Moment closure through spectral expansion in open stochastic systems

The derivation of dynamical laws for general observables (or moments) from the master equation for the probability distribution remains a challenging problem in statistical physics. Here, we present an alternative formulation of the general spectral expansion, which clarifies the connection between the relaxation dynamics of arbitrary moments and the intrinsic time scales of the system. Within this framework, we address the moment-closure problem in a way that streamline the conventional treatment of open systems. The effectiveness of the theory is illustrated by deriving analytical expressions for two representative cases: spectral amplification in stochastic resonance and the moment dynamics of a non-Gaussian system, namely the Bessel process with constant drift. We also identify a direct relationship between our theory and the Koopman operator approach. Finally, we apply our approach to the nonlinear and out-of-equilibrium mean-field description of interacting excitatory and inhibitory populations.