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  • Open Access

Impact of emergent order on energy efficiency in active matter: Insights from a minimal lattice model

Michael W. Jack*

  • *Contact author: michael.jack@otago.ac.nz
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Phys. Rev. Research 8, 023179 – Published 18 May, 2026

DOI: https://doi.org/10.1103/55xr-zqtp

Abstract

We investigate the nonequilibrium steady state of a one-dimensional (i.e., mechanical degree of freedom) run-and-tumble model of left-right oriented Brownian particles with an explicit coupling to a chemical degree of freedom and a macroscopic opposing force. This model enables us to explore the fundamental relationships between the energy process driving individual entities, the emergence of collective behavior, and macroscopic forces in active matter. Sufficiently large alignment interactions can break the right-left symmetry, leading to a macroscopic collective alignment of the particles, or flocking transition. Using a decoupling mean-field approximation, we find that collective alignment requires both sufficiently large alignment interactions and lattice hopping. Further, this collective alignment does not require chemical-to-mechanical coupling and can occur solely through passive mechanical hopping. If chemical-to-mechanical coupling is present, it can lead to a collective drift in the direction of particle alignment even against an opposing mechanical force. Interestingly, in the absence of an opposing force, the particles spontaneously align in a random direction, but in the presence of a force, they tend to align “uphill,” enabling macroscopic work output. The many-body system is formulated in terms of creation and annihilation operators for identical classical particles, which enables us to calculate a range of observables by exploiting techniques from many-body quantum systems.

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References (66)

