General solution of Pauli master equation and applications to diffusive transport
Stochastic non-equilibrium processes can serve as a driving force for a Brownian motor. In this case, the simplest description of an induced directional motion is carried out under the approximation of small changes in potential energy of a particle compared to the thermal energy (the high-temperature or low-energy approximation). Within this approximation, characteristics of the stochastic process are included into the final analytical expressions for the average velocity of a Brownian motor only through lower correlation functions. The purpose of this article is to obtain these correlation functions for Markov stochastic processes of a general form.
The Pauli master equation has been considered, which describes the kinetics of an N states system in terms of given transition rate constants. The general solution of this equation, expressed in terms of the eigenvalues and eigenfunctions of the transition rate matrix, has been specified for the cases of dichotomous fluctuations (N = 2) and fluctuations of transitions from the ground state into two symmetric excited ones (N = 3). To study the influence of a type of a correlation function on the average velocity of a Brownian motor, a spatially harmonic controlled signal is considered, with which the average motor velocity is proportional to the second-order correlation function.
The introduction of the additional state allowed us to obtain a bell-shaped frequency dependence of the average motor velocity, with the maximum and the width of the bell easily controllable by the transition rate constants and with the diffusion coefficient which depends on temperature, a particle size, and medium viscosity. Dichotomous process, which is a particular case of the model considered, is characterized by the highest values of the average velocity. A three-level model is preferable in case of a narrower width of the bell-shaped frequency dependence is required.
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