DISTRIBUTIVE PROCESSES IN DISCRETE-SYSTEMS

The concept of distributive process, recently developed as a description of systems which evolve through 'random' energy-transfer in binary collision complexes4) is here extended to the case of energy quanta exchanged similarly between discrete, degenerate internal energy-levels. A corresponding exact solution of the Master Equation is forthcoming, in which the eigenvectors are now orthogonal polynomials of the discrete variable, viz. the Meixner and Hahn types. The relaxation times of the discrete models prove to be identical with those of their continuous counterparts and the autocorrelation functions for equilibrium fluctuations are likewise of strictly exponential type. All results tend naturally to their continuous analogues as the quantity (hv/kBT) for the heat-bath tends to zero. A number of aspects of mathematical interest are pointed out. For example, when considering the spectral representation of the transition matrices for the discrete distributive processes we arrive inter alia at previously unknown Erdelyi-type bilinear expansion formulae for the Meixner and Hahn systems and thence a stochastic interpretation of these. Another point of interest is the occurrence of fractional sum and difference-operators, this giving a rare example of the fractional calculus in statistical physics. Apart from furnishing several exactly soluble Master equations on an infinite, discrete state-space, the 'distributive' transition probabilities would seem to be the first known examples, as yet, of transition probabilities which allow the consistent representation of multiple degrees of freedom in the transfer of vibrational energy between polyatomic molecules. © 1979.