Generalized Master Equation for Space-Time Coupled Continuous Time Random Walk

The generalized master equation for the space-time coupled continuous time random walk is derived analytically, in which the space-time coupling is considered through the correlated function g(t) \sim t^\gamma, 0 \le \gamma < 2, and the probability density function \omega(t) of a particle's waiting time t follows a power law form for large t: \omega(t) \sim t^-(1+\alpha), 0 < \alpha < 1. The results indicate that the expressions of the generalized master equation are determined by the correlation exponent \gamma and the long-tailed index \alpha of the waiting time. Moreover, the diffusion results obtained from the generalized master equation are in accordance with the previous known results and the numerical simulation results.