Dynamically emergent correlations in a Brownian gas with diffusing diffusivity

Abstract

We study a gas of N Brownian particles in the presence of a common stochastic diffusivity D(t) = B2(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 ∼ t. We show that because of the common stochastic diffusivity, the expanding gas becomes dynamically correlated, and the joint probability density function of the position of the particles has a conditionally independent and identically distributed (CIID) structure that was recently found in several other systems. This special CIID structure allows us to compute the average density profile of the gas, extreme and order statistics, the gap distribution between successive particles, and the full counting statistics (FCS) that describe the probability density function (PDF) H(κ, t) of the fraction of particles κ in a given region [−L,L]. Interestingly, the position fluctuations 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) ∼ (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→1, whereas β(t) becomes positive for t > tc, where tc is computed exactly. For t > tc, the scaling function H(κ, t) vanishes as κ→1, indicating that it is highly unlikely to have all the particles in the interval [−L,L]. Exactly at t = tc, β =0, indicating that the PDF approaches a non-zero constant as κ→1. 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.