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

TL;DR

This paper explores the fractional Langevin equation's phase diagram, identifying critical exponents that mark transitions between different dynamical behaviors such as damping, resonance, and double peak loss phases.

Contribution

It introduces four new critical exponents that delineate phase transitions in the fractional Langevin system under harmonic confinement and external driving.

Findings

Identified four critical exponents for phase transitions.

Mapped phase diagrams showing unique over-damped and under-damped behaviors.

Revealed a cage effect causing elastic-like friction in the medium.

Abstract

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.