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Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes.

Carlos Medel-Portugal1, Juan Manuel Solano-Altamirano2, José Luis E Carrillo-Estrada1

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This study introduces a new framework for understanding how classical and quantum gases reach equilibrium from an out-of-equilibrium state. It extends the H-theorem to inhomogeneous systems, providing a unified approach for diverse gas types.

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Area of Science:

  • Statistical Mechanics
  • Quantum Gases
  • Thermodynamics

Background:

  • Describing the approach to equilibrium for inhomogeneous classical and quantum gases is complex.
  • Existing models may not fully capture the behavior of systems initially out of equilibrium.

Purpose of the Study:

  • To develop a novel framework for the time-evolution of dilute classical and quantum gases towards equilibrium.
  • To extend the H-theorem to spatially inhomogeneous systems.

Main Methods:

  • Dividing the system into small cells and applying the local equilibrium hypothesis.
  • Defining a global functional as the sum of cell H-functionals.
  • Utilizing the variational method to prove the time-evolution relationship (dH/dt≤0).

Main Results:

  • A unified framework is presented for classical (Maxwell-Boltzmann) and quantum (Fermi-Dirac, Bose-Einstein) gases.
  • The H-functional is shown to decrease over time (dH/dt≤0), indicating a move towards equilibrium.
  • The H-functionals align with the correspondence principle and can be identified with system entropy.

Conclusions:

  • The proposed framework offers a generalized H-theorem for inhomogeneous systems.
  • It provides insights into the relaxation processes of out-of-equilibrium gases.
  • The approach unifies the description of diverse gas behaviors approaching equilibrium.