Dynamical transition for a particle in a squared Gaussian potential

Abstract

We study the problem of a Brownian particle diffusing in finite dimensions in a potential given by $\psi ={\varphi }^2/2$ where $\varphi$ is Gaussian random field. Exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power-law fashion at the dynamical transition temperature. Our results are confronted with numerical simulations where the Gaussian field is constructed, in a standard way, as a sum over random Fourier modes. We show that when the number of Fourier modes is finite the low temperature diffusion constant becomes non-zero and has an Arrhenius form. Thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition. In addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model.

18 pages, 4 figures .eps, JPA style

References in corpus (24)

• Models of traps and glass phenomenology
• Replica Field Theory for Deterministic Models (II): A Non-Random Spin Glass with Glassy Behavior
• Aging on Parisi's tree
• Geometrical Aspects of Aging and Rejuvenation in the Ising Spin Glass: A Numerical Study
• Replica Field Theory for Deterministic Models: Binary Sequences with Low Autocorrelation
• Separation of time and length scales in spin-glasses: temperature as a microscope
• Self Induced Quenched Disorder: A Model for the Glass Transition
• Sub-diffusion and localization in the one dimensional trap model
• Large times off-equilibrium dynamics of a particle in a random potential
• On Mean Field Glassy Dynamics out of Equilibrium
• Does the potential energy landscape of a supercooled liquid resemble a collection of traps?
• Anomalous diffusion, Localization, Aging and Sub-aging effects in trap models at very low temperature
• Surfing on a critical line: Rejuvenation without chaos, Memory without a hierarchical phase space
• Bouchaud's model exhibits two different aging regimes in dimension one
• Localization properties of the anomalous diffusion phase $x{t}^{\mu }$ in the directed trap model and in the Sinai diffusion with bias
• Non-linear Response of the trap model in the aging regime : Exact results in the strong disorder limit
• Linear and non linear response in the aging regime of the 1D trap model
• Freezing of dynamical exponents in low dimensional random media
• Continuum Derrida Approach to Drift and Diffusivity in Random Media
• Microscopic realizations of the Trap Model
• Perturbation theory for the effective diffusion constant in a medium of random scatterer
• On a non-linear Fluctuation Theorem for the aging dynamics of disordered trap models
• Effective diffusion constant in a two dimensional medium of charged point scatterers
• Effective drift and diffusivity in non-Gaussian random gradient flows

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