JUMP STATISTICS, SOJOURN TIMES, FLUCTUATION DYNAMICS AND ERGODIC BEHAVIOR FOR MARKOV-PROCESSES IN CONTINUOUS-TIME WITH A FINITE NUMBER OF STATES

A general approach is introduced for describing the time evolution of a Markov process in continuous time and with a finite number of states . The total number of transition events from one state to other states and of the total sojourn times of the systemin the different states a re used as additional state variables. The large time behavior of thes e two types of stochastic state variables is investigated analytically by using a stochastic Liouville equation. It is shown that the cumula nts of first and second order of the state variables increase asymptot ically linearly in time. A set of scaled sojourn times is introduced w hich in the limit of large times have a Gaussian behavior. For long ti mes, the total average sojourn times are proportional to the stationar y state probability of the process and, even though the relative fluct uations decrease to zero, the relative cross correlation functions ten d towards finite values. The results are used for investigating the co nnections with Van Kampen's approach for investigating the ergodic pro perties of Markov processes. The theory may be applied for studying fl uctuation dynamics in stochastic reaction diffusion systems and for co mputing effective rates and transport coefficients for non-equilibrium processes in systems with dynamical disorder.