Modeling bed-load transport by the master-equation approach

The stochastic analysis of sediment and bed-lend movements, initiated by Hans A. Einstein, represents a landmark; its significance and utility have been greatly enhanced by the successful development of tracer techniques. This expositional paper reveals that a well-developed stochastic algorithm capable of formulating the master equations of hydrological phenomena yields results sufficiently general as to reduce to those of the special cases previously reported. In the present paper the location of a particle in the discretized space of interest is considered to be a time-dependent random variable. The transition of the system from one state, i.e., particle location, to another is characterized by a set of intensity functions The master equations have been established through stochastic population balances of particles in the various states. The analytical solution, derived by the method of eigenvalues and eignevectors, is analogous to the continuous-state dispersion equation of Einstein. Numerical calculations of the evolutions of particle dispersion profiles demonstrate that these two approaches yield comparable results.