  1. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
  2. S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  3. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  4. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
  5. J. O’Byrne, Y. Kafri, J. Tailleur, and F. van Wijland, Time irreversibility in active matter, from micro to macro, Nat. Rev. Phys. 4, 167 (2022).
  6. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  7. D. Needleman and Z. Dogic, Active matter at the interface between materials science and cell biology, Nat. Rev. Mater. 2, 17048 (2017).
  8. M. J. Bowick, N. Fakhri, M. Cristina Marchetti, and S. Ramaswamy, Symmetry, thermodynamics, and topology in active matter, Phys. Rev. X 12, 010501 (2022).
  9. Y. Fily and M. C. Marchetti, Athermal phase separation of self-propelled particles with no alignment, Phys. Rev. Lett. 108, 235702 (2012).
  10. G. S. Redner, M. F. Hagan, and A. Baskaran, Structure and dynamics of a phase-separating active colloidal fluid, Phys. Rev. Lett. 110, 055701 (2013).
  11. J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo, and M. E. Cates, Continuum theory of phase separation kinetics for active Brownian particles, Phys. Rev. Lett. 111, 145702 (2013).
  12. R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, Scalar ϕ4 field theory for active-particle phase separation, Nat. Commun. 5, 4351 (2014).
  13. 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).
  14. M. E. Cates and J. Tailleur, Motility-induced phase separation, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  15. C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Goldstein, and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett. 93, 098103 (2004).
  16. A. Sokolov, I. S. Aranson, J. O. Kessler, and R. E. Goldstein, Concentration dependence of the collective dynamics of swimming bacteria, Phys. Rev. Lett. 98, 158102 (2007).
  17. J. Tailleur and M. E. Cates, Statistical mechanics of interacting run-and-tumble bacteria, Phys. Rev. Lett. 100, 218103 (2008).
  18. A. Sokolov and I. S. Aranson, Physical properties of collective motion in suspensions of bacteria, Phys. Rev. Lett. 109, 248109 (2012).
  19. T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, Spontaneous motion in hierarchically assembled active matter, Nature (London) 491, 431 (2012).
  20. S. J. DeCamp, G. S. Redner, A. Baskaran, M. F. Hagan, and Z. Dogic, Orientational order of motile defects in active nematics, Nat. Mater. 14, 1110 (2015).
  21. T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans, and B. Ladoux, Topological defects in epithelia govern cell death and extrusion, Nature (London) 544, 212 (2017).
  22. K. Kawaguchi, R. Kageyama, and M. Sano, Topological defects control collective dynamics in neural progenitor cell cultures, Nature (London) 545, 327 (2017).
  23. J. R. Howse, Richard A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Self-motile colloidal particles: From directed propulsion to random walk, Phys. Rev. Lett. 99, 048102 (2007).
  24. J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Living crystals of light-activated colloidal surfers, Science 339, 936 (2013).
  25. I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, and T. Speck, Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles, Phys. Rev. Lett. 110, 238301 (2013).
  26. A. P. Solon and J. Taileur, Revisiting the flocking transition using active spins, Phys. Rev. Lett. 111, 078101 (2013).
  27. A. G. Thompson, J. Taileur, M. E. Cates, and R. A. Blythe, Lattice models of nonequilibrium bacterial dynamics, J. Stat. Mech. (2011) P02029.
  28. R. Bebon, J. Robinson, and T. Speck, Thermodynamics of active matter: Tracking dissipation across scales, Phys. Rev. X 15, 021050 (2025).
  29. É. Fodor, C. Nardini, M. E. Cates, J. Tailleur, P. Visco, and F. van Wijland, How far from equilibrium is active matter? Phys. Rev. Lett. 117, 038103 (2016).
  30. C. Nardini, É. Fodor, E. Tjhung, F. van Wijland, J. Tailleur, and M. E. Cates, Entropy production in field theories without time-reversal symmetry: Quantifying the non-equilibrium character of active matter, Phys. Rev. X 7, 021007 (2017).
  31. C. Desgranges and J. Delhommelle, Mapping local dissipation and entropy production in complex active fluids, J. Phys. Chem. Lett. 16, 11405 (2025).
  32. D. Keller and C. Bustamante, The mechanochemistry of molecular motors, Biophys. J. 78, 541 (2000).
  33. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  34. B. Kolomeisky, Motor proteins and molecular motors: How to operate machines at the nanoscale, J. Phys.: Condens. Matter 25, 463101 (2013).
  35. P. Pietzonka and U. Seifert, Entropy production of active particles and for particles in active baths, J. Phys. A: Math. Theor. 51, 01LT01 (2018).
  36. T. Speck, Active Brownian particles driven by constant affinity, Europhys. Lett. 123, 20007 (2018).
  37. T. Markovich, E. Fodor, E. Tjhung, and M. E. Cates, Thermodynamics of active field theories: Energetic cost of coupling to reservoirs, Phys. Rev. X 11, 021057 (2021).
  38. T. Ekeh, M. E. Cates, and E. Fodor, Thermodynamic cycles with active matter, Phys. Rev. E 102, 010101 (2020).
  39. Y. Yating, E. Lei, Y.-H. Ma, Z. C. Tu, and G. Li, Thermodynamic geometric control of active matter, Phys. Rev. E 112, 054124 (2025).
  40. A. Sokolov, M. M. Apodaca, B. A. Grzybowski, and I. S. Aranson, Swimming bacteria power microscopic gears, Proc. Natl. Acad. Sci. USA 107, 969 (2010).
  41. R. Di Leonardo, L. Angelani, D. Dell’Arciprete, G. Ruocco, V. Iebba, S. Schippa, M. P. Conte, F. Mecarini, F. De Angelis, and E. Di Fabrizio, Bacterial ratchet motors, Proc. Natl. Acad. Sci. USA 107, 9541 (2010).
  42. G. Vizsnyiczai, G. Frangipane, C. Maggi, F. Saglimbeni, S. Bianchi, and R. Di Leonardo, Light controlled 3D micro-motors powered by bacteria, Nat. Commun. 8, 15974 (2017).
  43. P. Pietzonka, E. Fodor, C. Lohrmann, M. E. Cates, and U. Seifert, Autonomous engines driven by active matter: Energetics and design principles, Phys. Rev X. 9, 041032 (2019).
  44. Jean-François Derivaux, R. L. Jack, and M. E. Cates, Active–passive mixtures with bulk loading: A minimal active engine in one dimension, J. Stat. Mech. (2023) 083212.
  45. A. B. Slowman, M. R. Evans, and R. A. Blythe, Jamming and attraction of interacting run-and-tumble random walkers, Phys. Rev. Lett. 116, 218101 (2016).
  46. M. Kourbane-Houssene, C. Erignoux, T. Bodineau, and J. Tailleur, Exact hydrodynamic description of active lattice gases, Phys. Rev. Lett. 120, 268003 (2018).
  47. M. Doi, Second quantization representation for classical many-particle system, J. Phys. A: Math. Gen. 9, 1465 (1976).
  48. M. Doi, Stochastic theory of diffusion-controlled reaction, J. Phys. A: Math. Gen. 9, 1479 (1976).
  49. P. Grassberger and M. Scheunert, Fock-space methods for identical classical objects, Fortschr. Phys. 28, 547 (1980).
  50. D. C. Mattis and M. L. Glasser, The uses of quantum field theory in diffusion-limited reactions, Rev. Mod. Phys. 70, 979 (1998).
  51. Fernando A. N. Santos, H. Gadâlha, and E. A. Gaffney, Fock space, symbolic algebra, and analytical solutions for small stochastic systems, Phys. Rev. E 92, 062714 (2015).
  52. R. Garcia-Millan and G. Pruessner, Run-and-tumble motion in a harmonic potential: Field theory and entropy production, J. Stat. Mech. (2021) 063203.
  53. G. Pruessner and R. Garcia-Millan, Field theories of active particle systems and their entropy production, Rep. Prog. Phys. 88, 097601 (2025).
  54. A. L. Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, Dover, 2003).
  55. M. O. Magnasco, Molecular combustion motors, Phys. Rev. Lett. 72, 2656 (1994).
  56. K. J. Challis and M. W. Jack, Energy transfer in a molecular motor in the Kramers regime, Phys. Rev. E 88, 042114 (2013).
  57. F. Jülicher and J. Prost, Cooperative molecular motors, Phys. Rev. Lett. 75, 2618 (1995).
  58. N. J. López-Alamilla, K. J. Challis, A. G. Deaker, and M. W. Jack, The effect of futile chemical cycles on chemical-to-mechanical energy conversion in interacting motor protein systems, Physica A 615, 128608 (2023).
  59. M. W. Jack and A. Deaker, Nonequilibrium master equation for interacting Brownian particles in a deep-well periodic potential, Phys. Rev. E 105, 054150 (2022).
  60. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  61. K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan, Superfluid and Insulating Phases in an Interacting-Boson Model: Mean-Field Theory and the RPA, Europhys. Lett. 22, 257 (1993).
  62. E. Levine, D. Mukamel, and G. M. Schütz, Zero-range process with open boundaries, J. Stat. Phys. 120, 759 (2005).
  63. R. Rajesh and S. N. Majumdar, Conserved mass models and particle systems in one dimension, J. Stat. Phys. 99, 943 (2000).
  64. A. Deaker and M. W. Jack, Gutzwiller approximation for indistinguishable interacting Brownian particles on a lattice, Phys. Rev. E 107, 044109 (2023).
  65. C. Van den Broeck, N. Kumar, and K. Lindenberg, Efficiency of isothermal molecular machines at maximum power, Phys. Rev. Lett. 108, 210602 (2012).
  66. https://doi.org/10.5281/zenodo.1836542.

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Phys. Rev. Research 8, 023179– Published 18 May, 2026
  • Received 16 February 2026
  • Accepted 3 May 2026
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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.

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