Correlated continuous-time random walks in external force fields

Area of Science:

• Statistical Mechanics
• Nonlinear Dynamics
• Complex Systems

Background:

• Anomalous diffusion describes particle movement deviating from standard Brownian motion.
• Continuous time random walks (CTRWs) model non-Brownian transport.
• Correlated waiting times introduce memory effects in CTRWs.

Purpose of the Study:

• To investigate anomalous diffusion in an external force field using a correlated CTRW model.
• To analyze the influence of correlated waiting times on particle dynamics.
• To explore relaxation dynamics, Einstein relations, and stationary state properties.

Main Methods:

• Development of coupled Langevin equations to incorporate external forces.
• Analysis of relaxation dynamics under confining potentials.
• Investigation of generalized and second Einstein relations.
• Characterization of the stationary solution and aging phenomena.

Main Results:

• Relaxation dynamics exhibit power-law or stretched exponential behavior based on model parameters.
• The process adheres to generalized Einstein-Stokes-Smoluchowski and second Einstein relations.
• The stationary distribution follows a Boltzmann-Gibbs form.
• Aging and ergodicity breaking are observed.

Conclusions:

• Correlated waiting times in CTRWs significantly alter anomalous diffusion dynamics.
• The model provides a framework for understanding complex transport phenomena in external fields.
• The findings have implications for systems exhibiting memory effects and non-ergodicity.