A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle

We study the probability distribution $P(X_N=X,N)$ of the total displacement $X_N$ of an $N$-step run and tumble particle on a line, in presence of a constant nonzero drive $E$. While the central limit theorem predicts a standard Gaussian form for $P(X,N)$ 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.Comment: 35 pages, 5 figures. An algebraic error in Appendix B of the previous version of the manuscript has been corrected. A new argument for the location $z_c$ of the transition is reported in Appendix B.

Extreme statistics and index distribution in the classical 1d1d Coulomb gas

We consider a one-dimensional gas of $N$ charged particles confined by an external harmonic potential and interacting via the one-dimensional Coulomb potential. For this system we show that in equilibrium the charges settle, on an average, uniformly and symmetrically on a finite region centred around the origin. We study the statistics of the position of the rightmost particle $x_{\max}$ and show that the limiting distribution describing its typical fluctuations is different from the Tracy-Widom distribution found in the one-dimensional log-gas. We also compute the large deviation functions which characterise the atypical fluctuations of $x_{\max}$ far away from its mean value. In addition, we study the gap between the two rightmost particles as well as the index $N_+$, i.e., the number of particles on the positive semi-axis. We compute the limiting distributions associated to the typical fluctuations of these observables as well as the corresponding large deviation functions. We provide numerical supports to our analytical predictions. Part of these results were announced in a recent Letter, Phys. Rev. Lett. 119, 060601 (2017).Comment: 34 pages, 8 figure

Survival probability of an immobile target surrounded by mobile traps

We study analytically, in one dimension, the survival probability $P_{s}(t)$ up to time $t$ of an immobile target surrounded by mutually noninteracting traps each performing a continuous-time random walk (CTRW) in continuous space. We consider a general CTRW with symmetric and continuous (but otherwise arbitrary) jump length distribution $f(\eta)$ and arbitrary waiting time distribution $\psi(\tau)$. The traps are initially distributed uniformly in space with density $\rho$. We prove an exact relation, valid for all time $t$, between $P_s(t)$ and the expected maximum $E[M(t)]$ of the trap process up to time $t$, for rather general stochastic motion $x_{\rm trap}(t)$ of each trap. When $x_{\rm trap}(t)$ represents a general CTRW with arbitrary $f(\eta)$ and $\psi(\tau)$, we are able to compute exactly the first two leading terms in the asymptotic behavior of $E[M(t)]$ for large $t$. This allows us subsequently to compute the precise asymptotic behavior, $P_s(t)\sim a\, \exp[-b\, t^{\theta}]$, for large $t$, with exact expressions for the stretching exponent $\theta$ and the constants $a$ and $b$ for arbitrary CTRW. By choosing appropriate $f(\eta)$ and $\psi(\tau)$, we recover the previously known results for diffusive and subdiffusive traps. However, our result is more general and includes, in particular, the superdiffusive traps as well as totally anomalous traps