Conflict between trajectories and density description: the statistical source of disagreement

Abstract

We study an idealized version of intermittent process leading the fluctuations of a stochastic dichotomous variable $ξ$. It consists of an overdamped and symmetric potential well with a cusp-like minimum. The right-hand and left-hand portions of the potential corresponds to $ξ= W$ and $ξ= -W$, respectively. When the particle reaches this minimum is injected back to a different and randomly chosen position, still within the potential well. We build up the corresponding Frobenius-Perron equation and we evaluate the correlation function of the stochastic variable $ξ$, called $Φ_ξ(t)$. We assign to the potential well a form yielding $Φ_ξ(t) = (T/(t + T))^β$, with $β> 0$. We limit ourselves to considering correlation functions with an even number of times, indicated for concision, by $<12>$, $<1234>$ and, more, in general, by $<1 ... 2n>$. The adoption of a treatment based on density yields $<1 ... 2n > = < 1 2 > ... < (2n-1) 2n>$. We study the same dynamic problem using trajectories, and we establish that the resulting two-time correlation function coincides with that afforded by the density picture, as it should. We then study the four-times correlation function and we prove that in the non-Poisson case it departs from the density prescription, namely, from $<1234 > = < 12 > < 34>$. We conclude that this is the main reason why the two pictures yield two different diffusion processes, as noticed in an earlier work [M. Bologna, P. Grigolini, B.J. West, Chem. Phys. {\bf 284}, (1-2) 115-128 (2002)].