- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

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

Microcanonical ensemble out of equilibrium

R. Belousov1,*, J. Elliott1,2, F. Berger3, L. Rondoni4,5, and A. Erzberger1,2,†

  • *Contact author: roman.belousov@embl.de
  • †Contact author: erzberge@embl.de
PDF

Phys. Rev. E 113, 014133 – Published 23 January, 2026

DOI: https://doi.org/10.1103/r1cg-tpdb

Abstract

Introduced by Boltzmann under the name “monode,” the microcanonical ensemble serves as the fundamental representation of equilibrium thermodynamics in statistical mechanics by counting all possible realizations of a system's states. Ensemble theory connects this idea with probability and information theory, leading to the notion of Shannon-Gibbs entropy and, ultimately, to the principle of maximum caliber describing trajectories of systems—in and out of equilibrium. While the latter phenomenological generalization reproduces many results of nonequilibrium thermodynamics, given a proper choice of observables, its physical justification remains an open area of research. What is the microscopic origin and physical interpretation of this variational approach? What guides the choice of relevant observables? We address these questions by extending Boltzmann's method to a microcanonical caliber principle and counting realizations of a system's trajectories—all assumed equally probable. Maximizing the microcanonical caliber under the imposed constraints, we systematically develop generalized local detailed-balance relations, clarify the statistical origins of inhomogeneous transport, and provide an independent derivation of key equations from stochastic thermodynamics. This approach introduces a dynamical ensemble theory for nonequilibrium steady states in spatially extended and active systems. While verifying the equivalence of ensembles, e.g., those of Norton and Thévenin, our framework contests other common assumptions about nonequilibrium regimes, with supporting evidence provided by stochastic simulations. Our theory suggests further connections to the first principles of microscopic dynamics in classical statistical mechanics, which are essential for investigating systems where the necessary conditions for thermodynamic behavior are not satisfied.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (149)

