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.

arXiv comment24 pages, 10 figures

References in corpus (45)

• Novel type of phase transition in a system of self-driven particles
• Active Particles in Complex and Crowded Environments
• Collective motion
• The Mechanics and Statistics of Active Matter
• Motility-Induced Phase Separation
• Statistical Mechanics of Interacting Run-and-Tumble Bacteria
• Diffusion with Stochastic Resetting
• Stochastic Resetting and Applications
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Pressure is not a state function for generic active fluids
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Run-and-tumble particle in one-dimensional confining potential: Steady state, relaxation and first passage properties
• Extreme value statistics of correlated random variables: a pedagogical review
• Statistical mechanics of a nonlinear discrete system
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• The Nature of the Condensate in Mass Transport Models
• Canonical Analysis of Condensation in Factorised Steady State
• Statistical distribution of quantum entanglement for a random bipartite state
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Wealth Condensation in Pareto Macro-Economies
• Run-and-tumble particles in two-dimensions : Marginal position distributions
• The Single Big Jump Principle in Physical Modelling
• A First-Order Dynamical Transition in the displacement distribution of a Driven 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
• Stationary superstatistics distributions of trapped run-and-tumble particles
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Constraint driven condensation in large fluctuations of linear statistics
• Condensation and Extreme Value Statistics
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Exact spectral solution of two interacting run-and-tumble particles on a ring lattice
• Localization transition in the Discrete Non-Linear Schrödinger Equation: ensembles inequivalence and negative temperatures
• 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
• Relaxation and coarsening of weakly-interacting breathers in a simplified DNLS chain
• Signatures of motor susceptibility in the dynamics of a tracer particle in an active gel
• Condensation Transition in Polydisperse Hard Rods
• Finite-size localization scenarios in condensation transitions
• Participation ratio for constraint-driven condensation with superextensive mass
• Conditioned random walks and interaction-driven condensation
• General criterion for harmonicity