Revisiting Lévy flights on bounded domains: a Fock space approach

Araújo, H. A.Lukin, M. O.da Luz, M. G. E.Viswanathan, G. M.Santos, F. A. N.Raposo, E. P.  Journal of Statistical Mechanics: Theory and Experiment : 083202 (2020)  DIGITAL CSIC

The statistical description of a one-dimensional superdiffusive Lévy flier restricted to a finite domain is well known to be technically involving. For example, in this type of process the probability distribution P(x, t) and survival probability S(t) cannot be obtained from the method of images. Other methods, such as the fractional derivative approach, also find technical difficulties due to the long jumps combined with the presence of absorbing boundaries. Here we revisit this problem through a different point of view. We map the corresponding master equation to a Schrödinger-like equation and then describe the Lévy flier evolution in a Fock space. The system states are assigned to the available positions in the discrete space. The Hamiltonian-like matrix is calculated for any Lévy index α ∈ (0, 2]. For the system sizes studied here the computation of its eigenvalues and eigenvectors are performed using a symbolic computing software. This method allows to build the time evolution operator, the distribution P(x, t) and the survival probability S(t). We compare our results for P(x, t) with direct Monte Carlo simulations and find a good agreement for all α ∈ (0, 2]. Similarly, our results for S(t) nicely agree with the numerical simulations for any time t, including the short-term behavior. In the long-term asymptotic limit we identify the crossover between the power-law and exponential behaviors, which emerge respectively when only one or both boundaries are reached by the Lévy flier. Comparisons with some exact expressions for Lévy flights in the continuous space limit also display good agreement. We conclude our analysis by discussing the possibility of extending the present framework to general bounded random walks and flights.