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.

18 pages, 15 figures

References in corpus (2)

• Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving
• Fracture surfaces of heterogeneous materials: a 2D solvable model

Cited by in corpus (16)

• Review of Some Promising Fractional Physical Models
• Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion
• Thermodynamic Uncertainty Relation Bounds the Extent of Anomalous Diffusion
• Relaxation time distributions for an anomalously diffusing particle
• Emergent time crystal from a fractional Langevin equation with white and colored noise
• Observation of Super-ballistic Brownian Motion in Liquid
• Stretched exponential to power-law: crossover of relaxation in a kinetically constrained model
• Fractional Stochastic Loewner Evolution and Scaling Curves
• Nonergodic Brownian oscillator: High-frequency response
• Diffusion with preferential relocation in a confining potential
• Numerical stability of Grünwald-Letnikov method for time fractional delay differential equations
• 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
• Linear Fractionally Damped Oscillator
• Noise-induced resonant acceleration of a charge in an intermittent magnetic field: an exact solution for ergodic and non-ergodic fluctuations
• Nonergodic Brownian oscillator: Low-frequency response