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  • Open Access

First passage time distribution of active thermal particles in potentials

Benjamin Walter, Gunnar Pruessner, and Guillaume Salbreux
Phys. Rev. Research 3, 013075 – Published 22 January 2021
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Abstract
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Article Text
  • INTRODUCTION AND MAIN RESULTS
  • PERTURBATION THEORY
  • RESULTS FOR SIMPLE POTENTIALS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We introduce a perturbative method to calculate all moments of the first passage time distribution in stochastic one-dimensional processes which are subject to both white and colored noise. This class of non-Markovian processes is at the center of the study of thermal active matter, that is self-propelled particles subject to diffusion. The perturbation theory about the Markov process considers the effect of self-propulsion to be small compared to that of thermal fluctuations. To illustrate our method, we apply it to the case of active thermal particles (i) in a harmonic trap and (ii) on a ring. For both we calculate the first-order correction of the moment-generating function of first passage times, and thus to all its moments. Our analytical results are compared to numerics.

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    • Received 4 June 2020
    • Accepted 21 December 2020

    DOI:https://doi.org/10.1103/PhysRevResearch.3.013075

    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

    Physics Subject Headings (PhySH)

    1. Research Areas
    Nonequilibrium statistical mechanicsStochastic processes
    1. Physical Systems
    Living matter & active matter
    1. Techniques
    First passage problemsNon-Markovian processesTheories of collective dynamics & active matter
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    Benjamin Walter1,2,3, Gunnar Pruessner1,2, and Guillaume Salbreux4,5

    • 1Department of Mathematics, Imperial College London, 180 Queen's Gate, SW7 2AZ London, United Kingdom
    • 2Centre for Complexity & Networks, Imperial College London, SW7 2AZ London, United Kingdom
    • 3SISSA-International School for Advanced Studies, via Bonomea 265, 34135 Trieste, Italy
    • 4The Francis Crick Institute, 1 Midland Road, NW1 1AT London, United Kingdom
    • 5Department of Genetics and Evolution, University of Geneva, Quai Ernest-Ansermet 30, 1205 Geneva, Switzerland

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    Vol. 3, Iss. 1 — January - March 2021

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    • Figure 1
      Figure 1

      A particle in a potential (orange parabola) subject to both white and colored noise [see Eq. (1)]. While the white noise models a thermal environment whose timescale of correlation is negligibly small, the driving term models hidden degrees of freedom which are correlated over timescales comparable to those of the particle's stochastic dynamics. Those driving forces induce correlations (pink correlation kernel) in the particle's increments and therefore break its Markovianity. In this work, we study first passage times τx0,x1, the time such a random walker (blue rough path) takes to first reach x1 starting from x0 (dashed lines).

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    • Figure 2
      Figure 2

      First-order corrections to the probability distribution of first passage times as found by the framework presented in this work for two example processes. (a) First passage time distribution of active thermal Ornstein-Uhlenbeck process (ATOU) [cf. Eq. (76)] compared to numerical Laplace inversion of analytically obtained moment-generating function [cf. theoretical result in Eq. (98)] for various values of ν (solid lines). The plot marks indicate the distribution as sampled through Monte Carlo simulations with 5×106 runs. Simulation parameters are x0=0,x1=1,α=1,Dx=1,Dy=1,β=12 while ɛ is tuned to fix ν=Dyβɛ2/(Dxα) to values as indicated in the legend. The inset shows rescaled deviations to the undriven first passage time distribution [plot marks, cf. Eq. (104)] as compared to the first-order correction (solid lines, ν=0.1 omitted). Higher-order corrections appear for growing values of ν (cf. Fig. 3). See Sec. 3a2 for discussion. (b) First passage time distribution of active thermal Brownian motion (ATBM) [cf. Eq. (108)] compared to numerical Laplace inversion of analytically obtained moment-generating function [cf. theoretical result in Eq. (127)] for various values of ν (solid lines). The plot marks indicate the distribution as sampled through Monte Carlo simulations with 106 runs. Simulation parameters are x0=0,x1=π,r=1,Dx=1,Dy=1,β=12 while ɛ is tuned to fix ν=Dyr2βɛ2/Dx2 to values as indicated in legend. The inset shows rescaled deviations to the undriven first passage time distribution [cf. Eq. (104)] as compared to the first-order correction (solid lines, ν=0.1 omitted). Higher-order corrections appear for growing values of ν (cf. Fig. 6). See Sec. 3b2 for discussion.

