwicklungsmech. Org. 43, 20–27.
  196. Purcell, E. M., 1977, “Life at low Reynolds number,” Am. J. Phys. 45, 3–11.
  197. Quincke, G., 1896, “Ueber Rotationen im constanten electrischen Felde [On rotations in a constant electric field],” Ann. Phys. Chem. 295, 417–486.
  198. Quine, Willard Van Orman, 1940, Mathematical Logic (Harvard University Press, Cambridge, MA).
  199. Ramaswamy, Sriram, 2010, “The mechanics and statistics of active matter,” Annu. Rev. Condens. Matter Phys. 1, 323–345.
  200. Ramaswamy, Sriram, 2017, “Active matter,” J. Stat. Mech. 054002.
  201. Ramaswamy, Sriram, 2019, “Active fluids,” Nat. Rev. Phys. 1, 640–642.
  202. Ramaswamy, Sriram, and R. Aditi Simha, 2006, “The mechanics of active matter: Broken-symmetry hydrodynamics of motile particles and granular layers,” Solid State Commun. 139, 617–622.
  203. Raspe, Rudolf Erich, 1785, Baron Munchausen’s Narrative of His Marvellous Travels and Campaigns in Russia (London).
  204. Rayleigh, Lord, 1916, “On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side,” London Edinburgh Dublin Philos. Mag. J. Sci. 32, 529–546.
  205. Reimann, Peter, 2002, “Brownian motors: Noisy transport far from equilibrium,” Phys. Rep. 361, 57–265.
  206. Reisenbauer, Manuel, Henning Rudolph, Livia Egyed, Klaus Hornberger, Anton V. Zasedatelev, Murad Abuzarli, Benjamin A. Stickler, and Uroš Delić, 2024, “Non-Hermitian dynamics and non-reciprocity of optically coupled nanoparticles,” Nat. Phys. 20, 1629–1635.
  207. Reynolds, Craig W., 1987, “Flocks, herds and schools: A distributed behavioral model,” in Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, Anaheim, 1987, edited by Maureen C. Stone (Association for Computing Machinery, New York), pp. 25–34.
  208. Rieser, Jakob, Mario A. Ciampini, Henning Rudolph, Nikolai Kiesel, Klaus Hornberger, Benjamin A. Stickler, Markus Aspelmeyer, and Uroš Delić, 2022, “Tunable light-induced dipole-dipole interaction between optically levitated nanoparticles,” Science 377, 987–990.
  209. Rings, Daniel, Romy Schachoff, Markus Selmke, Frank Cichos, and Klaus Kroy, 2010, “Hot Brownian Motion,” Phys. Rev. Lett. 105, 090604.
  210. Romanczuk, P., M. Bär, W. Ebeling, B. Lindner, and L. Schimansky-Geier, 2012, “Active Brownian particles: From individual to collective stochastic dynamics,” Eur. Phys. J. Special Topics 202, 1–162.
  211. Rosenblatt, Frank, 1958, “The perceptron: A probabilistic model for information storage and organization in the brain.” Psych. Rev. 65, 386.
  212. Rudolph, Henning, Uroš Delić, Klaus Hornberger, and Benjamin A. Stickler, 2024, “Quantum theory of non-Hermitian optical binding between nanoparticles,” Phys. Rev. A 110, 063507.
  213. Rutherford, Ernest, 1912, “The chemical effects produced by the radiations from active matter,” Br. Med. J. 2, 371–373, https://www.jstor.org/stable/25298231.
  214. Saha, Suropriya, Jaime Agudo-Canalejo, and Ramin Golestanian, 2020, “Scalar Active Mixtures: The Nonreciprocal Cahn-Hilliard Model,” Phys. Rev. X 10, 041009.
  215. Saleh, Omar A., Byoung-jin Jeon, and Tim Liedl, 2020, “Enzymatic degradation of liquid droplets of DNA is modulated near the phase boundary,” Proc. Natl. Acad. Sci. U.S.A. 117, 16160–16166.
