Abstract
The precise determination of diffusive properties is presented for a system described by the generalized Langevin equation. The time-dependent fractional diffusion function and the Green-Kubo relation as well as the generalized Stokes-Einstein formula, in the spirit of ensemble averages, are reconfigured. The effective friction function is introduced as a measure of the influence of frequency-dependent friction on the evolution of the system. This is applied to the generalized Debye model, from which self-oscillation emerges as indicative of ergodicity that breaks due to high finite-frequency cutoff. Moreover, several inconsistent conclusions that have appeared in the literature are revised.
Highlights
In the early 20th century, Brownian motion became the subject of a theoretical investigation by Einstein, Langevin, Smoluchowski, and others [1,2,3]
The following questions arise: What forms do the generalized Green-Kubo and Stokes-Einstein relations take in regard to anomalous diffusion? Can the coefficient of diffusion and the exponent be measured accurately? How strong is the effect of the high finite-frequency cutoff on the dynamics? The answers will become clear in the present study
This work aims at furnishing a connection between velocity autocorrelation function (VACF) and frictional kernel in the generalized Langevin equation (GLE) framework for the diffusion dynamics
Summary
In the early 20th century, Brownian motion became the subject of a theoretical investigation by Einstein, Langevin, Smoluchowski, and others [1,2,3]. A few years later, Ivar Nordlune conceived a method for recording much longer time series This let him determine time average individual trajectories and avoid average ensembles of particles that were probably not identical [3]. In the anomalous diffusion sense, the generalization of the two basic statistical-dynamics laws—the Green-Kubo and the Stokes-Einstein relations—regarding measurable approaches for arbitrary frequency-dependent friction has importance. This aids our understanding of the characteristic behaviors of anomalous, diffusive, and nonergodic systems [7, 13]. The following questions arise: What forms do the generalized Green-Kubo and Stokes-Einstein relations take in regard to anomalous diffusion? Can the coefficient of diffusion and the exponent be measured accurately? How strong is the effect of the high finite-frequency cutoff on the dynamics? The answers will become clear in the present study
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