[Paper Review] First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension

Francesco Mori, Giacomo Gradenigo+1 more|arXiv (Cornell University)|Jan 1, 2021

Diffusion and Search DynamicsBiochemistry, Genetics and Molecular Biology70 references42 citations

TL;DR

This paper studies the position distribution of a one-dimensional run-and-tumble particle (RTP) with Gaussian-distributed velocities and exponentially distributed run times. It identifies a first-order dynamical phase transition at X ∼ N^{3/4}, where a single large displacement (condensate) dominates the trajectory, signaled by a discontinuous derivative in the rate function and a jump in the participation ratio, confirmed by exact analysis and numerical simulations.

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.

Motivation & Objective

• Understand the large-deviation statistics of the position distribution P(X, N) for a one-dimensional RTP with Gaussian velocities and exponential run times.
• Identify whether a first-order dynamical phase transition occurs in the absence of an external drive (E = 0), contrasting with previous E > 0 results.
• Characterize the nature of the transition by identifying the order parameter and analyzing its discontinuity at the critical point.
• Establish the scaling Xc ∼ N^{3/4} and confirm that the condensate size Xcond scales as N^{3/4}, matching the total displacement.
• Verify the transition mechanism and rate function via exact calculations and high-precision numerical simulations.

Proposed method

• Derive the large deviation form of the position distribution P(X, N) using the cumulant generating function and saddle-point approximation.
• Apply the formalism of real-space condensation in stochastic systems to identify the critical point Xc where the rate function F(z) exhibits a discontinuous derivative.
• Define the participation ratio of single-run displacements as the order parameter to detect the transition, showing its jump at X = Xc.
• Use constrained Markov chain Monte Carlo simulations to sample rare trajectories and validate the analytical predictions.
• Compute the condensate distribution pcond(x, N) and show that its integral over x is 1/N, confirming localization in a single run.
• Relate the critical behavior to the solution of transcendental equations involving the rate function and its second derivatives.

Experimental results

Research questions

• RQ1 What is the nature of the large-deviation behavior of the position distribution P(X, N) for a one-dimensional RTP with Gaussian velocities and exponential run times?
• RQ2 Does a first-order dynamical phase transition occur in the absence of an external drive (E = 0), and if so, what is the scaling of the critical position Xc?
• RQ3 How does the condensate—defined as a single dominant run—emerge in the trajectory, and what is its size scaling with N?
• RQ4 Is there an order parameter that exhibits a discontinuity at the transition point, signaling a first-order transition?
• RQ5 What is the exact form of the rate function F(z) for the large-deviation regime, and how does it reflect the non-analyticity at the critical point?

Key findings

• The position distribution P(X, N) for the RTP exhibits a first-order dynamical phase transition at X = ±Xc, where the rate function F(z) has a discontinuous derivative.
• The critical position scales as Xc ∼ N^{3/4}, corresponding to the large-deviation scaling α = 3/4 in the rate function F(z) = N^{2α−1}F(X/N^α).
• A single run of size Xcond ∼ N^{3/4} dominates the trajectory for |X| > Xc, forming a real-space condensate that contributes a finite fraction of the total displacement.
• The participation ratio of the single-run displacements acts as the order parameter and exhibits a jump discontinuity at X = Xc, confirming a first-order transition.
• Numerical simulations using a constrained Markov chain Monte Carlo algorithm confirm the analytical predictions, including the condensate localization and the scaling of the rate function.
• The integral of the condensate distribution pcond(x, N) over x is exactly 1/N, confirming that the condensate is localized in a single displacement among N runs.