  1. C. M. Kriebisch, O. Bantysh, L. Baranda Pellejero, A. Belluati, E. Bertosin, K. Dai, M. de Roy, H. Fu, N. Galvanetto, J. M. Gibbs, S. S. Gomez, G. Granatelli, A. Griffo, M. Guix, C. O. Gurdap, J. Harth-Kitzerow, I. S. Haugerud, G. Häfner, P. Jaiswal, S. Javed, et al., A roadmap toward the synthesis of life, Chem 11, 102399 (2025).
  2. A. Genthon, C. D. Modes, F. Jülicher, and S. W. Grill, Nonequilibrium transitions in a template copying ensemble, Phys. Rev. Lett. 134, 068402 (2025).
  3. I. S. Haugerud, P. Jaiswal, and C. A. Weber, Nonequilibrium wet–dry cycling acts as a catalyst for chemical reactions, J. Phys. Chem. B 128, 1724 (2024).
  4. Y. Wang, E. Lei, Y.-H. Ma, Z. C. Tu, and G. Li, Thermodynamic geometric control of active matter, Phys. Rev. E 112, 054124 (2025).
  5. J. Bauermann, S. Laha, P. M. McCall, F. Jülicher, and C. A. Weber, Chemical kinetics and mass action in coexisting phases, J. Am. Chem. Soc. 144, 19294 (2022).
  6. J. Bauermann, G. Bartolucci, J. Boekhoven, C. A. Weber, and F. Jülicher, Formation of liquid shells in active droplet systems, Phys. Rev. Res. 5, 043246 (2023).
  7. Z.-K. Liu, Thermodynamics and its prediction and calphad modeling: Review, state of the art, and perspectives, Calphad 82, 102580 (2023).
  8. R. Zakine and E. Vanden-Eijnden, Minimum-action method for nonequilibrium phase transitions, Phys. Rev. X 13, 041044 (2023).
  9. M. J. Bowick, N. Fakhri, M. C. Marchetti, and S. Ramaswamy, Symmetry, thermodynamics, and topology in active matter, Phys. Rev. X 12, 010501 (2022).
  10. 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).
  11. P. Gaspard and R. Kapral, Active matter, microreversibility, and thermodynamics, Research 2020, 9739231 (2020).
  12. T. Herpich, J. Thingna, and M. Esposito, Collective power: Minimal model for thermodynamics of nonequilibrium phase transitions, Phys. Rev. X 8, 031056 (2018).
  13. P. Sartori and S. Leibler, Lessons from equilibrium statistical physics regarding the assembly of protein complexes, Proc. Natl. Acad. Sci. USA 117, 114 (2019).
  14. F. Benoist and P. Sartori, High-speed combinatorial polymerization through kinetic-trap encoding, Phys. Rev. Lett. 134, 038402 (2025).
  15. S. C. Takatori and J. F. Brady, Towards a thermodynamics of active matter, Phys. Rev. E 91, 032117 (2015).
  16. R. Chetrite and H. Touchette, Nonequilibrium Markov processes conditioned on large deviations, Ann. Henri Poincaré 16, 2005 (2015).
  17. N. Freitas, J.-C. Delvenne, and M. Esposito, Stochastic thermodynamics of nonlinear electronic circuits: A realistic framework for computing around kT, Phys. Rev. X 11, 031064 (2021).
  18. N. I. Petridou, B. Corominas-Murtra, C.-P. Heisenberg, and E. Hannezo, Rigidity percolation uncovers a structural basis for embryonic tissue phase transitions, Cell 184, 1914 (2021).
  19. E. Tang and R. Golestanian, Quantifying configurational information for a stochastic particle in a flow-field, New J. Phys. 22, 083060 (2020).
  20. E. Tang, J. Agudo-Canalejo, and R. Golestanian, Topology protects chiral edge currents in stochastic systems, Phys. Rev. X 11, 031015 (2021).
  21. Y. I. Li, R. Garcia-Millan, M. E. Cates, and E. Fodor, Towards a liquid-state theory for active matter (a), Europhys. Lett. 142, 57004 (2023).
  22. F. Avanzini, T. Aslyamov, E. Fodor, and M. Esposito, Nonequilibrium thermodynamics of non-ideal reaction–diffusion systems: Implications for active self-organization, J. Chem. Phys. 161, 174108 (2024).
  23. C. Zheng and E. Tang, A topological mechanism for robust and efficient global oscillations in biological networks, Nat. Commun. 15, 6453 (2024).
  24. A. Raghu and I. Neri, Effective affinity for generic currents in Markov processes, J. Stat. Phys. 192, 50 (2025).
  25. G. Tkačik and P. R. T. Wolde, Information processing in biochemical networks, Annu. Rev. Biophys. 54, 249 (2025).
  26. C. Maes, Frenesy: Time-symmetric dynamical activity in nonequilibria, Phys. Rep. 850, 1 (2020).
  27. C. Maes, Local detailed balance, SciPost Phys. Lect. Notes 32 (2021).
  28. R. M. L. Evans, Detailed balance has a counterpart in non-equilibrium steady states, J. Phys. A: Math. Gen. 38, 293 (2005).
  29. G. Falasco and M. Esposito, Macroscopic stochastic thermodynamics, Rev. Mod. Phys. 97, 015002 (2025).
