Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

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) αc = 0.402 ± 0.002 marks a transition to a non-monotonic under-damped phase, (ii) αR = 0.441... marks a transition to a resonance phase when an external oscillating field drives the system, (iii) αχ 1 = 0.527... and (iv) αχ 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.

Generalized Langevin dynamics of a nanoparticle using a finite element approach: Thermostating with correlated noise

“…In the literature, pure power-law correlation functions have been employed to investigate the anomalous diffusion behavior of the particle that is related to the long time tail correlations. [35][36][37][38] Recently, Viñales and Despósito 39 have considered a Mittag-Leffler noise given by…”

Control of Stochastic Resonance in Overdamped Fractional Langevin Equation

“…In normal diffusion, the relation of mean-square displacement and time  indicate superdiffusion and subdiffusion [6]. The fractional Langevin equation (FLE) is mainly used to model such anomalous diffusion, that replacing the usual friction term by a power-law-type memory in Generalized Langevin equation (GLE) [7,8]. In recent decades, SR phenomenon is investigated in some systems described by FLE.…”

Velocity and displacement correlation functions for fractional generalized Langevin equations

“…It has been mentioned that in the case of internal white Gaussian noise, the GLE (1.1) corresponds to the classical Langevin equation. In many papers the GLE with an internal noise with a power law correlation functions of form C(t) = C λ t −λ Γ(1−λ) , where Γ(·) is the Euler-gamma function, C λ is a proportionality coefficient independent of time and which can depends on the exponent λ (0 < λ < 1 or 1 < λ < 2), has been used for modeling anomalous diffusion [29,2,56,53,47,10,30,31].…”

“…Several approaches to anomalous diffusion exist. Starting from the generalized Langevin equation (GLE) [1,26,29,39,20,2], introduced by Kubo [27], the fractional diffusion equation [34] (see also Refs. [41] and [52]), fractional Fokker-Planck equation [34,35,37], generalized Chapman-Kolmogorov equation [33], fractional generalized Langevin equation (FGLE) [12,28,15].…”