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Majorization and Dynamics of Continuous Distributions.

Ignacio S Gomez1, Bruno G da Costa2, Maike A F Dos Santos3

  • 1Instituto de Física, Universidade Federal da Bahia, Rua Barao de Jeremoabo, Salvador-BA 40170-115, Brazil.

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Summary

This study introduces majorization in continuous distributions to characterize mixing, diffusive, and quantum dynamics. Stationary states are identified as the infimum elements in these majorized ordered chains.

Keywords:
H-theoremcontinuous majorizationconvex functionsordered chain

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Area of Science:

  • Statistical Mechanics
  • Information Theory
  • Quantum Dynamics

Background:

  • Majorization is a concept from information theory used to compare probability distributions.
  • Understanding the evolution of dynamical systems is crucial in various scientific fields.

Purpose of the Study:

  • To demonstrate how majorization in continuous distributions can characterize mixing, diffusive, and quantum dynamics.
  • To connect majorization with the H-Boltzmann theorem.

Main Methods:

  • Utilizing the definition of majorization with a range of convex functions (ϕ).
  • Analyzing time evolution of dynamical systems as majorized ordered chains.
  • Deriving the H-Boltzmann theorem as a special case.

Main Results:

  • Mixing, generalized Fokker-Planck, and quantum dynamics are characterized as majorized ordered chains.
  • Stationary states are identified as the infimum elements in these chains.
  • The H-Boltzmann theorem is obtained for the specific convex function ϕ(x) = x ln x.

Conclusions:

  • Majorization provides a unified framework for characterizing diverse dynamical systems.
  • The choice of convex functions allows for a flexible study of dynamics.
  • This approach offers new insights into the behavior of systems approaching equilibrium.