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Stretched-exponential behavior and random walks on diluted hypercubic lattices.

N Lemke1, Ian A Campbell

  • 1Departamento de Física e Biofísica Instituto de Biociências de Botucatu UNESP-Universidade Estadual Paulista Distrito de Rubião Jr. s/n Botucatu, São Paulo 18618-970, Brazil. lemke@ibb.unesp.br

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 21, 2011
PubMed
Summary

This study models glassy relaxation using diffusion on diluted hypercubes. Large-scale simulations reveal relaxation functions follow stretched exponentials with a characteristic time that grows exponentially with hypercube dimension.

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Area of Science:

  • Statistical physics
  • Complex systems
  • Network science

Background:

  • Glassy relaxation is a complex phenomenon observed in disordered systems.
  • Stochastic processes on graphs provide a framework for modeling such dynamics.
  • Diluted hypercubes serve as a specific model for investigating glassy relaxation.

Purpose of the Study:

  • To numerically determine eigenvalue spectra for diffusion on diluted hypercubes.
  • To calculate the time evolution of autocorrelation functions and return probabilities at criticality.
  • To analyze relaxation behavior in hypercubes of dimensions up to N=28.

Main Methods:

  • Large-scale numerical simulations.
  • Analysis of eigenvalue spectra.
  • Calculation of time-dependent autocorrelation functions and return probabilities.

Main Results:

  • Relaxation functions are described by stretched exponentials with an exponent of 1/3 at long times.
  • A characteristic relaxation time exhibits exponential growth with hypercube dimension (N).
  • Numerical results align with analytic predictions for sparse network models.

Conclusions:

  • The diluted hypercube model effectively captures key aspects of glassy relaxation.
  • Stretched exponential relaxation with a 1/3 exponent is a significant finding.
  • The exponential dependence of relaxation time on dimension highlights the complexity of the system.