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Updated: May 22, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
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Published on: December 4, 2017

Subdiffusive master equation with space-dependent anomalous exponent and structural instability.

Sergei Fedotov1, Steven Falconer

  • 1School of Mathematics, The University of Manchester, Manchester, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 17, 2012
PubMed
Summary
This summary is machine-generated.

Subdiffusive fractional equations with varying anomalous exponents exhibit unpredictable long-time behavior. These systems are dominated by rare, low-exponent events, illustrating a "Black Swan" phenomenon in anomalous diffusion.

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Evolution of Staircase Structures in Diffusive Convection
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Last Updated: May 22, 2026

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Area of Science:

  • Statistical Physics
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Fractional master equations describe anomalous diffusion processes.
  • Subdiffusion is characterized by a constant anomalous exponent (μ).
  • The stability of these equations under parameter variations is not well understood.

Purpose of the Study:

  • To derive and analyze the fractional master equation with a space-dependent anomalous exponent.
  • To investigate the asymptotic behavior and stationary solutions of such systems.
  • To explore the impact of non-uniform anomalous exponent distributions on subdiffusive dynamics.

Main Methods:

  • Analytical derivation of the fractional master equation.
  • Analysis of asymptotic behavior using lattice models.
  • Monte Carlo simulations to validate analytical findings.

Main Results:

  • Subdiffusive fractional equations with constant μ are not structurally stable under non-homogeneous variations of μ.
  • The Gibbs-Boltzmann distribution ceases to be a stationary solution for fractional Fokker-Planck equations with space-varying exponents.
  • Random spatial distributions of μ lead to long-time behavior dominated by rare, low-exponent events, akin to a 'Black Swan'.

Conclusions:

  • Anomalous diffusion systems with spatially varying anomalous exponents exhibit 'Black Swan' characteristics.
  • The long-time dynamics are critically influenced by extremely low probability events.
  • Standard stationary solutions may not apply in complex, non-uniform subdiffusive environments.