  30. H. Touchette, Introduction to dynamical large deviations of Markov processes, Physica A 504, 5 (2018).
  31. H. Touchette and R. J. Harris, Large deviation approach to nonequilibrium systems, in Nonequilibrium Statistical Physics of Small Systems: Fluctuation Relations and Beyond, edited by R. Klages, W. Just, and C. Jarzynski (Wiley, New York, 2013).
  32. E. Smith, Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions, Rep. Prog. Phys. 74, 046601 (2011).
  33. H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478, 1 (2009).
  34. M. Hofer, J. Korbel, R. Hanel, and S. Thurner, Thermodynamics of driven systems with explicitly broken detailed balance, arXiv:2501.12192.
  35. P. Gaspard, Multivariate fluctuation relations for currents, New J. Phys. 15, 115014 (2013).
  36. D. Ruelle, A review of linear response theory for general differentiable dynamical systems, Nonlinearity 22, 855 (2009).
  37. J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571 (1976).
  38. K. Ghosh, P. D. Dixit, L. Agozzino, and K. A. Dill, The maximum caliber variational principle for nonequilibria, Annu. Rev. Phys. Chem. 71, 213 (2020).
  39. P. Pradhan, C. P. Amann, and U. Seifert, Nonequilibrium steady states in contact: Approximate thermodynamic structure and zeroth law for driven lattice gases, Phys. Rev. Lett. 105, 150601 (2010).
  40. A. Porporato and L. Rondoni, Deterministic engines extending Helmholtz thermodynamics, Physica A 640, 129700 (2024).
  41. F. Cardin and M. Favretti, On the Helmholtz-Boltzmann thermodynamics of mechanical systems, Continuum Mech. Thermodyn. 16, 15 (2004).
  42. M. Ding and M. E. Cates, Hamiltonian heat baths, coarse-graining and irreversibility: A microscopic dynamical entropy from classical mechanics, arXiv:2503.11334.
  43. S. Caruso, C. Giberti, and L. Rondoni, Dissipation function: Nonequilibrium physics and dynamical systems, Entropy 22, 835 (2020).
  44. A. Dhar and H. Spohn, Fourier's law based on microscopic dynamics, Comptes Rendus. Phys. 20, 393 (2019).
  45. A. Agarwal, J. Zhu, C. Hartmann, H. Wang, and L. D. Site, Molecular dynamics in a grand ensemble: Bergmann–Lebowitz model and adaptive resolution simulation, New J. Phys. 17, 083042 (2015).
  46. D. Ruelle, A mechanical model for Fourier's law of heat conduction, Commun. Math. Phys. 311, 755 (2012).
  47. J. L. Lebowitz and P. G. Bergmann, Irreversible Gibbsian ensembles, Ann. Phys. 1, 1 (1957).
  48. P. G. Bergmann and J. L. Lebowitz, New approach to nonequilibrium processes, Phys. Rev. 99, 578 (1955).
  49. J. Gibbs, Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundations of Thermodynamics (C. Scribner's Sons, New York, NY, 1902).
  50. H. Callen, Thermodynamics and an Introduction to Thermostatistics (Wiley, New York, NY, 1991).
  51. M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E 85, 041125 (2012).
  52. A. Puglisi, S. Pigolotti, L. Rondoni, and A. Vulpiani, Entropy production and coarse graining in Markov processes, J. Stat. Mech.: Theory Exp. (2010) P05015.
  53. P. E. Harunari, A. Dutta, M. Polettini, and E. Roldán, What to learn from a few visible transitions' statistics? Phys. Rev. X 12, 041026 (2022).
  54. B. Altaner and J. Vollmer, Fluctuation-preserving coarse graining for biochemical systems, Phys. Rev. Lett. 108, 228101 (2012).
  55. E. T. Jaynes, The minimum entropy production principle, Annu. Rev. Phys. Chem. 31, 579 (1980).
  56. E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).
  57. A. A. Filyukov and V. Y. Karpov, Description of steady transport processes by the method of the most probable path of evolution, J. Eng. Phys. 13, 326 (1967).
  58. A. A. Filyukov and V. Y. Karpov, Method of the most probable path of evolution in the theory of stationary irreversible processes, J. Eng. Phys. 13, 416 (1967).
  59. P. Attard, The second entropy: A general theory for non-equilibrium thermodynamics and statistical mechanics, Annu. Rep. Sect. C 105, 63 (2009).
  60. P. Attard, Non-Equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications (Oxford University Press, Oxford, UK, 2012).
  61. A. A. Filyukov, Compatibility property of steady systems, J. Eng. Phys. 14, 429 (1968).
  62. H. Haken, A new access to path integrals and Fokker Planck equations via the maximum calibre principle, Z. Phys. B 63, 505 (1986).
