Lévy random walks in finite systems

Lévy walks on finite intervals with absorbing boundaries are studied using analytic and Monte Carlo techniques. The integral equations for Lévy walks in infinite 1D systems are generalized to treat the evolution of the probability distribution on finite and semi-infinite intervals. In particular the near-boundary behavior of the probability distribution and also its properties at asymptotically large times are studied. The probability distribution is found to be discontinuous near the boundary for Lévy walks in finite and semi-infinite systems. Previous results for infinite systems, and a previous scaling for semi-infinite systems, are reproduced. The use of linear operator theory to solve the integral equations governing the evolution of the Lévy walk implies that the probability distribution decays exponentially at large times. For a jump distribution that satisfies $\psi \mathrm{}\left(x\right)\mathrm{}\sim |x{|}^{-\alpha }$ for large $|x|,$ the decay constant for the exponential decay is estimated and found to scale at large L as $L^{1-\alpha }$ for $2<\mathrm{}\alpha <3\mathrm{}$ and $L^{-1}$ for $1<\mathrm{}\alpha <2,$ in contrast to $L^{-2}$ for normal diffusion. For $2<\mathrm{}\alpha <3,$ the ratio of the decay constants of the first and second eigenfunctions is less than 4 for large $L,$ so that the second eigenfunction is relatively more important in describing the system’s large time behavior than the corresponding eigenfunction for normal diffusion. For $1<\mathrm{}\alpha <2\mathrm{}$ the ratio of the decay constants may be greater or less than 4. The shapes of the eigenfunctions for the Lévy processes are obtained numerically and the strong similarity between the first eigenfunction and its normal diffusion counterpart for $2\lesssim \alpha <3\mathrm{}$ indicate that it would be difficult experimentally to distinguish such a Lévy process on a finite interval from a normal diffusive system by considering only the asymptotic shape of the probability distribution. For $\alpha \lesssim 2$ we observe significant differences between the first and second eigenfunctions and their normal diffusion counterparts. For moderately large intervals, the first eigenfunction is flatter with large boundary discontinuities while the second eigenfunction can differ from its normal diffusion counterpart in both its symmetry properties and number of nodes.