Theoretical and Mathematical Physics

, Volume 94, Issue 3, pp 345–357 | Cite as

Random walks in disordered systems with long-range transitions. Asymptotically exactly solvable models

  • F. S. Dzheparov
  • V. E. Shestopal


Random-jump models of transport in disordered systems are studied. They are described by the master equationP=−AξP, whereA is the generator of a spatially and temporally uniform random walk on a regular lattice, ξ is a diagonal operator, and ξ xy x δ xy , where {ξ x } are independent non-negative bounded random variables having the same distribution. A detailed analysis is made of the case when the transition rates are due to an interaction of multipole type and ξ x have several negative first moments (model of isotropic random jumps with long-range transport). Methods are developed for constructing asymptotic expansions of the propagator for small values of the Laplace parameter and at large times. An expansion is also obtained by means of a functional integral. The influence of both the long-range interaction and the disorder of the medium on the establishment of the long-time asymptotic behavior is considered. A method of investigating systems with forced drift along a certain direction is suggested. Methods of transforming asymptotically exactly solvable problems and connections with other known systems and realistic models are discussed. An estimate is obtained of thel1 norm of the resolvent of a Markov process with countable set of states andl1-bounded generator.


Random Walk Asymptotic Expansion Markov Process Transition Rate Solvable Model 
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Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • F. S. Dzheparov
  • V. E. Shestopal

There are no affiliations available

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