Dynamically emergent correlations in a Brownian gas with diffusing diffusivity

TL;DR

This work analyzes a gas of $N$ non-interacting Brownian particles subjected to a common stochastic diffusivity $D(t)=B^2(t)$, revealing dynamic correlations that grow with time and drive ballistic expansion $x\sim t$. The authors derive the exact joint pdf, which exhibits a conditionally independent and identically distributed (CIID) structure, enabling closed-form calculations of macroscopic and microscopic observables such as the average density, order statistics, gaps, full counting statistics, and first-passage properties. A central result is that many observables share universal scaling forms governed by a Brownian functional $V(t)=2\int_0^t D(\tau) d\tau$, with its distribution $h(V,t)$ characterized by a scaling function $Q(z)$; crucial tails of $Q(z)$ control extreme-value and gap statistics, as well as FCS. The study uncovers a time-dependent shape transition in the full counting statistics near the upper bound of the observed region, and provides detailed asymptotics for single- and multi-particle first-passage and first-exit times, all derived from the same CIID framework. These results illuminate how a time-dependent, shared random environment induces strong, nontrivial correlations and rich temporal behavior in a simple Brownian gas, with potential extensions to other diffusivity models and confining potentials.

Abstract

We study a gas of $N$ Brownian particles in the presence of a common stochastic diffusivity $D(t)=B^2(t)$, where $B(t)$ represents a one-dimensional Brownian motion at time $t$. Starting from all the particles localized at the origin, the gas expands with a ballistic scaling $x\sim t$. We show that because of the common stochastic diffusivity, the expanding gas gets dynamically correlated, and the joint probability density function of the position of the particles has a CIID structure that was recently found in several other systems. The special structure allows us to compute the average density profile of the gas, extreme and order statistics, gap distribution between successive particles, and the full counting statistics (FCS) that describes the probability density function (PDF) $H(κ, t)$ of the fraction of particles $κ$ in a given region $[-L,L]$. Interestingly, the position fluctuation of the central particles and the average density profiles are described by the same scaling function. The PDF describing the FCS has an essential singularity near $κ=0$, indicating the presence of particles inside the box $[-L,L]$ at all times. Near the upper limit $κ=1$, the scaling function $H(κ,t)$ has a rather unusual behavior: $H(κ,t)\sim (1-κ)^{β(t)}$ where the exponent $β(t)$ changes continuously with time. At early times $β(t)$ is negative, indicating a divergence of $H(κ,t)$ as $κ\to 1$, whereas $β(t)$ becomes positive for $t>t_c$ where $t_c$ is computed exactly. Thus, as a function of $t$, the FCS exhibits an interesting shape transition. We also obtain the PDFs of the first-passage time to a given position $x$ and first-exit time from a box $[-L,L]$, by any one of the particles, and find that both PDFs are described by the same scaling function.

Figures (11)

• Figure 1: The PDF $Q(V/[\xi(t)]^2) \equiv [\xi(t)]^2 \, h(V,t)$ of the scaled variable $V/[\xi(t)]^2$. The points are from numerical simulations (detailed in \ref{['sec:sim_h_V']}) for the parameters $\Lambda=1$, $dt =0.01$, $t=10$, averaged over $10^8$ realizations. The left panel highlights the small $z$ behavior, while the right one highlights the tail behavior. Left: The solid lines are analytical plots for different truncations of the infinite series in equation \ref{['Q(z)']}. Right: The solid line plots the asymptotic tail given in \ref{['eq:Q large z']}.
• Figure 2: The scaled collapsed plot of the distribution of the average density profile at various times. The solid curve plots the closed form of the analytical scaling function $f(z) \equiv [2 \sqrt{2} \xi(t)] \, \rho(x,t)$ given in \ref{['single particle distribution']}. The parameters used in the simulation are $\Lambda=1$, $dt = 0.01$, and we average over $10^7$ realizations.
• Figure 3: The PDF of the scaled variable $M_k/[\xi(t) \theta_\alpha]$ for various $\alpha = k/N$. The points represent the simulation data for parameters $\Lambda =1$, $dt = 0.01$, $t=4$, $N= 10^6$, and averaged over $10^6$ realizations. The solid curve is the analytical scaling function $Q(z)$ described in equation \ref{['order_statistics equation']}. The inset shows the essential singularity of the distribution near zero.
• Figure 4: The PDF of the central particle with the appropriate scaling. The points represent the simulation data for the parameters $\Lambda =1$, $dt = 0.01$, $t=4$, $N= 10^6$, and averaged over $10^6$ realizations. The solid line plots the closed form of the scaling function $f(z) \equiv \,[\sqrt{N}/ (\, 2 \sqrt{\pi} \xi(t) \, )]^{-1} \, \mathrm{Prob}.(M_{1/2})$ given in equation \ref{['single particle distribution']}.
• Figure 5: Plot of the distribution of the scaled spacing variable $(M_k - M_{k+1})/\lambda_N(t)$. The points are from numerical simulations while the solid line plots the scaling function $F(z)$ in eq:Hx. The parameters used in the simulation are $\Lambda =1$, $dt=0.01$, $t =4$, $N = 10^6$, and averaged over $10^6$ realizations.