Dynamics of switching processes: general results and applications in intermittent active motion

Systems switching between different dynamical phases is a ubiquitous phenomenon. The general understanding of such a process is limited. To this end, we present a general expression that captures fluctuations of a system exhibiting a switching mechanism. Specifically, we obtain an exact expression of the Laplace-transformed characteristic function of the particle's position. Then, the characteristic function is used to compute the effective diffusion coefficient of a system performing intermittent dynamics. Furthermore, we employ two examples: (1) generalized run-and-tumble active particle, and (2) an active particle switching its dynamics between generalized active run-and-tumble motion and passive Brownian motion. In each case, explicit computations of the spatial cumulants are presented. Our findings reveal that the particle's position probability density function exhibit rich behaviours due to intermittent activity. Numerical simulations confirm our findings.