We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution P(X, N) of the position X of the particle after N runs, with N &gt;&gt; 1. We show that in the regime X ~ N^{3/4} the distribution P(X, N) has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value X = X_c &gt; 0. The same is true for X = −X_c due to the symmetry of P(X, N). We show that this singularity corresponds to a first-order condensation transition: for X &gt; X_c a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at X = X_c. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm.

### First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension

#### Abstract

We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution P(X, N) of the position X of the particle after N runs, with N >> 1. We show that in the regime X ~ N^{3/4} the distribution P(X, N) has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value X = X_c > 0. The same is true for X = −X_c due to the symmetry of P(X, N). We show that this singularity corresponds to a first-order condensation transition: for X > X_c a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at X = X_c. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm.
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2021
Run-and-Tumble particles, Active Matter, Large Deviations, Condensation Transition
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.12571/27128`
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