From generalized Langevin stochastic dynamics to anomalous diffusion.

Area of Science:

• Physics
• Statistical Mechanics
• Complex Systems

Background:

• Scaling methods are crucial in physics for understanding system behavior over different scales.
• Anomalous diffusion, deviating from standard Brownian motion, is prevalent in complex systems.
• Systems with memory exhibit temporal correlations affecting diffusion dynamics.

Purpose of the Study:

• To develop and validate a new scaling method for anomalous diffusion in systems with memory.
• To generalize the concept of the diffusion exponent and derive an analytical diffusion coefficient.
• To classify and identify new forms of diffusion, including nonergodic ballistic diffusion.

Main Methods:

• Generalization of the diffusion exponent.
• Asymptotic analysis to derive an analytical diffusion coefficient.
• Introduction and exact determination of a time scale factor, lambda(t).
• Comparison of analytical results with numerical simulations.

Main Results:

• An exact expression for the time scale factor lambda(t) was derived, enabling accurate description of diffusive processes.
• For large times, lambda(t) converges to a universal parameter dependent on the diffusion exponent.
• New forms of diffusion were classified, including the identification of nonergodic ballistic diffusion (alpha=2).
• Analytical and numerical results showed good agreement.

Conclusions:

• The proposed scaling method effectively describes anomalous diffusion in systems with memory for intermediate and long times.
• The method provides a universal framework for analyzing diffusion processes and classifying new diffusion regimes.
• The generalization of the diffusion exponent and the derived analytical expressions offer significant advancements in understanding complex stochastic systems.