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Entropy of classical systems with long-range interactions.

T M Rocha Filho1, A Figueiredo, M A Amato

  • 1Instituto de Física, Universidade de Brasília, CP: 04455, 70919-970 - Brasília, Brazil.

Physical Review Letters
|December 31, 2005
PubMed
Summary

We explore entropy in classical Hamiltonian systems with long-range interactions using the Vlasov equation. Stationary states are stable maxima of Boltzmann-Gibbs entropy, dependent on infinite Lagrange multipliers.

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Area of Science:

  • Statistical Mechanics
  • Plasma Physics
  • Dynamical Systems

Background:

  • Classical Hamiltonian systems with long-range interactions exhibit complex dynamics.
  • The Vlasov equation models the behavior of N-particle systems as N approaches infinity.
  • Understanding entropy is crucial for characterizing the equilibrium and stability of such systems.

Purpose of the Study:

  • To determine the form of entropy for classical Hamiltonian systems with long-range interactions.
  • To investigate the properties of stationary states within the Vlasov dynamics framework.
  • To analyze the stability criteria for these stationary states and their relation to entropy.

Main Methods:

  • Utilized the Vlasov equation to describe the N-particle limit dynamics.

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  • Analyzed the conserved quantities inherent in Vlasov dynamics for stationary states.
  • Applied extremum principles to the Boltzmann-Gibbs entropy to identify stable states.

    • Main Results:

    • Identified that stationary states possess infinite conserved quantities.
    • Demonstrated that stable stationary states correspond to a maximum of the Boltzmann-Gibbs entropy.
    • Showed that entropy is a functional of an infinite set of Lagrange multipliers tied to initial conditions.

    Conclusions:

    • The stability of stationary states in long-range interacting systems is intrinsically linked to entropy maximization.
    • The concept of ensemble inequivalence and temperature are discussed within this framework.
    • The findings provide insights into the statistical mechanics of systems with infinite degrees of freedom.