  63. R. Dewar, Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states, J. Phys. A: Math. Gen. 36, 631 (2003).
  64. P. Gaspard, Time-reversed dynamical entropy and irreversibility in Markovian random processes, J. Stat. Phys. 117, 599 (2004).
  65. R. C. Dewar, Maximum entropy production and the fluctuation theorem, J. Phys. A: Math. Gen. 38, L371 (2005).
  66. K. Ghosh, K. A. Dill, M. M. Inamdar, E. Seitaridou, and R. Phillips, Teaching the principles of statistical dynamics, Am. J. Phys. 74, 123 (2006).
  67. V. Lecomte, C. Appert-Rolland, and F. van Wijland, Thermodynamic formalism for systems with Markov dynamics, J. Stat. Phys. 127, 51 (2007).
  68. E. Seitaridou, M. M. Inamdar, R. Phillips, K. Ghosh, and K. Dill, Measuring flux distributions for diffusion in the small-numbers limit, J. Phys. Chem. B 111, 2288 (2007).
  69. D. Wu, K. Ghosh, M. Inamdar, H. J. Lee, S. Fraser, K. Dill, and R. Phillips, Trajectory approach to two-state kinetics of single particles on sculpted energy landscapes, Phys. Rev. Lett. 103, 050603 (2009).
  70. M. Otten and G. Stock, Maximum caliber inference of nonequilibrium processes, J. Chem. Phys. 133, 034119 (2010).
  71. G. Stock, K. Ghosh, and K. A. Dill, Maximum caliber: A variational approach applied to two-state dynamics, J. Chem. Phys. 128, 194102 (2008).
  72. S. Pressé, K. Ghosh, R. Phillips, and K. A. Dill, Dynamical fluctuations in biochemical reactions and cycles, Phys. Rev. E 82, 031905 (2010).
  73. C. Monthus, Non-equilibrium steady states: maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints, J. Stat. Mech.: Theory Exp. (2011) P03008.
  74. H. Ge, S. Pressé, K. Ghosh, and K. A. Dill, Markov processes follow from the principle of maximum caliber, J. Chem. Phys. 136, 064108 (2012).
  75. S. Pressé, K. Ghosh, and K. A. Dill, Modeling stochastic dynamics in biochemical systems with feedback using maximum caliber, J. Phys. Chem. B 115, 6202 (2011).
  76. K. Ghosh, Stochastic dynamics of complexation reaction in the limit of small numbers, J. Chem. Phys. 134, 195101 (2011).
  77. J. Lee and S. Pressé, A derivation of the master equation from path entropy maximization, J. Chem. Phys. 137, 074103 (2012).
  78. J. Lee and S. Pressé, Microcanonical origin of the maximum entropy principle for open systems, Phys. Rev. E 86, 041126 (2012).
  79. S. Pressé, K. Ghosh, J. Lee, and K. A. Dill, Principles of maximum entropy and maximum caliber in statistical physics, Rev. Mod. Phys. 85, 1115 (2013).
  80. M. J. Hazoglou, V. Walther, P. D. Dixit, and K. A. Dill, Communication: Maximum caliber is a general variational principle for nonequilibrium statistical mechanics, J. Chem. Phys. 143, 051104 (2015).
  81. G. Auletta, L. Rondoni, and A. Vulpiani, On the relevance of the maximum entropy principle in non-equilibrium statistical mechanics, Eur. Phys. J.: Spec. Top. 226, 2327 (2017).
  82. S. Chibbaro, L. Rondoni, and A. Vulpiani, Reductionism, Emergence and Levels of Reality: The Importance of Being Borderline (Springer International Publishing, Berlin, 2014).
  83. M. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation (Oxford University Press, Oxford, UK, 2023).
  84. E. M. Pearson, T. Halicioglu, and W. A. Tiller, Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble, Phys. Rev. A 32, 3030 (1985).
  85. R. M. L. Evans, Rules for transition rates in nonequilibrium steady states, Phys. Rev. Lett. 92, 150601 (2004).
  86. R. Evans, Driven steady states: Rules for transition rates, Physica A 340, 364 (2004).
  87. J. A. Pachter and K. A. Dill, Nonequilibrium statistical physics beyond the ideal heat bath approximation, Phys. Rev. E 107, 014131 (2023).
  88. J. Rombouts, J. Elliott, and A. Erzberger, Forceful patterning: Theoretical principles of mechanochemical pattern formation, EMBO Rep. 24, e57739 (2023).
  89. J. A. Davies and F. Glykofrydis, Engineering pattern formation and morphogenesis, Biochem. Soc. Trans. 48, 1177 (2020).
  90. D. Hua, R. Xiong, K. Braeckmans, B. Scheid, C. Huang, F. Sauvage, and S. C. De Smedt, Concentration gradients in material sciences: Methods to design and biomedical applications, Adv. Funct. Mater. 31, 2009005 (2021).
  91. J. Halatek and E. Frey, Rethinking pattern formation in reaction–diffusion systems, Nat. Phys. 14, 507 (2018).
