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Area of Science:

  • Statistical Mechanics
  • Theoretical Physics
  • Dynamical Systems

Background:

  • Hamiltonian systems are fundamental in physics.
  • Understanding statistical behavior under parameter changes is crucial.
  • Previous work suggested action increases under parametric kicks.

Purpose of the Study:

  • To analyze the change in adiabatic invariant (action) of 1D Hamiltonian systems under parametric kicks.
  • To test the conjecture that the action, and thus Gibbs entropy, increases on average.
  • To explore the conditions under which this conjecture holds.

Main Methods:

  • Studied one-dimensional Hamiltonian systems.
  • Assumed an initial microcanonical distribution.
  • Analyzed the system's change under a parametric kick (discontinuous parameter jump).
  • Conducted numerous case studies.

Main Results:

  • The conjecture that the change of action at mean energy increases was largely satisfied.
  • Exceptions were found for non-smooth potentials or energies near phase space separatrices.
  • The parametric kick approximation is valid for sufficiently fast parameter changes.

Conclusions:

  • The study supports the conjecture regarding the increase of action and Gibbs entropy in Hamiltonian systems under parametric kicks.
  • Identified specific conditions (potential smoothness, energy proximity to stationary points) where the conjecture fails.
  • Highlights the relevance of parametric kick approximation for fast, time-dependent systems in statistical mechanics.