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We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression for the Greens function (propagator) and investigate both analytically and numerically how this function and the associated moments behave. We also study first-passage properties in expanding hyperspherical geometries. We show that in all cases the behavior is determined to a great extent by the so-called Brownian conformal time $tau(t)$, which we define via the relation $dot tau=1/a^2$, where $a(t)$ is the expansion scale factor. If the medium expansion is driven by a power law [$a(t) propto t^gamma$ with $gamma>0$], we find interesting crossover effects in the mixing effectiveness of the diffusion process when the characteristic exponent $gamma$ is varied. Crossover effects are also found at the level of the survival probability and of the moments of the first passage-time distribution with two different regimes separated by the critical value $gamma=1/2$. The case of an exponential scale factor is analyzed separately both for expanding and contracting media. In the latter situation, a stationary probability distribution arises in the long time limit.
We consider a continuous random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates
It is pointed out that, for the fractional Fokker-Planck equation for subdiffusion proposed by Metzler, Barkai, and Klafter [Phys. Rev. Lett. 82 (1999) 3563], there are four types of infinitely many exact solutions associated with the newly discovere
We study a stochastic process $X_t$ related to the Bessel and the Rayleigh processes, with various applications in physics, chemistry, biology, economics, finance and other fields. The stochastic differential equation is $dX_t = (nD/X_t) dt + sqrt{2D
We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise
Microscopic theory of Brownian motion of a particle of mass $M$ in a bath of molecules of mass $mll M$ is considered beyond lowest order in the mass ratio $m/M$. The corresponding Langevin equation contains nonlinear corrections to the dissipative fo