Diffusion process of a random particle in a one-dimensional finite interval in the Fock space approach

Abstract

In this work we study the statistical properties of a random walker (RW) in a one-dimensional discrete lattice in different scenarios. In the simplest case, the steps of the RW have fixed length and equal probabilities to hop to the right or to the left, in a lattice with absorbing borders. We apply the Fock space approach to obtain a Schrödinger-like equation from a master equation that describes the transport mechanisms of the random particle and that allows us to write a quasi-Hamiltonian and compute the probability of the RW to be at any position in the lattice after some arbitrary time. The plot of the probability versus position shows, in this simplest case, an expected Gaussian distribution when the borders are still untouched, with a change occurring after the borders are reached. In addition, we compute some quantities like the mean value and standard deviation of the RW position. In a second scenario, we study the RW behavior under a power-law distribution of step lengths, with the power-law exponent controlling the diffusive properties of the RW, from the superdiffusive regime for this variable exponent in the interval (1,3) to the diffusive (normal, Gaussian-like) regime with values greater than 3. In a similar manner, we compute the probability distribution of the RW as a function of the position and time, and the mean value, standard deviation, and survival rate as a function of time. We show that after a few steps the behavior of the survival probability is proportional to the inverse of the square root of time, that obeys the Sparre-Andersen theorem, when the RW can reach only one border, and its long-term asymptotic behavior is given by an exponential decay behavior, when both absorbing borders are touched. Finally, we study the behavior of the RW with Lévy alfa-stable distribution of step lengths. The behavior of the probability distribution of the RW as a function of the position in the lattice is shown, as well as the survival rate and other relevant quantities. Overall, the statistical properties of the Lévy and power-law RWs are found to be similar, as expected. Indeed, the power-law distribution corresponds to the asymptotic limit of the Lévy distribution. Our findings are generally found to be in good agreement with previous results for these types of RW particles obtained under other approaches.