Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium

Hirschberg, Ori; Mukamel, David; Schütz, Gunter M.

doi:10.1088/1742-5468/2012/02/p02001

Cited by 28 publications

(30 citation statements)

References 61 publications

(266 reference statements)

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“…Formally, Eq. (7) with p interpreted as the position is related to the diffusion of a particle in a logarithmic potential [25][26][27], which can be used to model a range of physical systems [28][29][30]. In these cases, however, it is not clear what the physical equivalent of the potential U (x) is.…”

Section: Semiclassical Description Of Sisyphus Coolingmentioning

confidence: 99%

Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs: Confined laser-cooled atoms in a nonthermal state

et al. 2016

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The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for non-thermal systems such as cold atoms in optical lattices, where the heat bath is replaced by the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The non-equilibrium nature of the steady state is confounded by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density.

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Section: Semiclassical Description Of Sisyphus Coolingmentioning

confidence: 99%

Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs: Confined laser-cooled atoms in a nonthermal state

et al. 2016

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“…In Refs. [22,23], it was observed that these two classes of initial conditions lead to a qualitatively different time evolution of the velocity probability density. Our scaling Green-Kubo relation shows that for 2 < α < 3, the initial condition also persists in the diffusive behavior of the system in the form of different expressions for the anomalous diffusion coefficient.…”

Section: Fig 2 (Color Online)mentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function hvðt þ τÞvðtÞi asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, hx 2 ðtÞi ≃ 2D ν t ν . We show how both the anomalous diffusion coefficient D ν and the exponent ν can be extracted from this scaling form. Our scaling GreenKubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D ν is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

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“…For power-law initial conditions, a different behavior is expected; see Refs. [35,36]. For most potentials U (x) (e.g.…”

Section: Fokker-planck Equation For the Logarithmic Potentialmentioning

confidence: 99%

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

et al. 2012

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We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and we support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the super-aging behavior.

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Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium

Cited by 28 publications

References 61 publications

Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs: Confined laser-cooled atoms in a nonthermal state

Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs: Confined laser-cooled atoms in a nonthermal state

Scaling Green-Kubo Relation and Application to Three Aging Systems

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

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