Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension

Abstract

We consider an active run-and-tumble particle (RTP) in d dimensions, starting from the origin and evolving over a time interval [0, t]. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the x component does not change sign up to time t, showing that it does not depend on d. As a consequence of this result, we compute exactly other x-component properties, namely the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e. they do not depend on d. Moreover, we show that these universal results hold also if the speed v of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)].

Universal survival probability for a correlated random walk and applications to records

“…In the particular case of the exponential distribution where p f (τ ) = p(τ )/γ, one simply has that p f (τ )e −sτ /p f (s) = p(τ )e −sτ /p(s) and the (n + 1)-fold integral Q n+1 can be interpreted as the universal survival probability of an alternating random walk given by Q n+1 = S + n+1 (x = 0; q = 0) in equation (24). Using this result together with the inverse Laplace transform recovering the result of [32] (see also [39,40] for an extension to the survival probability in arbitrary dimension d 1). Here we used the Taylor expansion [46] I p (x) =…”

“…In this section, using the Sparre Andersen theorem [16], we derive the exact expression for the survival probability S + n (x = 0; q), showing that it is completely universal, i.e. independent of the step distribution p(η), for any finite n. The derivation below is based on the technique presented in [39,40], in which the survival probability of an RTP in d dimensions is investigated. In the general case of starting position x 0, one can show that S + n (x; q) and S − n (x; q) satisfy a set of coupled integral equations.…”

Universal survival probability for a correlated random walk and applications to records

“…(24). Using this result together with the inverse Laplace transform recovering the result of [32] (see also [39,40] for an extension to the survival probability in arbitrary dimension d ≥ 1). Here we used the Taylor expansion [46]…”

“…independent of the step distribution p(η), for any finite n. The derivation below is based on the technique presented in [39,40], in which the survival probability of an RTP in d dimensions is investigated. In the general case of starting position x ≥ 0, one can show that S + n (x; q) and S − n (x; q) satisfy a set of coupled integral equations.…”