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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Diffusion is a type of passive transport. In passive transport, a substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across the space. For example, take the diffusion of substances through the air. When someone opens a perfume bottle in a room filled with people, the perfume is at its highest concentration in the bottle and is at its lowest at the edges of the room. The perfume vapor will diffuse, or spread away, from the...
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Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
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Spherically Restricted Random Hyperbolic Diffusion.

Philip Broadbridge1, Alexander D Kolesnik2, Nikolai Leonenko3

  • 1Department of Mathematics and Statistics, La Trobe University, Melbourne VIC 3086, Australia.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary

This study analyzes hyperbolic diffusion equations with random initial conditions, providing approximations and analyzing solution properties like smoothness and dependence. Findings link solution continuity to initial condition spectral decay.

Keywords:
Hölder continuityapproximation errorscosmic microwave backgroundhyperbolic diffusion equationlong-range dependencespherical random fieldstochastic partial differential equations

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Area of Science:

  • Mathematical Physics
  • Stochastic Partial Differential Equations
  • Random Field Theory

Background:

  • Hyperbolic diffusion equations model various physical phenomena.
  • Understanding solutions with random initial conditions is crucial for complex systems.
  • Analysis often relies on spectral properties of initial data.

Purpose of the Study:

  • Investigate solutions to hyperbolic diffusion equations in 3D with random initial conditions.
  • Analyze the properties of these spatial-temporal random field solutions.
  • Develop and assess approximations for these solutions.

Main Methods:

  • Formulating assumptions using angular power spectrum and spectral measure.
  • Deriving approximations to exact solutions.
  • Obtaining upper bounds for mean-square convergence rates.
  • Investigating smoothness properties (Hölder-type continuity).

Main Results:

  • Solutions are characterized as spatial-temporal random fields.
  • Hölder-type continuity depends on the decay of the angular power spectrum.
  • Conditions for short- or long-range dependence based on spectral measure are identified.
  • Numerical studies validate theoretical findings.

Conclusions:

  • The spectral properties of random initial conditions dictate solution behavior.
  • Approximation methods provide reliable estimates for solution fields.
  • Theoretical results are confirmed through numerical simulations.