A Random Walk to a Non-Ergodic Equilibrium Concept
arXiv:cond-mat/0506338 · doi:10.1103/PhysRevE.73.016125
Abstract
Random walk models, such as the trap model, continuous time random walks, and comb models exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is: what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this manuscript a non-ergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point, approaches U or W shaped distributions related to the arcsin law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the non-ergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. Relation of our work with single molecule experiments is briefly discussed.
14 pages, 6 figures
References in corpus (17)
- Models of traps and glass phenomenology
- Fractional calculus and continuous-time finance II: the waiting-time distribution
- Statistical Aging and Non Ergodicity in the Fluorescence of Single Nanocrystals
- Stochastic Ergodicity Breaking: a Random Walk Approach
- The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times
- Non-ergodic Intensity Correlation Functions for Blinking Nano Crystals
- Exact Solution for the Influence of Spectral Diffusion on Single-Molecule Photon-Statistics
- Aging Continuous Time Random Walks
- The Local and the Occupation Time of a Particle Diffusing in a Random Medium
- Power-law tail distributions and nonergodicity
- Aging in the random energy model
- Large deviations and nontrivial exponents in coarsening systems
- Breakdown of the Onsager principle as a sign of aging
- Residence Time Distribution for a Class of Gaussian Markov Processes
- Sign-time distributions for interface growth
- Stationary definition of persistence for finite temperature phase ordering
- Critical dimensions of the diffusion equation
Cited by in corpus (28)
- Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
- Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary
- Simplicity of Completion Time Distributions for Common Complex Biochemical Processes
- Weakly non-ergodic Statistical Physics
- Weakly nonergodic dynamics in the Gross--Pitaevskii lattice
- Photon Counting Statistics For Blinking CdSe-ZnS Quantum Dots: A Levy Walk Process
- A fractional Feynman-Kac equation for weak ergodicity breaking
- Continuous time random walk for open systems: Fluctuation theorems and counting statistics
- Residence Time Statistics for Normal and Fractional Diffusion in a Force Field
- Dynamical glass in weakly non-integrable Klein-Gordon chains
- Universality of market superstatistics
- Specificity and Completion Time Distributions of Biochemical Processes
- Linear Response and Fluctuation Dissipation Theorem for non-Poissonian Renewal Processes
- Ergodic properties of heterogeneous diffusion processes in a potential well
- Different anomalous diffusion regimes measured in the dynamics of tracer particles in actin networks
- Anomalous Infiltration
- A Unified Approach to Gated Reactions on Networks
- Nonergodicity of -dimensional generalized Lévy walks and their relation to other space-time coupled models
- Weak ergodicity breaking induced by global memory effects
- Fluctuations around equilibrium laws in ergodic continuous-time random walks
- A continuous-time random walk extension of the Gillis model
- Experimental verification of Arcsine laws in mesoscopic non-equilibrium and active systems
- Ergodic transitions in continuous-time random walks
- Non-Markovian Stochastic Liouville equation and its Markovian representation. Extensions of the continuous time random walk approach
- Lévy Flights and Leaky Boxes: Anomalous Diffusion of Cosmic Rays
- Non-normalizable quasi-equilibrium states under fractional dynamics
- Boundary conditions at a thin membrane that generate non--Markovian normal diffusion
- Super-extreme event's influence on a Weierstrass-Mandelbrot Continuous-Time Random Walk