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) α c = 0.402 ± 0.002 marks a transition to a nonmonotonic underdamped phase, (ii) α R = 0.441 … marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) α χ 1 = 0.527 … and (iv) α χ 2 = 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.