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Nonergodicity, fluctuations, and criticality in heterogeneous diffusion processes.

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This study analyzes heterogeneous diffusion processes, finding that fluctuations in individual particle paths increase with the scaling exponent. At critical values, these fluctuations diverge, revealing similarities to continuous time random walk processes.

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Area of Science:

  • Statistical Physics
  • Anomalous Diffusion
  • Stochastic Processes

Background:

  • Heterogeneous diffusion processes exhibit anomalous diffusion, characterized by a power-law dependence of the diffusion coefficient D(x) ∼ |x|(α).
  • Understanding the stochastic behavior and ergodicity breaking is crucial for characterizing these complex systems.

Purpose of the Study:

  • To investigate the stochastic behavior of heterogeneous diffusion processes with power-law diffusion coefficients.
  • To analyze the degree of irreproducibility and fluctuations between individual realizations of the diffusion process.
  • To compare the behavior of heterogeneous diffusion with continuous time random walk (CTRW) processes.

Main Methods:

  • Analysis of statistical measures including amplitude scatter of time-averaged mean-squared displacement.
  • Calculation of ergodicity breaking and non-Gaussianity parameters.
  • Examination of the probability density function P(x,t) and its evolution.

Main Results:

  • Fluctuations between individual realizations increase with the modulus of the scaling exponent |α|.
  • Fluctuations diverge near the critical value α = 2, exhibiting a transition in mean-squared displacement behavior.
  • At criticality (α = 2), mean-squared displacement shows exponential growth, and fluctuations do not converge.

Conclusions:

  • Heterogeneous diffusion processes display weakly nonergodic behavior with significant irreproducibility of individual paths.
  • The observed behavior, particularly at criticality, shows striking similarities to subdiffusive CTRW processes with power-law waiting times.
  • The scaling exponent α critically influences the fluctuations and statistical properties of anomalous diffusion.