  216. Saleh, Omar A., Sam Wilken, Todd M. Squires, and Tim Liedl, 2023, “Vacuole dynamics and popping-based motility in liquid droplets of DNA,” Nat. Commun. 14, 3574.
  217. Samanta, Avik, Lorena Baranda Pellejero, Marcos Masukawa, and Andreas Walther, 2024, “DNA-empowered synthetic cells as minimalistic life forms,” Nat. Rev. Chem. 8, 454–470.
  218. Sánchez, Alejandro D., Juan M. López, and Miguel A. Rodriguez, 2002, “Nonequilibrium Phase Transitions in Directed Small-World Networks,” Phys. Rev. Lett. 88, 048701.
  219. Saw, Thuan Beng, Wang Xi, Benoit Ladoux, and Chwee Teck Lim, 2018, “Biological tissues as active nematic liquid crystals,” Adv. Mater. 30, 1802579.
  220. Schäfer, Lars V., and Lukas S. Stelzl, 2025, “Deciphering driving forces of biomolecular phase separation from simulations,” Curr. Opin. Struct. Biol. 92, 103026.
  221. Scheibner, Colin, Anton Souslov, Debarghya Banerjee, Piotr Surówka, William T. M. Irvine, and Vincenzo Vitelli, 2020, “Odd elasticity,” Nat. Phys. 16, 475–480.
  222. Schilling, Tanja, 2022, “Coarse-grained modelling out of equilibrium,” Phys. Rep. 972, 1–45.
  223. Schimansky-Geier, Lutz, Michaela Mieth, Helge Rosé, and Horst Malchow, 1995, “Structure formation by active Brownian particles,” Phys. Lett. A 207, 140–146.
  224. Schliwa, Manfred, and Günther Woehlke, 2003, “Molecular motors,” Nature (London) 422, 759–765.
  225. Schnitzer, M. J., 1993, “Theory of continuum random walks and application to chemotaxis,” Phys. Rev. E 48, 2553–2568.
  226. Schrödinger, Erwin, 1944, What is Life? (Cambridge University Press, Cambridge, England).
  227. Seifert, Udo, 2012, “Stochastic thermodynamics, fluctuation theorems and molecular machines,” Rep. Prog. Phys. 75, 126001.
  228. Sellitto, Mauro, and Jorge Kurchan, 2005, “Shear-Thickening and Entropy-Driven Reentrance,” Phys. Rev. Lett. 95, 236001.
  229. Shaebani, M. Reza, Adam Wysocki, Roland G. Winkler, Gerhard Gompper, and Heiko Rieger, 2020, “Computational models for active matter,” Nat. Rev. Phys. 2, 181–199.
  230. Shankar, Suraj, and M. Cristina Marchetti, 2018, “Hidden entropy production and work fluctuations in an ideal active gas,” Phys. Rev. E 98, 020604.
  231. Shankar, Suraj, Anton Souslov, Mark J. Bowick, M. Cristina Marchetti, and Vincenzo Vitelli, 2022, “Topological active matter,” Nat. Rev. Phys. 4, 380–398.
  232. Shi, Zhan, Zhiyuan Zhang, Justus Schnermann, Stephan C. F. Neuhauss, Nitesh Nama, Raphael Wittkowski, and Daniel Ahmed, 2025, “Ultrasound-driven programmable artificial muscles,” Nature (London) 646, 1096–1104.
  233. Shin, Yongdae, and Clifford P. Brangwynne, 2017, “Liquid phase condensation in cell physiology and disease,” Science 357, eaaf4382.
  234. Sieberer, Lukas M., Michael Buchhold, and Sebastian Diehl, 2016, “Keldysh field theory for driven open quantum systems,” Rep. Prog. Phys. 79, 096001.
  235. Sieberer, Lukas M., Michael Buchhold, Jamir Marino, and Sebastian Diehl, 2025, “Universality in driven open quantum matter,” Rev. Mod. Phys. 97, 025004.
