First passage time distribution of active thermal particles in potentials

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 coloured noise. This class of non-Markovian processes is at the centre 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 (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.

Extinctions of coupled populations, and rare event dynamics under non-Gaussian noise

“…Colored, and in particular non-Gaussian noise is often generated by an autonomous process, see e.g. [20][21][22][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. A prototypical model of this family features a "reaction coordinate" x + driven both by Gaussian noise and an additional non-Gaussian colored noise…”

“…The broader class of problems amenable to this formalism includes many systems experiencing colored and in particular non-Gaussian noise, e.g. [20][21][22][36][37][38][39][40][41][42][43][44][45][46][47], sometimes appearing in conjunction with a noise-induced stabilization effect [14, 16, 20-22, 41, 48-50]. Within this class, works on specific models provided numerical or partial analytic results [16,21,37,39,42,49], while others [36,38,[45][46][47][48] employed features of the specialized form of certain models to gain analytical insight, yet without describing the dynamics leading to the rare event or providing a general methodology.…”

Field Theory of free Active Ornstein-Uhlenbeck Particles

“…The external potential V (x) in Eq. (1a) has been moved to the perturbative part of the action, as it is usually not easily integrated, except for special cases, such as a harmonic one, similar to the effective friction potential of the velocity here [22,23]. More traditionally, one can recast the process (1) in a master equation on a lattice [17,24] and obtain Eq.…”

“…Because of the harmonic potential term τ −1 ∂ v (vφ) and the mixed term ivk, we cannot diagonalise the action by simply Fourier transforming in v as well. Instead we follow similar considerations for the harmonic potential acting on a particle's position using Hermite polynomials [22,23] and the eigenvalue problem of the Kramer's equation [18].…”

Run-and-tumble motion: field theory and entropy production

“…with eigenvalue λ n = −n/L 2 , where H n (x) is the n-th Hermite polynomial, see Appendix A [36,37,30]. Defining the set of functions…”