Distributive processes and combinatorial dynamics

Abstract

We describe a new family of Markov processes, a prototype for which is in the statistics of a test molecule undergoing “random” energy transfer in collisional complexes with heat-bath particles. Master equations for several versions of this process are constructed and solved exactly under purely statistical prescriptions of the mechanism and degrees of freedom available. Their eigenfunctions, arising through a natural factorization of the transition kernels, prove to be classical polynomials of Laguerre and Jacobi type; the relaxation times are given by simple terminating series in the degree-of-freedom parameters. Moreover, the spectral representations of such kernels prove to be Erdelyi-type bilinear expansion in the respective eigenfunctions, giving these little-known formulas a previously unsuspected physical interpretation. A remarkable property of the solutions is that they are both exact and parametrized over the whole range of behavior from effective “Brownian motion” at one extreme to virtually purely random processes at the other. Autocorrelation functions for equilibrium fluctuations in the same ensembles are also obtained and shown to be strictly exponential. Applications of such “distributive processes” are discussed with reference to both the physics of energy transfer and possible alternative realizations, e.g., in operations research. Some related mathematical topics, notably the role of fractional integral-operators in the master equation, are pointed out.