  236. Siebers, Frank, Robin Bebon, Ashreya Jayaram, and Thomas Speck, 2024, “Collective Hall current in chiral active fluids: Coupling of phase and mass transport through traveling bands,” Proc. Natl. Acad. Sci. U.S.A. 121, e2320256121.
  237. Singh, Rajesh, and M. E. Cates, 2019, “Hydrodynamically Interrupted Droplet Growth in Scalar Active Matter,” Phys. Rev. Lett. 123, 148005.
  238. Slack, G. W., 1963, “Sedimentation of compact clusters of uniform spheres,” Nature (London) 200, 466–467.
  239. Smrek, Jan, and Kurt Kremer, 2017, “Small Activity Differences Drive Phase Separation in Active-Passive Polymer Mixtures,” Phys. Rev. Lett. 118, 098002.
  240. Sokolov, Andrey, Mario M. Apodaca, Bartosz A. Grzybowski, and Igor S. Aranson, 2010, “Swimming bacteria power microscopic gears,” Proc. Natl. Acad. Sci. U.S.A. 107, 969–974.
  241. Solon, Alexandre P., and Julien Tailleur, 2013, “Revisiting the Flocking Transition Using Active Spins,” Phys. Rev. Lett. 111, 078101.
  242. Stark, Holger, 2018, “Artificial chemotaxis of self-phoretic active colloids: Collective behavior,” Acc. Chem. Res. 51, 2681–2688.
  243. Sweatman, Martin B., and Leo Lue, 2019, “The giant SALR cluster fluid: A review,” Adv. Theory Simul. 2, 1900025.
  244. Szamel, Grzegorz, 2014, “Self-propelled particle in an external potential: Existence of an effective temperature,” Phys. Rev. E 90, 012111.
  245. Takasan, Kazuaki, Kyosuke Adachi, and Kyogo Kawaguchi, 2024, “Activity-induced ferromagnetism in one-dimensional quantum many-body systems,” Phys. Rev. Res. 6, 023096.
  246. Tan, Tzer Han, Alexander Mietke, Junang Li, Yuchao Chen, Hugh Higinbotham, Peter J. Foster, Shreyas Gokhale, Jörn Dunkel, and Nikta Fakhri, 2022, “Odd dynamics of living chiral crystals,” Nature (London) 607, 287–293.
  247. Tarama, Sonja, Stefan U. Egelhaaf, and Hartmut Löwen, 2019, “Traveling band formation in feedback-driven colloids,” Phys. Rev. E 100, 022609.
  248. ten Hagen, Borge, Raphael Wittkowski, Daisuke Takagi, Felix Kümmel, Clemens Bechinger, and Hartmut Löwen, 2015, “Can the self-propulsion of anisotropic microswimmers be described by using forces and torques?,” J. Phys. Condens. Matter 27, 194110.
  249. te Vrugt, Michael, 2021, “The five problems of irreversibility,” Stud. Hist. Philos. Sci. 87, 136–146.
  250. te Vrugt, Michael, 2022, “Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism,” Eur. J. Philos. Sci. 12, 41.
  251. te Vrugt, Michael, Jens Bickmann, and Raphael Wittkowski, 2023, “How to derive a predictive field theory for active Brownian particles: A step-by-step tutorial,” J. Phys. Condens. Matter 35, 313001.
  252. te Vrugt, Michael, Tobias Frohoff-Hülsmann, Eyal Heifetz, Uwe Thiele, and Raphael Wittkowski, 2023, “From a microscopic inertial active matter model to the Schrödinger equation,” Nat. Commun. 14, 1302.
  253. te Vrugt, Michael, Max Philipp Holl, Aron Koch, Raphael Wittkowski, and Uwe Thiele, 2022, “Derivation and analysis of a phase field crystal model for a mixture of active and passive particles,” Model. Simul. Mater. Sci. Eng. 30, 084001.
