Родион Телятник » Публикация

Аннотация The classical problem of random walks (RW), first formulated by Pearson in 1905 [1, 2], is the question about the probability to find a random walker at certain distance from the start point after n steps, that can occur in general at random direction and random distance with given probability distributions (the simplest case is the isotropic walks with constant step length λ). In most cases of arbitrary distributions, numbers n and space dimensions solution can’t be expressed in elementary functions [2]. For n→∞ and λ→0 Rayleigh found Gaussian asymptotic solution for similar problem [3] that reflects in fact Central Limit Theorem about sum of random independent identically distributed values, while Einstein obtained [4, 5] this asymptotic solution as Green function for diffusion equation that he derived from the recurrent chain equation of RW [6]. In this article we treat 3D isotropic RW with exponential distribution on free paths with mean value λ and mean free time τ [7, 8]. For the centrally symmetric system with known concentration ρ(r,t) of particles we found new concentration ρ(r,t+τ) after one step of RW and ρ(r,t+nτ) approximately. Solution is expressed in terms of exponential integrals that can be evaluated only computationally. Solution was also applied to the problem of the droplet growth in surrounding vapor as generalization to the case of transient Knudsen number, when λ is comparable to the radius of a droplet and the mass exchange between drop and vapor can be accounted neither in the limit of diffusive flow nor in the limit of free molecular flow, but the mixing of these flows causes exponential Langmuir jump of vapor concentration near the surface of a drop [9]. Obtained solutions were approved by direct simulations of RW.