The Critical Exponent of the Fractional Langevin Equation is $\alpha_c\approx 0.402$

TL;DR

This paper identifies critical exponents in the fractional Langevin equation that mark dynamical phase transitions, including a transition to a non-monotonic phase at approximately 0.402 and a resonance phase at about 0.441, revealing complex behaviors in fractional systems.

Contribution

The study determines specific critical exponents in the fractional Langevin equation that delineate different dynamical phases, expanding understanding of fractional stochastic systems.

Findings

Critical exponent for transition to non-monotonic phase: 0.402

Critical exponent for resonance transition: 0.441

Fractional oscillator behaviors differ significantly from classical models.

Abstract

We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent $\alpha_c= 0.402\pm 0.002$ marks a transition to a non-monotonic under-damped phase. The critical exponent $\alpha_{R}=0.441...$ marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the under-damped, the over-damped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.