A class of solvable reaction–diffusion processes on a Cayley tree

Abstract

Considering the most general one-species reaction-diffusion processes on a Cayley tree, it has been shown that there exist two integrable models. In the first model, the reactions are the various creation processes, i.e.and in the second model, only the diffusion process •• → •• exists. For the first model, the probabilities P l (m; t), of finding m particles on l-th shell of Cayley tree, have been found exactly, and for the second model, the functions P l (1; t) have been calculated. It has been shown that these are the only integrable models, if one restricts himself to L + 1-shell probabilities P (m0,

Extracting the parton distribution functions evolution equations using the stochastic modeling in the non-equilibrium statistical mechanics

“…The evolution equations obtained from stochastic modeling (the equation ( 17)) are the same as the DGLAP evolution equations, but obtained much easier. It should be noted that according to the evolution equation ( 17), which is derived from (1) the non-equilibrium statistical mechanics, and (2) the results of reference [14] in relation to the boundary conditions governing this evolution equation, we hope that in the near future, by solving methods of the master equation (for example by the coordinate Bethe ansatz method, see the reference [17], and the recursive method, see the reference [21]), the new solutions for the P DF are presented.…”

The eigenpairs of a Sylvester–Kac type matrix associated with a simple model for one-dimensional deposition and evaporation

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