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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

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Continuous time random walk with generic waiting time and external force.

Kwok Sau Fa¹, K G Wang

¹Departamento de Física, Universidade Estadual de Maringá, Av. Colombo 5790, 87020-900 Maringá, PR, Brazil. kwok@dfi.uem.br

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 28, 2010

Summary

This summary is machine-generated.

This study introduces a new diffusion equation for continuous time random walks, applicable to various waiting times and forces. It demonstrates that the generalized Einstein relation holds universally for any waiting time distribution.

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Area of Science:

Physics
Statistical Mechanics
Physical Chemistry

Background:

Continuous Time Random Walks (CTRWs) are fundamental models for anomalous diffusion.
Existing diffusion equations often rely on specific assumptions about waiting time distributions.
Understanding diffusion under external forces requires a generalized theoretical framework.

Purpose of the Study:

To derive a generalized integrodifferential diffusion equation for CTRRWs.
To analyze diffusion behavior for various waiting time probability density functions (WTPDFs) under a harmonic trap.
To investigate the applicability of the second Einstein relation for arbitrary WTPDFs.

Main Methods:

Derivation of an integrodifferential diffusion equation for CTRRWs.
Analysis of specific WTPDFs: exponential, power law + Mittag-Leffler, and sum of exponentials.
Study of diffusion under a harmonic external potential.

Main Results:

The derived equation is valid for generic WTPDFs and external forces.
Exponential and power law + Mittag-Leffler WTPDFs reproduce standard and fractional diffusion results.
A sum of exponentials WTPDF exhibits complex diffusion patterns.
The second Einstein relation is shown to hold universally for any WTPDF.

Conclusions:

The generalized diffusion equation provides a unified framework for CTRRWs.
The choice of WTPDF significantly impacts diffusion dynamics.
The universality of the second Einstein relation is confirmed, offering broad applicability.