Linear response characteristics of time-dependent time fractional Fokker-Planck equation systems

The time fractional Fokker-Planck equation approach is an important tool for modeling subdiffusion. When the external field is time modulated, two types of time-dependent time fractional Fokker-Planck equations have been proposed, both reduced to the same time-dependent time fractional Fokker-Planck equation when the external field is time uncorrelated. The first type is strictly deduced as the continuous limit of the continuous time random walk with time modulated Boltzmann jumping weight, while the second type is derived by ideally assuming that the jump probabilities can be evaluated at the start of the waiting time prior to jumping. For the first time we obtain the linear response characteristic for the first type of the time fractional Fokker-Planck equation systems, and for a comparison we revisit the corresponding result for the second type of the time fractional Fokker-Planck equation systems, and the similarity and difference between them is discussed with an application example. The investigation not only helps in understanding the competition between subdiffusion and time-dependent modulation, but also has significance in accessing the spectral properties of spontaneous fluctuation and the linear dynamic susceptibility of external perturbation in subdiffusive processes.