Aging and diffusion in low dimensional environments

Abstract

We study out of equilibrium dynamics and aging for a particle diffusing in one dimensional environments, such as the random force Sinai model, as a toy model for low dimensional systems. We study fluctuations of two times $(t_w,t)$ quantities from the probability distribution $Q(z,t,t_w)$ of the relative displacement $z=x(t)-x(t_w)$ in the limit of large waiting time $t_w\to \infty$ using numerical and analytical techniques. We find three generic large time regimes: (i) a quasi-equilibrium regime (finite $\tau =t-t_w$) where $Q(z,\tau )$ satisfies a general FDT equation (ii) an asymptotic diffusion regime for large time separation where $Q(z)dz\sim \bar{Q}[L(t)/L(t_w)]dz/L(t)$ (iii) an intermediate ``aging'' regime for intermediate time separation ($h(t)/h(t_w)$ finite), with $Q(z,t,t^{\prime })=f(z,h(t)/h(t^{\prime }))$. In the unbiased Sinai model we find numerical evidence for regime (i) and (ii), and for (iii) with $\overline{Q(z,t,t^{\prime })}=Q_0(z)f(h(t)/h(t^{\prime }))$ and $h(t)\sim \ln t$. Since $h(t)\sim L(t)$ in Sinai's model there is a singularity in the diffusion regime to allow for regime (iii). A directed model, related to the biased Sinai model is solved and shows (ii) and (iii) with strong non self-averaging properties. Similarities and differences with mean field results are discussed. A general approach using scaling of next highest encountered barriers is proposed to predict aging properties, $h(t)$ and $f(x)$ in landscapes with fast growing barriers. We introduce a new exactly solvable model, with barriers and wells, which shows clearly diffusion and aging regimes with a rich variety of functions $h(t)$.

43 pages, 19 .eps figures, RevTeX

References in corpus (1)

• Dynamics of particles and manifolds in a quenched random force field

Cited by in corpus (28)

• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Random Walkers in 1-D Random Environments: Exact Renormalization Group Analysis
• Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
• Aging Continuous Time Random Walks
• Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Real-time non-equilibrium dynamics of quantum glassy systems
• Anomalous Diffusion in Aperiodic Environments
• Correlation function and generalized master equation of arbitrary age
• Anomalous diffusion, Localization, Aging and Sub-aging effects in trap models at very low temperature
• Localization of thermal packets and metastable states in Sinai model
• Aging Dynamics of an Elastic String Diffusing in a Disordered Media
• Reaction Diffusion Models in One Dimension with Disorder
• Langevin simulations of the out-of-equilibrium dynamics of the vortex glass in high-temperature superconductors
• Eigenvalue method to compute the largest relaxation time of disordered systems
• Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems
• Freezing of dynamical exponents in low dimensional random media
• Aging dynamics and density relaxation in granular compaction
• Energy dynamics in the Sinai model
• Statistical and dynamical properties of the discrete Sinai model at finite times
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Dynamic fluctuations of elastic lines in random environments
• Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass
• Volterra-Prabhakar derivative of distributed order and some applications
• A minimal model for slow dynamics: Compaction of granular media under vibration or shear
• Dynamics of Ising models near zero temperature : Real Space Renormalization Approach
• Temperature shifts in the Sinai model: static and dynamical effects
• Local equilibrium properties of ultraslow diffusion in the Sinai model