Cusp of non - Gaussian density of particles for a diffusing diffusivity model

  Mario Hidalgo-Soria  ,  Eli Barkai  ,  Stanislav Burov  
Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel

We study a two state ``jumping diffusivity'' model for a Brownian process alternating between two different diffusion constants,  D_{+}>D_{-},  with random waiting times in both states  whose distribution is rather general. In the limit of long measurement times  Gaussian behavior with an effective diffusion coefficient is recovered. We show that for equilibrium initial conditions and when the limit of the diffusion coefficient D_{-} -> 0 is taken, the short time  behavior leads to a cusp, namely a non - analytical behavior, in the distribution of the displacements P(x,t) for x ->0. Visually this cusp, or tent-like shape, resembles  similar behavior  found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D_{+} to a delta-function. The short time behavior  of the same quantity converges  to a uniform distribution, which leads to the non - analyticity in P(x,t).