- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

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
  • Access by Xinjiang University

Memory-induced active particle ratchets: Mean currents and large deviations

Venkata D. Pamulaparthy* and Rosemary J. Harris†

  • *Contact author: venkata.pamulaparthy.22@ucl.ac.uk
  • †Contact author: rosemary.j.harris@ucl.ac.uk
PDF

Phys. Rev. E 114, 024143 – Published 21 August, 2026

DOI: https://doi.org/10.1103/7w8f-5l87

Abstract

We analyze a continuous-time random walk model with stochastic reversals of direction. There is no external potential but the reorientation mechanism generates a nonzero current from asymmetry in the forward and backward waiting-time distributions (even when they have the same mean); the system can therefore be considered as a type of active particle ratchet. We derive an explicit expression for the mean ratchet current with exponentially distributed reorientation times and also develop a general renewal-theory framework to obtain the full large deviations, using this to comment on the possibility of dynamical phase transitions.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (68)

  1. R. P. Feynman, R. B. Leighton, and M. Sands, Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1996).
  2. P. Reimann, Brownian motors: Noisy transport far from equilibrium, Phys. Rep. 361, 57 (2002).
  3. S. Denisov, S. Flach, and P. Hänggi, Tunable transport with broken space–time symmetries, Phys. Rep. 538, 77 (2014).
  4. M. O. Magnasco, Forced thermal ratchets, Phys. Rev. Lett. 71, 1477 (1993).
  5. D. Cubero, V. Lebedev, and F. Renzoni, Current reversals in a rocking ratchet: Dynamical versus symmetry-breaking mechanisms, Phys. Rev. E 82, 041116 (2010).
  6. S. N. Ethier and J. Lee, The flashing Brownian ratchet and Parrondo's paradox, R. Soc. Open Sci. 5, 171685 (2018).
  7. L. Angelani, A. Costanzo, and R. D. Leonardo, Active ratchets, Europhys. Lett. 96, 68002 (2011).
  8. C. Roberts and Z. Zhen, Run-and-tumble motion in a linear ratchet potential: Analytic solution, power extraction, and first-passage properties, Phys. Rev. E 108, 014139 (2023).
  9. K. Białas, J. Łuczka, and J. Spiechowicz, Periodic potential can enormously boost free-particle transport induced by active fluctuations, Phys. Rev. E 107, 024107 (2023).
  10. V. D. Pamulaparthy and R. J. Harris, Towards neural reinforcement learning for large deviations in non-equilibrium systems with memory, J. Stat. Mech. (2025) 073404.
  11. W. De Roeck and C. Maes, Symmetries of the ratchet current, Phys. Rev. E 76, 051117 (2007).
  12. D. Lacoste and K. Mallick, Fluctuation theorem for the flashing ratchet model of molecular motors, Phys. Rev. E 80, 021923 (2009).
  13. R. S. Ellis, Large deviation property and asymptotics of integrals, in Entropy, Large Deviations, and Statistical Mechanics (Springer, Berlin, 2006), pp. 30–58.
  14. B. Derrida, Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current, J. Stat. Mech. (2007) P07023.
  15. H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478, 1 (2009).
  16. F. Jülicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Mod. Phys. 69, 1269 (1997).
  17. R. D. Leonardo, L. Angelani, D. Dell'Arciprete, G. Ruocco, V. Iebba, S. Schippa, M. P. Conte, F. Mecarini, F. D. Angelis, and E. D. Fabrizio, Bacterial ratchet motors, Proc. Natl. Acad. Sci. USA 107, 9541 (2010).
  18. A.-K. Pumm, W. Engelen, E. Kopperger, J. Isensee, M. Vogt, V. Kozina, M. Kube, M. N. Honemann, E. Bertosin, M. Langecker, R. Golestanian, F. C. Simmel, and H. Dietz, A DNA origami rotary ratchet motor, Nature (London) 607, 492 (2022).
  19. M. Wiśniewski and J. Spiechowicz, Memory-induced current reversal of Brownian motors, Phys. Rev. E 111, 024130 (2025).
