Master equation description of external poisson white noise in finite systems

Abstract

We consider finite systems with random control parameters. A theory for a unified description of internal fluctuations and external noise is presented. Internal fluctuations are modeled by a one-step Markovian master equation. External noise is introduced by random parameters in the master equation. It is modeled by a Poisson white noise. The unified description of fluctuations features a Markovian master equation with nonvanishing transition probabilities for all steps in the state space. Alternative formulations are given in terms of the generating function, Poisson representation and the equations for the factorial moments. An expansion around the thermodynamic limit is considered. The theory permits the calculation of finite-size effects. It predicts the existence of a coupling of the two types of fluctuations leading to “crossed-fluctuation” contributions. Two examples are considered: (i) a Poisson counting process with fluctuating parameter, (ii) a creation and annihilation process with source terms and fluctuations in each of the creation, annihilation, and source parameters. In the second example a complete analysis is given for the stationary distribution and associated moments for a finite system and also in the thermodynamic limit. The different role of the fluctuations of the three parameters is discussed. Explicit “crossed-fluctuations” contributions are found. The effect of the system size on the type of transitions induced by external noise in the thermodynamic limit is discussed.