Subdiffusion and stable laws

This paper examines particle diffusion in N-dimensional Euclidean space wit h traps of the return type. Under the assumption that the random continuous -diffusion time has a finite mean value, it is established that subdiffusio n (which is characterized by an increase in the width of the diffusion pack et with time according to the t(alpha)-law, where alpha < 1; for normal dif fusion alpha=1) emerges if and only if the distribution density of the rand om time a particle spends in a trap has a tail of the power-law type propor tional to t(alpha-1). In these conditions the asymptotic expression for the distribution density of a diffusing particle is found in terms of the dens ity of a one-sided stable law with a characteristic exponent alpha. It is s hown that the density is a solution of subdiffusion equations in fractional derivatives. The physical meaning of the solution is discussed, and so are the properties of the solution and its relation to the results of other re searchers in the field of anomalous-diffusion theory. Finally, the results of numerical calculations are discussed. (C) 1999 American Institute of Phy sics. [S1063-7761(99)01606-6].