  20. H. C. Berg and D. A. Brown, Chemotaxis in Escherichia coli analysed by three-dimensional tracking, Nature (London) 239, 500 (1972).
  21. M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E 48, 2553 (1993).
  22. A. E. Patteson, A. Gopinath, M. Goulian, and P. E. Arratia, Running and tumbling with E. coli in polymeric solutions, Sci. Rep. 5, 15761 (2015).
  23. O. Farago and N. R. Smith, Confined run-and-tumble particles with non-Markovian tumbling statistics, Phys. Rev. E 109, 044121 (2024).
  24. F. Detcheverry, Generalized run-and-turn motions: From bacteria to Lévy walks, Phys. Rev. E 96, 012415 (2017).
  25. I. Santra, K. S. Olsen, and D. Gupta, Dynamics of switching processes: General results and applications in intermittent active motion, Soft Matter 20, 9360 (2024).
  26. E. W. Montroll and G. H. Weiss, Random walks on lattices. II, J. Math. Phys. 6, 167 (1965).
  27. Such two-channel models, in fact, have a long history; for example, see Ref. [61], where the mean and diffusivity of a related temporally correlated random walk are calculated in the context of an application to superionic conductivity. At least for some special cases there may also be a connection to the more recent concept of lifted Markov chains popularized in computer science [62].
  28. M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A: Math. Theor. 53, 193001 (2020).
  29. Y. Aharonov, L. Davidovich, and N. Zagury, Quantum random walks, Phys. Rev. A 48, 1687 (1993).
  30. D. R. Cox, A use of complex probabilities in the theory of stochastic processes, Math. Proc. Cambridge Philos. Soc. 51, 313 (1955).
  31. O. Aalen, Phase-type distributions in survival analysis, Scand. J. Stat. 22, 447 (2005).
  32. M. Esposito and K. Lindenberg, Continuous-time random walk for open systems: Fluctuation theorems and counting statistics, Phys. Rev. E 77, 051119 (2008).
  33. V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger, Generalized master equations for continuous-time random walks, J. Stat. Phys. 9, 45 (1973).
  34. M. F. Shlesinger, Asymptotic solutions of continuous-time random walks, J. Stat. Phys. 10, 421 (1974).
  35. W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, New York, 1966), Vol. 2.
  36. J. Metzger, S. Ro, and J. Tailleur, Revisiting the ratchet principle: When hidden conservation laws prevent directed currents in stochastic systems, Phys. Rev. Lett. 136, 257102 (2026).
  37. D. Cox, Renewal Theory (Methuen, London, 1962).
  38. S. Pal, L. Dagdug, D. Ghosh, D. Boyer, and A. Pal, Universal criterion for selective outcomes under stochastic resetting, Phys. Rev. E 112, 034116 (2025).
  39. G. Mercado-Vásquez and D. Boyer, Stochastic synthesis-degradation processes: First-passage properties and connections with resetting, arXiv:2602.11095.
  40. S. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52, 1 (1929).
  41. Z. Yang and J. F. Tian, Monotonicity rules for the ratio of two Laplace transforms with applications, J. Math. Anal. Appl. 470, 821 (2019).
  42. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1 (2000).
  43. M. Meerschaert and H. P. Scheffler, Continuous time random walks and space-time fractional differential equations, in Handbook of Fractional Calculus with Applications Volume 1 Basic Theory, edited by A. Kochubei and Y. Luchko (De Gruyter, Berlin, 2019), pp. 385–406.
  44. R. Gorenflo and F. Mainardi, Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: Mathematical aspects, in Anomalous Transport, edited by R. Klages, G. Radons, and I. M. Sokolov (John Wiley & Sons, Weinheim, 2008), Chap. 4, pp. 93–127.
  45. R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51, R848(R) (1995).
  46. T. E. Huillet, On Mittag-Leffler distributions and related stochastic processes, J. Comput. Appl. Math. 296, 181 (2016).
