Random walks and related diffusion equations

Theoretical and Mathematical Physics, 1997

A particle moving in inhomogeneous 1D media is considered. Its speed performs abrupt changes of direction at Poisson times. For such process backward and forward Kolmogorov's equations are derived. The explicit formulas for the probability distributions of this process are obtained as well as for similar processes in presence of reflecting and absorbing barriers.

Exact expressions for the arrival probabilities with direction are obtained for correlated walks on an infinite line. The probability distribution exhibits a diffusive maximum, similar to that characteristic of random walks, and a runaway component which is associated with free passage (no scattering). For symmetric step probabilities, the arrival probabilities for a finite line bounded by reflecting walls are expressible in terms of free-space probabilities. The evolution of the system of probabilities is studied in terms of the Boltzmann H function. The system approaches equilibrium monotonically. In general, there exists an optimum degree of correlation between successive steps at which the randomization in space and direction proceeds most rapidly. At lower correlation the system moves like a wave packet with dissipation. The randomization in space is aided by the reflecting walls (and by the periodic boundary).

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1982

The discrete random walk problem for the unrestricted particle formulated in the double diffusion model given in Hill [2] is solved explicitly. In this model it is assumed that a particle moves along two distinct horizontal paths, say the upper path I and lower path 2. For i = 1, 2, when the particle is in path i, it can move at each jump in one of four possible ways, one step to the right with probability pi, one step to the left with probability qi, remains in the same position with probability ri, or exchanges paths but remains in the same horizontal position with probability si (pi + qi + ri + si = 1). Using generating functions, the probability distribution of the position of an unrestricted particle is derived. Finally some special cases are discussed to illustrate the general result.

Microelectronic Engineering, 2003

In this paper a rigorous probabilistic two-probability-parameter model of a diffusion barrier is investigated that describes comprehensively reflection, absorption, and segregation phenomena at a diffusion barrier. As a special case, a rigorous analysis of counting paths for 1D random walk in the presence of a reflecting barrier is presented. This paper defines and makes distinction between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. A combinatorial formula is derived showing that in the presence of a totally reflecting barrier (at m 50) the probability of a particle departing from position m 5 2j and arriving at position m 5 2k on the positive b j axis after N 5 2M steps is given by W (2k,N) 5 [C(2M, M 2 j 1 k) 2 C(2M, M 2 j 2 k 2 l)] / [C(2M,M) 1 2 o 2j i 50 C(2M,M 1 i)], where C(n,m) denotes the binomial coefficient. This formula enables easy computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps N 5 2M as 1 /(M 1 1) and that the peak of the probability distribution is moving away from the barrier with the increasing ] OE number of steps as 4M. Thus, the subsurface region is progressively depleted. The present analysis has bearing on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.

Discrete Mathematics, 1982

A one-dimensional random walk with unequa.1 step lengths restricted by tin absorbing barrier is considered as follows: (1) ezmmeration of the number of non-decreasing paths in a non-negative quadrant of the integral square lattice and in the inside of a polygon, (2) evaluatiion of trarlsient (or absorption) probabilities for tbe random wblk.

Journal of Physics A: Mathematical and Theoretical, 2008

The present work studies continuous time random walks (CTRWs) in a finite domain. A broad class of boundary conditions, of which absorbing and reflecting boundaries are particular cases, is considered. It is shown how any CTRW in this class can be mapped to a CTRW in an infinite domain. This may allow applying well-known techniques for infinite CTRWs to the problem of obtaining the fluid limit for finite domain CTRWs, where the fluid limit (or hydrodynamic limit) refers to the partial differential equation describing the long time and large distance behaviour of the system. As an illustration, the fluid limit equation and its propagator are obtained explicitly in the case of purely reflecting boundaries. We also derive the modification of the Riemann-Liouville fractional differential operators implementing the reflecting boundary conditions.

Physical Review Letters, 1987

Analysis of Monte Carlo enumerations for diAusion on the fractal structure generated by the random walk on a two-dimensional lattice allows us to predict a behavior &r)n "(1nn)' with v=0. 325~0.01 and a =0.35~0.03. This leads to the conjecture that v=a = -, ' . This value of v, and the presence of logarithmic corrections, are strongly supported by heuristic arguments based on Flory theory and on plausible assumptions.

Physica A: Statistical Mechanics and its Applications, 1990

1"he random walk of a particle in a one-dimensional random mcdmm is examined by means of the cquiwflent transfer rates technique, in the discrcle as well as m the continuous version of lhe model.

Journal of Physics A: Mathematical and General, 2000

When a large number N of independent random walkers diffuse on a d-dimensional Euclidean substrate, what is the expectation value t 1,N of the time spent by the first random walker to cross a given distance r from the starting place? We here explore the relationship between this quantity and the number of different sites visited by N random walkers all starting from the same origin. This leads us to conjecture that t 1,N ≈ (r 2 /4D ln N)[1 + ∞ n=1 (ln N) −n n m=0 a (n) m (ln ln N) m ] for d 2, large N and r ln N , where a (n) m are constants (some of which we estimate numerically) and D is the diffusion constant. We find this conjecture to be compatible with computer simulations.

Journal of Experimental and Theoretical Physics, 2012

Based on the random trap model and using the mean field approximation, we derive an equation that allows the distribution of a functional of the trajectory of a particle making random walks over inhomo geneous lattice site to be calculated. The derived equation is a generalization of the Feynman-Kac equation to an inhomogeneous medium. We also derive a backward equation in which not the final position of the par ticle but its position at the initial time is used as an independent variable. As an example of applying the derived equations, we consider the one dimensional problem of calculating the first passage time distribu tion. We show that the average first passage times for homogeneous and inhomogeneous media with identical diffusion coefficients coincide, but the variance of the distribution for an inhomogeneous medium can be many times larger than that for a homogeneous one.