- 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

Exact results in stochastic processes with division, death, and diffusion: Spatial correlations, marginal entropy production, and macroscopic currents

Samuel Cameron and Elsen Tjhung

  • School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom

PDF

Phys. Rev. E 112, 034109 – Published 2 September, 2025

DOI: https://doi.org/10.1103/b9n6-mynw

Abstract

We consider a generic class of stochastic particle-based models whose state at an instant in time is described by a set of continuous degrees of freedom (e.g., positions), and the length of this set changes stochastically in time due to birth-death processes. Using a master equation formalism, we write the dynamics of the corresponding (infinite) set of probability distributions: this takes the form of coupled Fokker-Planck equations with model-dependent source and sink terms. We derive the general expression of the entropy production rate for this class of models in terms of path irreversibility. To demonstrate the practical use of this framework, we analyze a biologically motivated model incorporating division, death, and diffusion, where spatial correlations arise through the division process. By systematically integrating out excess degrees of freedom, we obtain the marginal probability distribution, enabling exact calculations of key statistical properties such as average density and correlation functions. We validate our analytical results through numerical Brownian dynamics simulations, finding excellent agreement between theory and simulation. Our method thus provides a powerful tool for tackling previously unsolved problems in stochastic birth-death dynamics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. G. Gompper et al., J. Phys.: Condens. Matter 32, 193001 (2020).
  2. J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Phys. Rev. Lett. 99, 048102 (2007).
  3. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013).
  4. O. Hallatschek et al., Nat. Rev. Phys. 5, 407 (2023).
  5. M. Delarue et al., Nat. Phys. 12, 762 (2016).
  6. P. Gniewek, C. F. Schreck, and O. Hallatschek, Phys. Rev. Lett. 122, 208102 (2019).
  7. E. Tjhung and L. Berthier, Phys. Rev. Res. 2, 043334 (2020).
  8. H. Fujikawa and M. Matsushita, J. Phys. Soc. Jpn. 58, 3875 (1989).
  9. A. Barabási and H. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995).
  10. I. Klapper and J. Dockery, SIAM Rev. 52, 221 (2010).
  11. M. R. Mattei et al., J. Math. Biol. 76, 945 (2018).
  12. R. J. Allen and B. Waclaw, Rep. Prog. Phys. 82, 016601 (2019).
  13. N. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland Personal Library (Elsevier, North-Holland, Amsterdam, 2011).
  14. R. A. Fisher, Ann. Eugenics 7, 355 (1937).
  15. A. Kolmogorov, I. Petrovskii, and N. Piscunov, Byul. Moskovskogo Gos. Univ. 1, 1 (1937).
  16. D. T. Gillespie, J. Phys. Chem. 81, 2340 (1977).
  17. M. J. del Razo, S. Winkelmann, R. Klein, and F. Höfling, J. Math. Phys. 64, 013304 (2023).
  18. R. Garcia-Millan, J. Pausch, B. Walter, and G. Pruessner, Phys. Rev. E 98, 062107 (2018).
  19. R. Garcia-Millan and G. Pruessner, J. Stat. Mech. (2021) 063203.
  20. U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, Cambridge, 2014).
  21. M. J. del Razo et al., Lett. Math. Phys. 112, 49 (2022).
  22. L. Peliti, J. Phys. France 46, 1469 (1985).
  23. M. Doi, J. Phys. A: Math. Gen. 9, 1479 (1976).
  24. M. Kardar, Statistical Physics of Particles (Cambridge University Press, Cambridge, 2007).
  25. C. A. Vargas-Garcia, M. Soltani, and A. Singh, IEEE Life Sci. Lett. 2, 47 (2016).
  26. M. Xia and T. Chou, J. Phys. A: Math. Theor. 54, 385601 (2021).
  27. U. Seifert, Rep. Prog. Phys. 75, 126001 (2012).
  28. P. Gaspard, J. Stat. Phys. 117, 599 (2004).
  29. L. Cocconi, R. Garcia-Millan, Z. Zhen, B. Buturca, and G. Pruessner, Entropy 22, 1252 (2020).
  30. T. Speck, Europhys. Lett. 123, 20007 (2018).
  31. R. D. Astumian and M. Bier, Phys. Rev. Lett. 72, 1766 (1994).
  32. P. Gaspard and E. Gerritsma, J. Theor. Biol. 247, 672 (2007).
  33. L. Hong and H. Qian, Phys. Rev. E 104, 044113 (2021).
  34. E. Boksenbojm and B. Wynants, J. Phys. A: Math. Theor. 42, 445003 (2009).
  35. L. Cocconi, G. Salbreux, and G. Pruessner, Phys. Rev. E 105, L042601 (2022).
  36. L. Defaveri, E. Barkai, and D. A. Kessler, Phys. Rev. E 107, 024122 (2023).
  37. T. Agranov, R. L. Jack, M. E. Cates, and E. Fodor, New J. Phys. 26, 063006 (2024).
  38. F. Caballero and M. E. Cates, Phys. Rev. Lett. 124, 240604 (2020).
  39. P. Pietzonka and U. Seifert, J. Phys. A: Math. Theor. 51, 01LT01 (2018).
  40. See Sec. 6.9 in [13].
  41. A. V. Skorokhod, Theory Probab. Appl. 9, 445 (1964).
  42. J. R. Silvester, Math. Gaz. 84, 460 (2000).
  43. A. P. Thompson et al., Comput. Phys. Commun. 271, 108171 (2022).
  44. S. Cameron (2024), https://github.com/samueljmcameron/lammps.
  45. E. Mitchell and E. Tjhung, Soft Matter 18, 1082 (2022).
  46. P. S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, and P. Talkner, ChemPhysChem 10, 45 (2009).
  47. H. L. Michael te Vrugt and R. Wittkowski, Adv. Phys. 69, 121 (2020).
  48. H. Risken and T. Frank, The Fokker-Planck Equation: Methods of Solution and Applications, Springer Series in Synergetics (Springer, Berlin, 2012).
  49. C. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Proceedings in Life Sciences (Springer-Verlag, Berlin, 1985).
  50. K. J. Wiese, Phys. Rev. Lett. 133, 067103 (2024).
  51. H. Hinrichsen, Adv. Phys. 49, 815 (2000).

Outline

Information

Phys. Rev. E 112, 034109– Published 2 September, 2025
  • Received 18 March 2025
  • Accepted 21 July 2025
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