A first-order dynamical transition in the displacement distribution of a driven run-and-tumble particle

We study the probability distribution P(XN  =  X,N) of the total displacement XN of an N-step run and tumble particle on a line, in the presence of a constant nonzero drive E. While the central limit theorem predicts a standard Gaussian form for near its peak, we show that for large positive and negative X, the distribution exhibits anomalous large deviation forms. For large positive X, the associated rate function is nonanalytic at a critical value of the scaled distance from the peak where its first derivative is discontinuous. This signals a first-order dynamical phase transition from a homogeneous 'fluid' phase to a 'condensed' phase that is dominated by a single large run. A similar first-order transition occurs for negative large fluctuations as well. Numerical simulations are in excellent agreement with our analytical predictions.

We present here a revised version of the appendices of Gradenigo and Majumdar (2019 J. Stat. Mech. 053206). Some minor corrections are introduced and a new simplified argument to obtain the critical value of rc, the control parameter for the transition, is presented. The overall scenario and the description of the transition mechanism depicted in Gradenigo and Majumdar (2019 J. Stat. Mech. 053206) remains completely untouched, the only relevant dierence being the value of rc fixed to rc = 21/3 = 1.259 92 ... rather than rc = 1.3805 .... This dierence also implies a small quantitative changes in figures 2 and 4; a new version of both figures is reported here. A couple of other typos discovered in the paper are pointed out and the correct version of the expressions are reported.