Resonance behavior for a generalized Mittag-Leffler fractional Langevin equation with hydrodynamic interactions

The phenomenological model for the heavy tracers in viscoelastic media modeled by a generalized Mittag-Leffler fractional Langevin equation with the generalized Stokes force, the Basset force, the Hookean force, and the thermal force has been revisited. Under the fluctuation-dissipation relation, the generalized Stokes force describes the viscoelastic media by a Mittag-Leffler (ML) memory kernel. Furthermore, based on the background of ML function, the generalized Mittag-Leffler fractional derivative is introduced. Moreover, the exact expression of stationary first moment and the expression of spectral amplification (SPA) of a tracer model have been deserved by the generalized form of Shapiro-Loginov formula. The generalized stochastic resonance (GSR) phenomena has been systematically studied. Moreover, the GSR, reverse stochastic resonance (SR) phenomenon, bona fide SR, stochastic multi-resonance (SMR) phenomena, increasing multi-resonance and decreasing multi-resonance have been found. Especially, the periodic resonance phenomenon could be induced by the generalized Mittag-Leffler (GML) noise, which has been few observed in the previous literatures.