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 $N\gg 1$. We show that in the regime $X\sim N^{3/4}$ 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.

24 pages, 10 figures

References in corpus (16)

• Novel type of phase transition in a system of self-driven particles
• Motility-Induced Phase Separation
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Statistical distribution of quantum entanglement for a random bipartite state
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Interaction driven real-space condensation
• Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap
• Condensation and Extreme Value Statistics
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Survival probability of a run-and-tumble particle in the presence of a drift
• Condensation transition in joint large deviations of linear statistics
• Condensation transition and ensemble inequivalence in the Discrete Nonlinear Schrödinger Equation
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Participation ratio for constraint-driven condensation with superextensive mass
• Conditioned random walks and interaction-driven condensation