Hamiltonian systems adiabatic invariant

For isolated Hamiltonian systems the width of the work distribution remains finite even in the limit of infinitely slow switching. This is a consequence of the so-called adiabatic invariants [16],... 

B. Adiabatic Invariant for a Simple Hamiltonian Chaotic System... 

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. 

In this section we will consider an nonergodic adiabatic invariant for a chaotic Hamiltonian system. Specifically, we are interested in the mixed phase space, in which tori and chaotic seas coexist. 

Now we consider the adiabatic invariant for a simple Hamiltonian chaotic system. The Hamiltonian is defined as... 

With regard to the microcanonical equilibrium distribution and the extension of the fluctuation-dissipation theorem, we considered a nonergodic adiabatic invariant in a simple Hamiltonian chaotic system. We numerically demonstrated the breaking of the nonergodic adiabatic invariant in the mixed phase space. The variance of the nonergodic adiabatic invariant can be considered as a measure for complexity of the mixed phase space. 

In this chapter we considered only a small Hamiltonian system whose Poincare map is the standard map defined on the unit square. It is interesting to consider Hamiltonian systems in a large phase space in which diffusion appears. Specifically, we are interested how the accelerator mode, which causes the anomalous diffusion in the standard map, affects the breaking of the adiabatic invariant. We will continue this study in a forthcoming article [21]. 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

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