Basics of Counting Statistics

In this paper, we briefly review the scheme of counting statistics, in which a probability of the number of monitored or target transitions in a Markov jump process is evaluated. It is generally easy to construct a master equation for the Markov jump process, and the counting statistics enables us to straightforwardly obtain basic equations of the counting statistics from the master equation; the basic equation is used to calculate the cumulant generating function of the probability of the number of target transitions. For stationary cases, the probability is evaluated from the eigenvalue analysis. As for the nonstationary cases, we review a numerical integration scheme to calculate the statistics of the number of transitions. key words: stochasticity, numerical method, generating function approach, path-integral approach