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

TL;DR

This work analyzes the large-deviation fluctuations of the local time density $\rho_T$ of a $d$-dimensional Brownian particle at a unit sphere, framing the problem through the tilted generator $\mathcal{L}_k=\nabla^2 + k\delta(r-1)$ and its dominant eigenvalue $\lambda(k)$. By solving the radial eigenproblem in terms of modified Bessel functions and enforcing proper boundary and matching conditions, the authors derive the SCGF $\lambda(k)$ and obtain the rate function $I(\rho)$ via the Legendre-Fenchel transform, uncovering a dimension-dependent dynamical phase transition. For $d\le 4$, $I(\rho)$ is analytic with a second-order singularity in $\lambda(k)$ at $k_c^{(d)}=d-2$ for $2<d\le 4$, while for $d>4$ a first-order transition appears with a linear branch $I(\rho)=(d-2)\rho$ for $0<\rho<\rho_c^{(d)}$ and $\rho_c^{(d)}=\frac{d(d-4)}{2d-4}$, signaling temporal phase separation in trajectory ensembles. The predictions are corroborated by rare-event simulations using multiple histogram reweighting, demonstrating the practical relevance of dimension-driven dynamical phase transitions in diffusion observables.

Abstract

We study the fluctuation properties of the local time density, ${ρ_T} = \frac{1}{T}\int_0^T {δ( {r(t) - 1} )} dt$, 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 $ρ_T$ obeys the large deviation principle, $P(ρ_T= ρ) \sim e^{-T I(ρ)}$, where the rate function $I(ρ)$ is analytic everywhere for $d\leq 4$. In contrast, for $d>4$, $I(ρ)$ becomes nonanalytic at a specific point $ρ=ρ_c^{(d)}$, where $ρ_c^{(d)}=d(d-4)/(2d-4)$ 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.

Figures (5)

• Figure 1: The results for $d=1$ and $d=2$: The SCGF $\lambda(k)$ (a) and the rate function $I(\rho)$ (b) of local time density $\rho$.
• Figure 2: The results for $d=3$ and $d=4$: The SCGF $\lambda(k)$ (a) and the rate function $I(\rho)$ (b) of local time density $\rho$. The solid circles in (a) denote the point $k=k_c^{(d)}$ where $\lambda(k)$ shows a singularity at this point.
• Figure 3: The results for $d=5$ and $d=6$: The SCGF $\lambda(k)$ (a) and the rate function $I(\rho)$ (b) of local time density $\rho$. The solid circles in (a) denote the point $k=k_c^{(d)}$, at which the first derivative of $\lambda(k)$ is discontinuous. The solid circles in (b) mark the phase transition point $\rho _c^{(d)}=\lambda'(k_c^{(d)}+0^{+})$ below which $I(\rho)$ is linear with the slope $k_c^{(d)}$.
• Figure 4: Sample 100 radial trajectories of the Brownian motion for $d=5$ (grey lines). For each trajectory, the local time density $\rho$ is constrained to be a constant: $\rho=0.7<\rho _c^{(5)}$ (a), $\rho=0.85 \approx \rho _c^{(5)}$ (b), and $\rho=1.2>\rho _c^{(5)}$ (c). The red lines indicate the average values of 100 samples. Parameters: $T=5$, $\Delta t=0.05$, and $\epsilon =0.1$.
• Figure 5: Simulation verifications for the rate functions for $d=1,\cdots,6$ (from (a) to (f), respectively). Simulation results are indicated by symbols, and the rate functions predicted by our theory are shown by lines. In simulations, we set $T=10$, $\Delta t=0.01$ and $\epsilon=0.05$.