Dr Francesco Mori

Abstract:

We study the position distribution P(R[over ⃗],N) of a run-and-tumble particle (RTP) in arbitrary dimension d, after N runs. We assume that the constant speed v>0 of the particle during each running phase is independently drawn from a probability distribution W(v) and that the direction of the particle is chosen isotropically after each tumbling. The position distribution is clearly isotropic, P(R[over ⃗],N)→P(R,N) where R=|R[over ⃗]|. We show that, under certain conditions on d and W(v) and for large N, a condensation transition occurs at some critical value of R=R_{c}∼O(N) located in the large-deviation regime of P(R,N). For RR_{c} is typically dominated by a "condensate," i.e., a large single run that subsumes a finite fraction of the total displacement (supercritical condensed phase). Focusing on the family of speed distributions W(v)=α(1-v/v_{0})^{α-1}/v_{0}, parametrized by α>0, we show that, for large N, P(R,N)∼exp[-Nψ_{d,α}(R/N)], and we compute exactly the rate function ψ_{d,α}(z) for any d and α. We show that the transition manifests itself as a singularity of this rate function at R=R_{c} and that its order depends continuously on d and α. We also compute the distribution of the condensate size for R>R_{c}. Finally, we study the model when the total duration T of the RTP, instead of the total number of runs, is fixed. Our analytical predictions are confirmed by numerical simulations, performed using a constrained Markov chain Monte Carlo technique, with precision ∼10^{-100}.

Abstract:

We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic Brownian motion of duration t and diffusion constant D, in the presence of resetting to the origin at a constant rate r. We show that for any t, the mean perimeter is given by 〈L(t)〉=2πsqrt[D/r]f_{1}(rt) and the mean area is given by 〈A(t)〉=2πD/rf_{2}(rt) where the scaling functions f_{1}(z) and f_{2}(z) are computed explicitly. For large t≫1/r, the mean perimeter grows extremely slowly as 〈L(t)〉∝ln(rt) with time. Likewise, the mean area also grows slowly as 〈A(t)〉∝ln^{2}(rt) for t≫1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.

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 noninstantaneous. 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)10.1103/PhysRevLett.124.090603].

Abstract:

We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference τ=t_{min}-t_{max} between the time t_{min} of the global minimum and the time t_{max} of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of τ, as well as the covariance of t_{max} and t_{min}, showing that these times are anticorrelated. We demonstrate that the distribution of τ for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n→∞. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of τ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "τ=n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1+1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.200201].