Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

The dynamical phase diagram of the fractional Langevin equation is investigated for a harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) αc =0.402±0.002 marks a transition to a nonmonotonic underdamped phase, (ii) αR =0.441... marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) α χ1 =0.527... and (iv) α χ2 =0.707... mark transitions to a double-peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over and underdamped regimes, with or without resonances, show behaviors different from normal.