Modeling the Jump-like Diffusion Motion of a Brownian Motor by a Game- Theory Approach: Deterministic and Stochastic Models

Methods of paradoxical games are used to construct a stochastic hopping model of Brownian ratchets which extends the well-known analogous deterministic model. The dependencies of the average displacements of a Brownian particle in a stochastic ratchet system on a discrete time parameter are calculated, as well as the dependencies of the average ratchet velocity on the average lifetimes of the states of the governing dichotomous process. The results obtained are compared with both the results of modeling a similar deterministic model and the results of a known analytic description. While for the hopping analogue of the deterministic on-off ratchet, the time dependence of the displacement contains periodically repeated hopping changes when the potential is switched on and plateau of the diffusion stage of the motion when it is switched off, the stochastic dependencies, that are of an averaged character, are monotonous and do not contain jumps. It is shown that, with other things being equal, the difference in the results for the hopping ratchet model driven by the stochastic and deterministic dichotomous process of switching the potential profiles (game selection) is more pronounced at short lifetimes of the dichotomous states and vanishes with their increase.