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Self-adjoint integral operator for bounded nonlocal transport.

J E Maggs1, G J Morales1

  • 1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA.

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Summary
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A new integral operator models nonlocal transport using truncated Lévy stable distributions. This method accounts for wall effects and generates self-consistent inward flow, crucial for understanding complex systems.

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Area of Science:

  • Physics
  • Applied Mathematics
  • Transport Phenomena

Background:

  • Nonlocal transport phenomena are critical in various physical systems.
  • Existing models often simplify boundary conditions and transport mechanisms.
  • Lévy stable distributions offer a powerful framework for describing anomalous diffusion and jumps.

Purpose of the Study:

  • To develop an integral operator for nonlocal transport in bounded 1D systems.
  • To incorporate the effects of bounding walls on Lévy α-stable distributions.
  • To analyze the resulting self-consistent convective inward transport (pinch) and its properties.

Main Methods:

  • Development of a novel integral operator based on truncated Lévy α-stable distributions.
  • Analysis of the operator's properties with respect to Lévy parameters [α,γ] and wall conductivity.
  • Utilizing a self-adjoint formulation to handle spatial variations in Lévy parameters.
  • Demonstration using cold-pulse propagation examples in nonlocal systems.

Main Results:

  • The integral operator accurately describes nonlocal transport with wall truncation.
  • A self-consistent convective inward transport (pinch) term emerges due to truncation.
  • The operator recovers local transport features when α=2.
  • Spatial variations in Lévy parameters induce internally generated flows.

Conclusions:

  • The developed integral operator provides a robust framework for nonlocal transport in bounded systems.
  • The inclusion of truncated Lévy distributions and wall effects captures essential physical phenomena.
  • The methodology offers a versatile tool for simulating and understanding complex transport behaviors.