  254. te Vrugt, Michael, Julian Jeggle, and Raphael Wittkowski, 2021, “Jerky active matter: A phase field crystal model with translational and orientational memory,” New J. Phys. 23, 063023.
  255. te Vrugt, Michael, Julian Jeggle, and Raphael Wittkowski, 2024, “Passive and active field theories for disease spreading,” J. Phys. A 57, 315003.
  256. te Vrugt, Michael, Hartmut Löwen, and Raphael Wittkowski, 2020, “Classical dynamical density functional theory: From fundamentals to applications,” Adv. Phys. 69, 121–247.
  257. te Vrugt, Michael, and Raphael Wittkowski, 2020, “Projection operators in statistical mechanics: A pedagogical approach,” Eur. J. Phys. 41, 045101.
  258. te Vrugt, Michael, and Raphael Wittkowski, 2025, “Metareview: A survey of active matter reviews,” Eur. Phys. J. E 48, 12.
  259. Thiele, A. A., 1973, “Steady-State Motion of Magnetic Domains,” Phys. Rev. Lett. 30, 230.
  260. Tjhung, Elsen, Cesare Nardini, and Michael E. Cates, 2018, “Cluster Phases and Bubbly Phase Separation in Active Fluids: Reversal of the Ostwald Process,” Phys. Rev. X 8, 031080.
  261. Trinschek, Sarah, Karin John, Sigoléne Lecuyer, and Uwe Thiele, 2017, “Continuous versus Arrested Spreading of Biofilms at Solid-Gas Interfaces: The Role of Surface Forces,” Phys. Rev. Lett. 119, 078003.
  262. Turing, Alan Mathison, 1952, “The chemical basis of morphogenesis,” Phil. Trans. R. Soc. B 237, 37–72.
  263. Vicsek, Tamás, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet, 1995, “Novel Type of Phase Transition in a System of Self-Driven Particles,” Phys. Rev. Lett. 75, 1226.
  264. Vitral, Eduardo, Saikat Mukherjee, Perry H. Leo, Jorge Viñals, Mark R. Paul, and Zhi-Feng Huang, 2020, “Spiral defect chaos in Rayleigh-Bénard convection: Asymptotic and numerical studies of azimuthal flows induced by rotating spirals,” Phys. Rev. Fluids 5, 093501.
  265. Voß, Johannes, and Raphael Wittkowski, 2020, “On the shape-dependent propulsion of nano- and microparticles by traveling ultrasound waves,” Nanoscale Adv. 2, 3890–3899.
  266. Vossel, Maximilian, Bert L. de Groot, and Aljaž Godec, 2025, “Allosteric Lever: Toward a Principle of Specific Allosteric Response,” Phys. Rev. X 15, 021097.
  267. Walther, Andreas, and Axel H. E. Müller, 2008, “Janus particles,” Soft Matter 4, 663–668.
  268. Walther, Andreas, and Axel H. E. Müller, 2013, “Janus particles: Synthesis, self-assembly, physical properties, and applications,” Chem. Rev. 113, 5194–5261.
  269. Wan, Haixiao, Zheng Jiao, Jiaqi Li, Xiaobin Dai, Jianfeng Li, and Li-Tang Yan, 2025, “Dynamic interplay between deformability and activity in cell entry of soft active nanoparticles,” Nano Lett. 25, 6797–6802.
  270. Wang, Wei, Luz-Angelica Castro, Mauricio Hoyos, and Thomas E. Mallouk, 2012, “Autonomous motion of metallic microrods propelled by ultrasound,” ACS Nano 6, 6122–6132.
  271. Weber, Christoph A., David Zwicker, Frank Jülicher, and Chiu Fan Lee, 2019, “Physics of active emulsions,” Rep. Prog. Phys. 82, 064601.
  272. Wensink, H. H., and H. Löwen, 2008, “Aggregation of self-propelled colloidal rods near confining walls,” Phys. Rev. E 78, 031409.
