Goodness of ergodic adiabatic invariants

Journal Article · Sat Oct 31 23:00:00 EST 1987 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01009347· OSTI ID:5300499

Brown, R; Ott, E; Grebogi, C

For a slowly time-dependent Hamiltonian system exhibiting chaotic motion that ergodically covers the energy surface, the phase space volume enclosed inside this surface is an adiabatic invariant. In this paper we examine, both numerically and theoretically, how the error in this ergodic adiabatic invariant scales with the slowness of the time variation of the Hamiltonian. It is found that under certain circumstances, the error is diffusive and scales like T/sup -1/2/, where T is the characteristic time over which the Hamiltonian changes. On the other hand, for other cases (where motion in the Hamiltonian has a long-time 1/t tail in a certain correlation function), the error scales like (T/sup -1/lN(T))/sup 1/2/. Both of these scalings are verified by numerical experiments. In the situation where invariant tori exist amid chaos, the motion may not be fully ergodic on the entire energy surface. The ergodic adiabatic invariant may still be useful in this case and the circumstances under which this is so are investigated numerically (in particular, the islands have to be small enough).

• Research Organization: Univ. of Maryland, College Park (USA)
• OSTI ID: 5300499
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 49:3/4; ISSN JSTPB
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ADIABATIC INVARIANCE
COLLISIONS
COMPUTERIZED SIMULATION
CORRELATION FUNCTIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
EQUATIONS OF MOTION
ERGODIC HYPOTHESIS
FUNCTIONS
HAMILTONIANS
HYPOTHESIS
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE MODELS
PHASE SPACE
PLASMA SIMULATION
QUANTUM MECHANICS
QUANTUM OPERATORS
RANDOMNESS
SIMULATION
SPACE
STATISTICAL MECHANICS
TIME DEPENDENCE