  92. L. M. R. Keil, F. M. Möller, M. Kieß, P. W. Kudella, and C. B. Mast, Proton gradients and pH oscillations emerge from heat flow at the microscale, Nat. Commun. 8, 1897 (2017).
  93. J. R. Manning, Diffusion in a chemical concentration gradient, Phys. Rev. 124, 470 (1961).
  94. K. Sharp and F. Matschinsky, Translation of Ludwig Boltzmann's paper “On the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium” sitzungberichte der kaiserlichen akademie der wissenschaften. Mathematisch-naturwissen classe. Abt. II, LXXVI 1877, pp. 373–435 (Wien. Ber. 1877, 76:373-435). Reprinted in Wiss. Abhandlungen, Vol. II, reprint 42, p. 164–223, Barth, Leipzig, 1909, Entropy 17, 1971 (2015).
  95. A. H. Carter, Classical and Statistical Thermodynamics (Pearson, Upper Saddle River, NJ, 2001).
  96. R. Belousov, S. Savino, P. Moghe, T. Hiiragi, L. Rondoni, and A. Erzberger, Poissonian cellular Potts models reveal nonequilibrium kinetics of cell sorting, Phys. Rev. Lett. 132, 248401 (2024).
  97. M. Kardar, Statistical Physics of Particles (Cambridge University Press, Cambridge, UK, 2007).
  98. P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10, 51 (1942).
  99. M. L. Huggins, Some properties of solutions of long-chain compounds. J. Phys. Chem. 46, 151 (1942).
  100. C. A. Weber, D. Zwicker, F. Jülicher, and C. F. Lee, Physics of active emulsions, Rep. Prog. Phys. 82, 064601 (2019).
  101. S. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (CRC Press, Boca Raton, FL, 2018).
  102. S. Katz, J. L. Lebowitz, and H. Spohn, Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors, J. Stat. Phys. 34, 497 (1984).
  103. D. A. Wolf-Gladrow, Lattice Gas Cellular Automata and Lattice Boltzmann Models (Springer, Berlin, Heidelberg, 2000).
  104. S. A. Janowsky and J. L. Lebowitz, Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process, Phys. Rev. A 45, 618 (1992).
  105. B. Derrida, An exactly soluble non-equilibrium system: The asymmetric simple exclusion process, Phys. Rep. 301, 65 (1998).
  106. O. Golinelli and K. Mallick, The asymmetric simple exclusion process: An integrable model for non-equilibrium statistical mechanics, J. Phys. A: Math. Gen. 39, 12679 (2006).
  107. T. Chou, K. Mallick, and R. K. P. Zia, Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport, Rep. Prog. Phys. 74, 116601 (2011).
  108. M. R. Hilário, D. Kious, and A. Teixeira, Random walk on the simple symmetric exclusion process, Commun. Math. Phys. 379, 61 (2020).
  109. J. Elliott, H. Shah, R. Belousov, G. Dey, and A. Erzberger, Repulsive particle interactions enable selective information processing at cellular interfaces, Phys. Rev. Lett. 135, 218403 (2025).
  110. H. J. Bowlden, Detailed balance in nonvertical transitions, J. Phys. Chem. Solids 3, 115 (1957).
  111. T. S. Nguyen, R. Nanguneri, and J. Parkhill, How electronic dynamics with Pauli exclusion produces Fermi-Dirac statistics, J. Chem. Phys. 142, 134113 (2015).
  112. S. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Books on Physics (Dover Publications, Mineola, NY, 2013).
  113. 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).
  114. A. Svetlov, M. Vasiliev, E. Kononov, O. Petrov, and F. Trukhachev, 3D active Brownian motion of single dust particles induced by a laser in a DC glow discharge, Molecules 28, 1790 (2023).
  115. K. J. M. Bishop, S. L. Biswal, and B. Bharti, Active colloids as models, materials, and machines, Annu. Rev. Chem. Biomol. Eng. 14, 1 (2023).
  116. B. Kavčič and G. Tkačik, Token-driven totally asymmetric simple exclusion processes, Phys. Rev. E 111, 054122 (2025).
  117. L. Abbaspour and S. Klumpp, Enhanced diffusion of a tracer particle in a lattice model of a crowded active system, Phys. Rev. E 103, 052601 (2021).
  118. L. Meng, Y. Jin, Y. Guan, J. Xu, and J. Lin, Diffusion enhancement in bacterial cytoplasm through an active random force, Phys. Rev. Res. 5, L032018 (2023).
  119. D. J. Evans and J. F. Ely, Viscous flow in the stress ensemble, Mol. Phys. 59, 1043 (1986).
  120. F. Müller-Plathe, Reversing the perturbation in nonequilibrium molecular dynamics: An easy way to calculate the shear viscosity of fluids, Phys. Rev. E 59, 4894 (1999).
  121. R. Chetrite and H. Touchette, Nonequilibrium microcanonical and canonical ensembles and their equivalence, Phys. Rev. Lett. 111, 120601 (2013).
