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

Abstract

The equation which describes a particle diffusing in a logarithmic potential arises in diverse physical problems such as momentum diffusion of atoms in optical traps, condensation processes, and denaturation of DNA molecules. A detailed study of the approach of such systems to equilibrium via a scaling analysis is carried out, revealing three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ \sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii) the scaling exponents and scaling functions corresponding to both regimes are selected by the initial condition; and (iii) this dependence on the initial condition manifests a "phase transition" from a regime in which the scaling solution depends on the initial condition to a regime in which it is independent of it. The selection mechanism which is found has many similarities to the marginal stability mechanism which has been widely studied in the context of fronts propagating into unstable states. The general scaling forms are presented and their practical and theoretical applications are discussed.

42 pages

References in corpus (9)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Sequence sensitivity of breathing dynamics in heteropolymer DNA
• Loop Dynamics in DNA Denaturation
• Solution of the Fokker-Planck equation with a logarithmic potential
• First-passage and first-exit times of a Bessel-like stochastic process
• Approach to equilibrium of diffusion in a logarithmic potential
• Structure factor and dynamics of the helix-coil transition
• Finite-size effects on the dynamics of the zero-range process
• Probing Non-Integer Dimensions

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• From the area under the Bessel excursion to anomalous diffusion of cold atoms
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• Anomalous statistics of laser-cooled atoms in dissipative optical lattices
• Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials
• Logarithmic oscillators: ideal Hamiltonian thermostats
• Brownian motion in time-dependent logarithmic potential: Exact results for dynamics and first-passage properties
• Heavy-tailed phase-space distributions beyond Boltzmann-Gibbs and equipartition: Statistics of confined cold atoms
• Current fluctuations in the Dyson Gas
• Entropy of Sharp Restart
• Deviations from generalized Equipartition in Confined, Laser Cooled Atoms
• A Unified View of Transport Equations
• Descending from infinity: Convergence of tailed distributions