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    • Figure 3
      Figure 3

      Numerical validation of first-order correction to FPT moment-generating function M1 [cf. Eq. (4)] of ATOU [see Eq. (76) and Sec. 3a for discussion]. The result is calculated in Eq. (100). Numerical simulations are shown for various values of 0.1≤ν≤0.8 (plot marks). The moment-generating functions were sampled for x0=0,x1=1,Dx=1,Dy=1,α=1,β=0.1 and ɛ suitably chosen to fix ν=Dyɛ2β/(Dxα). For small values of ν agreement with theoretical first-order correction (black line) is excellent. For larger values of ν the deviation increases. The rescaled deviation M̃2 [see Eq. (102)], (inset) collapse and thus confirm that these deviations are systematic higher-order corrections. See Sec. 3a2 for further results and discussion.

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    • Figure 4
      Figure 4

      First-order correction to first and second moments of active thermal Ornstein-Uhlenbeck process (ATOU). Simulation parameters are x0=0,Dx=1,Dy=1,α=1,β=12 and ɛ suitably chosen to fix ν=Dyβɛ2/(Dxα). Averages were taken over 106 samples. (a) Correction to mean first passage time of ATOU [cf. Eq. (76)] as obtained from Eq. (106) versus target positions x1, x0=0 fixed, and various values of ν (plot marks) compared to theoretical result to first order in ν (black line) using Eq. (105) and the result obtained in (100). The inset shows the mean first passage time τx0,x1 as measured vs x1 for values of ν=0 to 0.8. Correction due to active driving noise increases MFPT for x1≲1.6 and decreases MFPT for x1≳1.6. This behavior is fully captured by the analytic result. (b) Correction to mean-squared first passage time of ATOU [cf. Eq. (76)] as obtained from Eq. (107) versus target positions x1, x0=0 fixed, and various values of ν (plot marks) compared to theoretical result to first order in ν (black line) using Eq. (105) and the result obtained in (100). The inset shows the mean-squared first passage time τx0,x1 as measured vs x1 for values of ν=0 to 0.8.

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    • Figure 5
      Figure 5

      A particle on a circle of radius r is driven by both white (thermal) and colored (active) noise [cf. Eq. (108)]. We study the first passage time distribution from x0 to x1 as a function of the angle θ∈[0,2π).

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    • Figure 6
      Figure 6

      Numerical validation of first-order correction to FPT moment-generating function M1 of ATBM [cf. Eq. (108) and Sec. 3b for discussion] for various values of 0.1≤ν≤0.8. The moment-generating functions were sampled for x0=0,x1=π, Dx=1, α=1, Dy=1, β=1 and ɛ suitably chosen to fix ν=Dyr2βɛ2/(Dx2) (plot marks). For small values of ν, agreement with theoretical first-order correction (black line) is very good. For larger values of ν, the deviation increases. The rescaled deviations M̃2 [see Eq. (134)] (inset) collapse and thus confirm that these deviations are systematic higher-order corrections. See Sec. 3b2 for further results and discussion.

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    • Figure 7
      Figure 7

      First-order correction to first and second moments of first passage times of active thermal Brownian motion (ATBM) with periodic boundary conditions. Simulation parameters are x0=0,Dx=1,r=1,Dy=1,β=12 and ɛ suitably chosen to fix ν=Dyr2βɛ2/(Dx2). (a) Correction to mean first passage time of ATBM on a ring [cf. Eq. (108)] as obtained from Eq. (106) versus target position x1 and various values of ν (plot marks). This is compared to theoretical result of Eq. (131) (solid black line). The inset shows the measured mean first passage time versus a varying target position x1 and different values of ν. (b) Correction to mean-squared first passage time of ATBM on a ring [cf. Eq. (108)] as obtained from Eq. (107) versus target position x1 and various values of ν (plot marks). This is compared to theoretical result of twice differentiating Eq. (129) (solid black line). The inset shows the measured mean-squared first passage time versus a varying target position x1 and different values of ν.

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