Ergodicity Hamiltonian systems

Definition 5.1 A Hamiltonian system is said to be ergodic on an energy surface X e (microcanonically ergodic) if, for almost all trajectories F emanating from initial conditions on He, and for any observable g, the following holds... 

It is important to note that a system may be chaotic without being ergodic on a particular level set of energy. For additional discussion of chaotic Hamiltonian systems and their trajectories, see [64,174,386]. 

The other distribution is the microcanonical equilibrium distribution. More than 15 years ago, Ott-Brown-Grebogi pointed out fractional scaling of deviation from ergodic adiabatic invariants in Hamiltonian chaotic systems [16, 17]. We will reconsider not only ergodic adiabatic invariants but also nonergodic adiabatic invariants, which are important in the mixed phase space. We will show results of our numerical simulation in which a nonergodic adiabatic invariant corresponding to uniform distribution is broken in the mixed phase space. 

Rigorous proof that a given Hamiltonian dynamical system is ergodic (or mixing) is extremely difficult for even simple cases, and is likely to be dependent on the choice of the potentials, themselves. 

The equivalence of these two averages is the essence of a very important system property called ergodicity, which deserves a separate subsection further in this chapter. pA hat is used here to emphasize that, in quantum mechanics, the Hamiltonian is an operator acting on the h function and not just a function of particle coordinates and momenta, as in classical mechanics. 

The Hamiltonian is a constant of the phase space sampled by the simulation. A mechanical property is one that is defined at each point in phase space, i.e., it can be computed for each configuration of the system. The macroscopic observable Fobs is then computed as the ensemble average (Monte Carlo) or time average (molecular dynamics) of the mechanical property F, according to the ergodic hypothesis, these should be equivalent. 

Here H(p, q) is the Hamiltonian (the sum of the potential and kinetic energy) of the system and p,- are the generalized momenta conjugate to qt. If the generalized coordinates represent the atomic coordinates of a molecular system, this approach is referred to as molecular dynamics. If one makes the assumption that the resulting trajectories cover phase space (or more specifically, are ergodic) then they generate a statistical mechanical ensemble. ... 

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