Density approach to ballistic anomalous diffusion: An exact analytical treatment

Abstract

This paper addresses the problem of deriving the probability distribution density of a diffusion process generated by a nonergodic dichotomous fluctuation using the Liouville equation (density method). The velocity of the diffusing particles fluctuates from the value of 1 to the value of −1, and back, with the distribution density of time durations τ of the two states proportional to 1/τμ in the asymptotic time limit. The adopted density method allows us to establish an exact analytical expression for the probability distribution density of the diffusion process generated by these fluctuations. Contrary to intuitive expectations, the central part of the diffusion distribution density is not left empty when moving from μ>2 (ergodic condition) to μ<2 (nonergodic condition). The intuitive expectation is realized for μ<μcr, with μcr≈1.6. For values of μ>μcr, the monomodal distribution density with a minimum at the origin is turned into a bimodal one, with a central bump whose intensity increases for μ→2. The exact theoretical treatment applies to the asymptotic time limit, which establishes for the diffusion process the ballistic scaling value δ=1. To assess the time evolution toward this asymptotic time condition, we use a numerical approach which relates the emergence of the central bump at μ=μcr with the generation of the ordinary scaling δ=0.5, which lasts for larger and larger times for μ coming closer and closer to the critical value μ=2. We assign to the waiting time distribution density two different analytical forms: one derived from the Manneville intermittence (MI) theory and one from the Mittag-Leffler (ML) survival probability. The adoption of the ML waiting time distribution density generates an exact analytical prediction, whereas the MI method allows us to get the same asymptotic time limit as the ML one for μ<2 as a result of an approximation. The joint adoption of these two waiting time distribution densities sheds light into the critical nature of the condition μ=2 and into why this is the critical point for the MI process, representing the phase transition from the nonergodic to the ergodic regime. Our main result can be interpreted as a new derivation of Lamperti distribution.

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