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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics.

Aljaž Godec¹, Ralf Metzler

¹Institute for Physics and Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany. aljaz.godec@ki.si

Physical Review Letters

|February 7, 2013

Summary

Superdiffusive Lévy walk processes exhibit unique scaling exponent scatter and finite-time amplitude depression, deviating from subdiffusive behaviors. These findings are crucial for accurately evaluating superdiffusion dynamics.

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Area of Science:

Statistical Mechanics
Complex Systems
Non-equilibrium Physics

Background:

Superdiffusion, characterized by faster-than-Fickian spreading, is observed in various physical and biological systems.
Lévy walk processes are a key model for superdiffusion, exhibiting anomalous scaling of mean squared displacement.
Ergodic properties are fundamental to understanding the relationship between time averages and ensemble averages in stochastic processes.

Purpose of the Study:

To investigate the ergodic properties of spatiotemporally coupled Lévy walk processes.
To analyze the behavior of the time-averaged mean squared displacement (δx²) for finite trajectory durations.
To quantify the impact of finite measurement time on the ensemble-averaged mean squared displacement.

Main Methods:

Analysis of scaling exponents for the time-averaged mean squared displacement (δx²).
Investigation of the dependence of δx² on finite measurement time.
Comparison of long-time averages with ensemble-averaged mean squared displacement.

Main Results:

A distinct scatter in scaling exponents of δx² was observed, ranging from ballistic motion to subdiffusion (1<α<2).
Significant finite-time amplitude depression in the ensemble-averaged δx² was found.
Ultraweak ergodicity breaking was identified, with long-time averages differing from ensemble means by a constant factor.

Conclusions:

The scatter in scaling exponents and finite-time amplitude depression are critical for quantitative evaluation of superdiffusive processes.
Lévy walk processes demonstrate distinct ergodic behaviors compared to subdiffusive processes.
The study reveals nuances in ergodicity for superdiffusive systems, particularly under finite observation times.