Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors

Details Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors Fractional Langevin Equation: Over-Damped, Under-Damped and Critical Behaviors

2008-02-26

arxiv.org/abs/0802.3777v1

Physics

Condensed Matter

Statistical Mechanics

18 pages, 15 figures

Scientific paper

10.1103/PhysRevE.78.031112

The dynamical phase diagram of the fractional Langevin equation is investigated for 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) $\alpha_c=0.402\pm 0.002$ marks a transition to a non-monotonic under-damped phase, (ii) $\alpha_R=0.441...$ marks a transition to a resonance phase when an external oscillating field drives the system, (iii) $\alpha_{\chi_1}=0.527...$ and (iv) $\alpha_{\chi_2}=0.707...$ marks transition 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-damped, under-damped regimes, motion and resonances, show behaviors different from normal.

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