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 ) = 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 ∼ 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="s