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.
Highlights
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
Active thermal Ornstein-Uhlenbeck process (ATOU). In this example we study the case of a particle in a harmonic potential driven by white and colored noise described by the Langevin equation xt = −αxt + ξt + εyt with driving noise correlator [see Eq (68)]
The process driven by an additional “active” term εyt. We refer to this process as active thermal Ornstein-Uhlenbeck process (ATOU)
Summary
Understanding the statistical properties of first passage times (FPT), the time a stochastic process takes to first reach a prescribed target has enjoyed increased attention over the last two decades [1–3] since it is a key characteristic of complex systems, such as chemical reactions [4], polymer synthesis [5], intracellular events [6], neuronal activity [7], or financial systems [8]. The formulas we obtain order by order are exact, and the results we obtain for two systems, as an illustration, are in excellent agreement with numerical simulations This allows for a detailed analysis of the qualitative changes of the FPT distribution induced by correlations in stochastic forces. We assume that the particle is subject to a second stochastic force which may model either self-propulsion or hidden internal degrees of freedom We refer to this force as active, as we equate “activeness” with the presence of colored noise (e.g., telegraphic noise or an Ornstein-Uhlenbeck noise), following, e.g., [29–31]. Understanding the FPT in active thermal matter is relevant to, e.g., neural activity [40], transport in living cells [41], and molecular motors [42,43] where the influence of the thermal environment cannot necessarily be neglected
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