  122. D. Evans and G. Morriss, Statistical Mechanics of Nonequilibrium Liquids, DOAB Directory of Open Access Books (ANU Press, Canberra, Australia, 2007).
  123. B. N. Miller and P. M. Larson, Heat flow in a linear harmonic chain: An information-theoretic approach to the nonequilibrium stationary state, Phys. Rev. A 20, 1717 (1979).
  124. A. Kato and D. Jou, Breaking of equipartition in one-dimensional heat-conducting systems, Phys. Rev. E 64, 052201 (2001).
  125. L. Conti, P. D. Gregorio, G. Karapetyan, C. Lazzaro, M. Pegoraro, M. Bonaldi, and L. Rondoni, Effects of breaking vibrational energy equipartition on measurements of temperature in macroscopic oscillators subject to heat flux, J. Stat. Mech.: Theory Exp. (2013) P12003.
  126. B. Hille, Ionic channels in excitable membranes. Current problems and biophysical approaches, Biophys. J. 22, 283 (1978).
  127. Q. Ren, Q. Cui, K. Chen, J. Xie, and P. Wang, Salinity-gradient power harvesting using osmotic energy conversion with designed interfacial nanostructures under thermal modulation, Desalination 535, 115802 (2022).
  128. P. S. Senanayake, A. Lugo, M. F. Ahmed, Z. Stoll, N. E. Moe, J. Barber, W. Shane Walker, P. Xu, and H. Wang, Electrodialysis modeling for desalination and resource recovery, Curr. Opin. Chem. Eng. 47, 101081 (2025).
  129. Source code, which implements numerical simulations in Sec. 4, is available at https://git.embl.de/rbelouso/microcaliber.
  130. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  131. L. Peliti and S. Pigolotti, Stochastic Thermodynamics: An Introduction (Princeton University Press, Princeton, NJ, 2021).
  132. R. Belousov, M. N. Qaisrani, A. Hassanali, and E. Roldán, First-passage fingerprints of water diffusion near glutamine surfaces, Soft Matter 16, 9202 (2020).
  133. R. Belousov, A. Hassanali, and E. Roldán, Statistical physics of inhomogeneous transport: Unification of diffusion laws and inference from first-passage statistics, Phys. Rev. E 106, 014103 (2022).
  134. S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15, 1 (1943).
  135. Z. Burda, J. Duda, J. M. Luck, and B. Waclaw, Localization of the maximal entropy random walk, Phys. Rev. Lett. 102, 160602 (2009).
  136. P. D. Dixit and K. A. Dill, Inferring microscopic kinetic rates from stationary state distributions, J. Chem. Theory Comput. 10, 3002 (2014).
  137. P. D. Dixit, Stationary properties of maximum-entropy random walks, Phys. Rev. E 92, 042149 (2015).
  138. B. Bijnens and C. Maes, Pushing run-and-tumble particles through a rugged channel, J. Stat. Mech.: Theory Exp. (2021) 033206.
  139. R. C. Tolman, The Principles of Statistical Mechanics (Courier Corporation, North Chelmsford, MA, 1979).
  140. G. Gallavotti, Statistical Mechanics: A Short Treatise (Springer Science & Business Media, Cham, 1999).
  141. D. J Evans and G. P Morriss, Statistical Mechanics of Nonequilbrium Liquids (ANU Press, Canberra, Australia, 2007).
  142. D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Probability of second law violations in shearing steady states, Phys. Rev. Lett. 71, 2401 (1993).
  143. D. J. Evans, S. R. Williams, D. J. Searles, and L. Rondoni, On typicality in nonequilibrium steady states, J. Stat. Phys. 164, 842 (2016).
  144. W. G. Hoover, Nonequilibrium molecular dynamics, in Molecular Dynamics, edited by H. Araki, J. Ehlers, K. Hepp, H. A. Weidenmüller, and J. Zittartz, Lecture Notes in Physics Vol. 258 (Springer, Berlin, Germany, 1986), p. 92.
  145. B. Hafskjold, T. Ikeshoji, and S. K. Ratkje, On the molecular mechanism of thermal diffusion in liquids, Mol. Phys. 80, 1389 (1993).
  146. F. Müller-Plathe, A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity, J. Chem. Phys. 106, 6082 (1997).
  147. H. Haken, Application of the maximum information entropy principle to selforganizing systems, Z. Phys. B 61, 335 (1985).
  148. H. Haken, The maximum entropy principle for non-equilibrium phase transitions: Determination of order parameters, slaved modes, and emerging patterns, Z. Phys. B 63, 487 (1986).
  149. J. E. Mayer and E. Montroll, Molecular distribution, J. Chem. Phys. 9, 2 (1941).

Outline

Information

Phys. Rev. E 113, 014133– Published 23 January, 2026
  • Received 30 May 2025
  • Revised 6 November 2025
  • Accepted 2 January 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