  47. For more on the properties of relevant Mittag-Leffler functions, see Ref. [63]. CTRWs with generic power-law waiting times can be rescaled to CTRWs with Mittag-Leffler waiting-time distributions [64]. Phase behavior and nonequilibrium stationary states induced by spatial resetting in such CTRWs and more general anomalous-diffusion processes have also been studied (see e.g., Refs. [65, 66, 67]).
  48. R. N. Pillai, On Mittag-Leffler functions and related distributions, Ann. Inst. Stat. Math. 42, 157 (1990).
  49. H. Touchette and R. J. Harris, Large deviation approach to nonequilibrium systems, in Nonequilibrium Statistical Physics of Small Systems, edited by R. Klages, W. Just, and C. Jarzynski (John Wiley & Sons, New York, 2013), Chap. 11, pp. 335–360.
  50. R. J. Harris and H. Touchette, Current fluctuations in stochastic systems with long-range memory, J. Phys. A: Math. Theor. 42, 342001 (2009).
  51. R. L. Jack and R. J. Harris, Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory, Phys. Rev. E 102, 012154 (2020).
  52. R. J. Harris and H. Touchette, Phase transitions in large deviations of reset processes, J. Phys. A: Math. Theor. 50, 10LT01 (2017).
  53. D. Poland and H. A. Scheraga, Phase transitions in one dimension and the helix-coil transition in polyamino acids, J. Chem. Phys. 45, 1456 (1966).
  54. D. Poland and H. A. Scheraga, Occurrence of a phase transition in nucleic acid models, J. Chem. Phys. 45, 1464 (1966).
  55. J. M. Meylahn, S. Sabhapandit, and H. Touchette, Large deviations for Markov processes with resetting, Phys. Rev. E 92, 062148 (2015).
  56. Note that if the hopping in both channels were in the same direction, then switching could give an overall mean current larger than the natural (nonswitching) current in either channel; this is effectively a realization of Parrondo's paradox [68].
  57. P. Reimann, M. Grifoni, and P. Hänggi, Quantum ratchets, Phys. Rev. Lett. 79, 10 (1997).
  58. F. J. Sevilla and A. Valdés-Hernández, Dynamics of closed quantum systems under stochastic resetting, J. Phys. A: Math. Theor. 56, 034001 (2023).
  59. C. A. Brown, K. Macieszczak, and R. L. Jack, Unraveling metastable Markovian open quantum systems, Phys. Rev. A 109, 022244 (2024).
  60. V. D. Pamulaparthy, Memory–induced active particle ratchets: Mean currents and large deviations, GitHub, 2026, https://github.com/Venkata-Dhruva-Pamulaparthy/ratchets.git.
  61. M. F. Shlesinger, Correlation effects on frequency dependent conductivity: Application to superionic conductors, Solid State Commun. 32, 1207 (1979).
  62. P. Diaconis, S. Holmes, and R. M. Neal, Analysis of a nonreversible Markov chain sampler, Ann. Appl. Probab. 10, 726 (2000).
  63. F. Mainardi, On some properties of the Mittag-Leffler function Eα(−tα), completely monotone for t>0 with 0<α<1, Discrete Contin. Dyn. Syst. Ser. B 19, 2267 (2014).
  64. R. Gorenflo, Mittag-Leffler waiting time, power laws, rarefaction, continuous time random walk, diffusion limit, arXiv:1004.4413.
  65. W. Wang, A. G. Cherstvy, R. Metzler, and I. M. Sokolov, Restoring ergodicity of stochastically reset anomalous- diffusion processes, Phys. Rev. Res. 4, 013161 (2022).
  66. V. Méndez, A. Masó-Puigdellosas, T. Sandev, and D. Campos, Continuous time random walks under Markovian resetting, Phys. Rev. E 103, 022103 (2021).
  67. A. S. Bodrova and I. M. Sokolov, Continuous-time random walks under power-law resetting, Phys. Rev. E 101, 062117 (2020).
  68. J. Parrondo and L. Dinís, Brownian motion and gambling: From ratchets to paradoxical games, Contemp. Phys. 45, 147 (2004).

Outline

Information

Phys. Rev. E 114, 024143– Published 21 August, 2026
  • Received 27 February 2026
  • Accepted 15 July 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