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Subdiffusion in time-averaged, confined random walks.

Thomas Neusius1, Igor M Sokolov, Jeremy C Smith

  • 1Computational Molecular Biophysics, Universität Heidelberg, Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 8, 2009
PubMed
Summary
This summary is machine-generated.

Time-averaged mean-squared displacement (MSD) in confined systems differs from unbounded systems. For confined continuous time random walks (CTRWs), time-averaged MSD scales sublinearly with lag time, unlike unbounded CTRWs.

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Area of Science:

  • Physics
  • Physical Chemistry
  • Statistical Mechanics

Background:

  • Diffusive processes are often characterized by time averages (e.g., single-particle tracking) rather than ensemble averages.
  • The ensemble-averaged mean-squared displacement (MSD) for unbounded continuous time random walks (CTRWs) with broad waiting times shows subdiffusion.
  • However, the time-averaged MSD (delta2) for unbounded CTRWs is linear with lag time, not subdiffusive.

Purpose of the Study:

  • To investigate the behavior of the time-averaged MSD for CTRWs within a finite volume.
  • To contrast this behavior with the established results for unbounded CTRWs.
  • To enable the interpretation of experimental time-averaged quantities using CTRW models.

Main Methods:

  • Theoretical analysis of continuous time random walks (CTRWs).
  • Derivation of the time-averaged mean-squared displacement (delta2) for a walker confined to a finite volume.
  • Comparison of confined CTRW results with unbounded CTRW theory.

Main Results:

  • The time-averaged MSD (delta2) for a CTRW in a finite volume exhibits sublinear dependence on lag time (Delta).
  • Specifically, for long lag times, delta2 scales as approximately Delta^(1-alpha).
  • This contrasts with the linear dependence of delta2 on Delta observed in unbounded CTRWs.

Conclusions:

  • The confinement of a diffusing particle significantly alters the scaling of the time-averaged mean-squared displacement.
  • The sublinear scaling of delta2 in confined CTRWs provides a new framework for analyzing experimental data from techniques like single-particle tracking.
  • This work bridges the gap between theoretical models and time-averaged experimental measurements in diffusive systems.