Abstract
We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.
Similar content being viewed by others
References
Balakrishnan V 1980 inProc. meeting on spin-glass alloys, University of Roorkee, Roorkee, India (to be published)
Balakrishnan V and Venkataraman G 1981Pramana 16 109
Cox D R 1967Renewal theory (London: Methuen)
Feller W 1966An introduction to probability theory and its applications (New York: Wiley) Vols 1 & 2
Katsura S, Morita T, Inawashiro S, Horiguchi T and Abe Y 1971J. Math. Phys. 12 892
Kehr K W and Haus J W 1978Physica A 93 412
Mannari I and Kawabe T 1970Prog. Theor. Phys. (Jpn.) 44 359
Montroll E W and Weiss G H 1965J. Math. Phys. 6 167
Saleh B 1978Photoelectron statistics (New York: Springer-Verlag)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Balakrishnan, V. Solvable models of temporally correlated random walk on a lattice. Pramana - J. Phys. 17, 55–68 (1981). https://doi.org/10.1007/BF02872037
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02872037