Stochastic Analysis of Discrete Markovian Systems. I. Steady State Fluctuations of Scalar and Vectorial Quantities

Abstract

Subject of this paper is the stochastic analysis of the steady state fluctuations of scalar and vectorial quantities in discrete Markovian systems. The scalar variables are introduced as step functions assuming discrete values which describe the state of the system. The vectorial variables are assigned to the transitions between different states of the systems. Under the assumption that the transition times are short compared with the characteristic times of the system, these variables are defined by <5-pulse sequences. As fundamental stochastic equation a master equation is introduced. The fluctuation analysis of the scalar state variables is performed with the use of the classical master equation approach to fluctuations. For the stochastic analysis of the transitions we have developed a new approach. It makes possible the fluctuation analysis of vectorial processes (e. g. electric current) at nonequilibrium steady states. In this paper the theoretical foundations of this approach are given. Furthermore, it is explicitly shown that via rigorous balance equations the results for the steady state fluctuations of scalar variables as known from the classical master equation approach can be derived from the new formalism for vectorial quantities. IntroductionMany kinetic processes as chemical reactions or biological transport processes can be represented by Markovian processes which are continuous in time and discrete in state space (e.g. [1][2][3][4][5][6][7]). Ttie stochastic behaviour of these systems is usually described by master equations. The treatment of fluctuations around steady states, which may be equilibrium or nonequilibrium, can be performed by the well known master equation approach to fluctuations ([8, 3, 4]). More than twenty years ago in a classical paper a possible extension of the fluctuationdissipation (Nyquist) relation, relating the equilibrium fluctuations to the macroscopic response properties of the system, to nonequilibrium steady states was

Monte-Carlo-simulations of voltage fluctuations in biological membranes in the case of small numbers of transport units

“…A recently developed theory of voltage noise Solleder 1985, 1986;Solleder and Frehland 1986;Frehland 1988) …”