Fractional Langevin Equation: Over-Damped, Under-Damped 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) ${\alpha }_c=0.402\pm 0.002$ marks a transition to a non-monotonic under-damped phase, (ii) ${\alpha }_R=\mathrm{0.441...}$ marks a transition to a resonance phase when an external oscillating field drives the system, (iii) ${\alpha }_{{\chi }_1}=\mathrm{0.527...}$ and (iv) ${\alpha }_{{\chi }_2}=\mathrm{0.707...}$ 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.

arXiv comment18 pages, 15 figures

References in corpus (8)

• Fractional Langevin equation
• Aging properties of an anomalously diffusing particle
• The Critical Exponent of the Fractional Langevin Equation is ${\alpha }_c\approx 0.402$
• Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
• Anomalous escape governed by thermal 1/f noise
• Linear response to perturbation of non-exponential renewal processes
• Chaotic and pseudochaotic attractors of perturbed fractional oscillator
• Fracture surfaces of heterogeneous materials: a 2D solvable model

Cited by in corpus (28)

• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• Review of Some Promising Fractional Physical Models
• Foundation of Fractional Langevin Equation: Harmonization of a Many Body Problem
• Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement
• Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion
• Subdiffusive behavior in a trapping potential: mean square displacement and velocity autocorrelation function
• Physical descriptions of the bacterial nucleoid at large scales, and their biological implications
• Transient ageing in fractional Brownian and Langevin equation motion
• Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
• Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion
• Subdiffusive Brownian ratchets rocked by a periodic force
• Viscoelastic subdiffusion in a random Gaussian environment
• Relaxation time distributions for an anomalously diffusing particle
• Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions
• Resonant behavior of the generalized Langevin system with tempered Mittag-Leffler memory kernel
• Fractional electron transfer kinetics and a quantum breaking of ergodicity
• Emergent time crystal from a fractional Langevin equation with white and colored noise
• Fractional Stochastic Loewner Evolution and Scaling Curves
• Nonergodic Brownian oscillator: High-frequency response
• Diffusion with preferential relocation in a confining potential
• Observation of Super-ballistic Brownian Motion in Liquid
• Stretched exponential to power-law: crossover of relaxation in a kinetically constrained model
• Numerical stability of Grünwald-Letnikov method for time fractional delay differential equations
• Linear Fractionally Damped Oscillator
• Nonergodic Brownian oscillator: Low-frequency response
• Noise-induced resonant acceleration of a charge in an intermittent magnetic field: an exact solution for ergodic and non-ergodic fluctuations
• Beyond the Markovian limit: Exact solutions for active motion in a power-law viscoelastic bath
• Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion