Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion

Abstract

We study the fluctuation properties of the local time density, $\ddot{I}{}_T=\frac{1}{T}{\int }_0^T\hat{I}´(r(t)-1)dt$, spent by a $d$-dimensional Brownian particle at a spherical shell of unit radius, where $r(t)$ denotes the radial distance from the particle to the origin. In the large observation time limit, $T\to \infty$, the local time density $\ddot{I}{}_T$ obeys the large deviation principle, $P(\ddot{I}{}_T=\ddot{I})\sim e^{-TI(\ddot{I})}$, where the rate function $I(\ddot{I})$ is analytic everywhere for $d\le 4$. In contrast, for $d>4$, $I(\ddot{I})$ becomes nonanalytic at a specific point $\ddot{I}=\ddot{I}{}_c^{(d)}$, where $\ddot{I}{}_c^{(d)}=d(d-4)/(2d-4)$ depends solely on dimensionality. The singularity signals the occurrence of a first-order dynamical phase transition in dimensions higher than four. Such a transition is accompanied by temporal phase separations in the large deviations of Brownian trajectories. Finally, we validate our theoretical results using a rare-event simulation approach.

11 pages, 5 figures