Condensation transition in the late-time position of a run-and-tumble particle,Physical Review E

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Condensation transition in the late-time position of a run-and-tumble particle
Physical Review E ( IF 2.4 ) Pub Date : 2021-06-21 , DOI: 10.1103/physreve.103.062134
Francesco Mori ₁ , Pierre Le Doussal ₂ , Satya N Majumdar ₁ , Grégory Schehr ₃

Affiliation

We study the position distribution

P (\vec{R}, 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 (\vec{R}, N) \to P (R, N)

where

R = | \vec{R} |

. 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} \sim O (N)

located in the large-deviation regime of

P (R, N)

. For

R < R_{c}

(subcritical fluid phase), all runs are roughly of the same size in a typical trajectory. In contrast, an RTP trajectory with

R > R_{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) \sim exp [- N ψ_{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