Anomalous diffusion: Exact solution of the generalized Langevin equation for harmonically bounded particle

Viñales, A. D.; Despósito, M. A.

doi:10.1103/physreve.73.016111

Cited by 92 publications

(93 citation statements)

References 19 publications

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“…Here we explore a similar scenario for the fractional oscillator, and find rich types of physical behaviors. It is known [12] that in the long time limit all solutions x (i.e. any 0 < α, 0 < γ and 0 < ω ) decay monotonically, some what like the over-damped behavior of the usual oscillator, however now the decay is of power law type.…”

Section: Introductionmentioning

confidence: 99%

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

2008

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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.

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Section: Introductionmentioning

confidence: 99%

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

2008

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“…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…”

Section: Thermostat Described By An 1d Glementioning

confidence: 99%

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

Batra

Swaminathan

Ayyaswamy

et al. 2011

The Journal of Chemical Physics

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A direct numerical simulation (DNS) procedure is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian stationary fluid medium with the generalized Langevin approach. We consider both the Markovian (white noise) and non-Markovian (Ornstein-Uhlenbeck noise and Mittag-Leffler noise) processes. Initial locations of the particle are at various distances from the bounding wall to delineate wall effects. At thermal equilibrium, the numerical results are validated by comparing the calculated translational and rotational temperatures of the particle with those obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation functions and mean square displacements with analytical results. Numerical predictions of wall interactions with the particle in terms of mean square displacements are compared with analytical results. In the non-Markovian Langevin approach, an appropriate choice of colored noise is required to satisfy the power-law decay in the velocity autocorrelation function at long times. The results obtained by using non-Markovian Mittag-Leffler noise simultaneously satisfy the equipartition theorem and the long-time behavior of the hydrodynamic correlations for a range of memory correlation times. The Ornstein-Uhlenbeck process does not provide the appropriate hydrodynamic correlations. Comparing our DNS results to the solution of an one-dimensional generalized Langevin equation, it is observed that where the thermostat adheres to the equipartition theorem, the characteristic memory time in the noise is consistent with the inherent time scale of the memory kernel. The performance of the thermostat with respect to equilibrium and dynamic properties for various noise schemes is discussed.

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“…(12) and (13), it follows G(0) = 0 and g(0) = I. Furthermore, similar to the one-dimensional case [30,31], it can be shown that the relaxation function g(t) is identical to the long-time behavior of the normalized velocity autocorrelation function (see the appendix for the detailed derivation):…”

Section: Generalized Langevin Equation With Two Coupled Coordinatesmentioning

confidence: 72%

“…(40)], which features subdiffusive dynamics [31]. Quantities in the WN coordinate, in contrast, exhibit an exponential equilibrium rate [as seen in Eq.…”

Section: Particle Bounded By Harmonic Potentialmentioning

confidence: 99%

Langevin dynamics of correlated subdiffusion and normal diffusion

Wang

Zhao²,

Yan³

2012

Phys. Rev. E

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We analyze the dissipative dynamics of a particle governed by a two-dimensional generalized Langevin equation with coupled fractional Gaussian noise and white noise in its respective coordinates, assuming the lowest-order coupling form. Two situations are studied: In the first the particle is free from external force and in the second the particle is subject to a two-dimensional harmonic potential. We derive the general expressions for the mean values, variances, and velocity autocorrelation function and evaluate their temporal evolutions via the numerical Laplace inversion technique. Through the analytical results of the short-time and long-time behaviors, we also explicitly elucidate the effects of fluctuation correlation coupling and interoscillator coupling on the dynamic behaviors of the particle. It is shown that in both situations the couplings do not affect the short-time behavior of self-diffusions in each coordinate, and the subdiffusive and normal diffusive features of these processes resemble those in a one-dimensional system with fractional Gaussian noise and white noise, respectively. However, over a long time period, the fluctuation correlation extends the characteristic time scales for the self-diffusions of a free particle; while only the interoscillator coupling induces a retardation of the relaxation processes of a bounded particle toward equilibrium. Moreover, both couplings generate a cross diffusion, whose long-time approximation has two possible forms, the selection of which depends on the relevant time scales of self-diffusions in each coordinate.

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Anomalous diffusion: Exact solution of the generalized Langevin equation for harmonically bounded particle

Cited by 92 publications

References 19 publications

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

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

Langevin dynamics of correlated subdiffusion and normal diffusion

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