  273. Wittgenstein, Ludwig, 1953, Philosophical Investigations, edited by G. E. M. Anscombe (Blackwell, Oxford).
  274. Wittkowski, Raphael, and Hartmut Löwen, 2011, “Dynamical density functional theory for colloidal particles with arbitrary shape,” Mol. Phys. 109, 2935–2943.
  275. Wittkowski, Raphael, Adriano Tiribocchi, Joakim Stenhammar, Rosalind J. Allen, Davide Marenduzzo, and Michael E. Cates, 2014, “Scalar ϕ4 field theory for active-particle phase separation,” Nat. Commun. 5, 4351.
  276. Wittkowski, Raphael, and Michael te Vrugt, 2026, Theoretische Physik für Dummies [Theoretical Physics for Dummies] (to be published).
  277. Wittmann, René, and Joseph M. Brader, 2016, “Active Brownian particles at interfaces: An effective equilibrium approach,” Europhys. Lett. 114, 68004.
  278. Wittmann, René, G. H. Philipp Nguyen, Hartmut Löwen, Fabian J. Schwarzendahl, and Anupam Sengupta, 2023, “Collective mechano-response dynamically tunes cell-size distributions in growing bacterial colonies,” Commun. Phys. 6, 331.
  279. Yamagishi, Manami, Naomichi Hatano, Ken-Ichiro Imura, and Hideaki Obuse, 2023, “Proposal of multidimensional quantum walks to explore Dirac and Schrödinger systems,” Phys. Rev. A 107, 042206.
  280. Yamagishi, Manami, Naomichi Hatano, and Hideaki Obuse, 2024, “Proposal of a quantum version of active particles via a nonunitary quantum walk,” Sci. Rep. 14, 28648.
  281. Young, Jeremy T., Alexey V. Gorshkov, Michael Foss-Feig, and Mohammad F. Maghrebi, 2020, “Nonequilibrium Fixed Points of Coupled Ising Models,” Phys. Rev. X 10, 011039.
  282. Zakine, Ruben, Jérôme Garnier-Brun, Antoine-Cyrus Becharat, and Michael Benzaquen, 2024, “Socioeconomic agents as active matter in nonequilibrium Sakoda-Schelling models,” Phys. Rev. E 109, 044310.
  283. Zeh, Hans-Dieter, 1989, The Physical Basis of the Direction of Time (Springer, Berlin).
  284. Zhang, Jing, Xu Zheng, Haihang Cui, and Zhanhua Silber-Li, 2017, “The self-propulsion of the spherical Pt-SiO2 Janus micro-motor,” Micromachines 8, 123.
  285. Zhang, Yiwei, and Étienne Fodor, 2023, “Pulsating Active Matter,” Phys. Rev. Lett. 131, 238302.
  286. Zippo, Emanuele, Dorothee Dormann, Thomas Speck, and Lukas S. Stelzl, 2025, “Molecular simulations of enzymatic phosphorylation of disordered proteins and their condensates,” Nat. Commun. 16, 4649.
  287. Zöttl, Andreas, and Holger Stark, 2016, “Emergent behavior in active colloids,” J. Phys. Condens. Matter 28, 253001.
  288. Zöttl, Andreas, and Holger Stark, 2023, “Modeling active colloids: From active Brownian particles to hydrodynamic and chemical fields,” Annu. Rev. Condens. Matter Phys. 14, 109–127.
  289. Zwanzig, Robert, 2001, Nonequilibrium Statistical Mechanics, 3rd ed. (Oxford University Press, New York).
  290. Zwicker, David, Markus Decker, Steffen Jaensch, Anthony A. Hyman, and Frank Jülicher, 2014, “Centrosomes are autocatalytic droplets of pericentriolar material organized by centrioles,” Proc. Natl. Acad. Sci. U.S.A. 111, E2636–E2645.

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Rev. Mod. Phys. 98, 031001– Published 13 July, 2026
  • Received 20 May 2024
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