Scaled Hamiltonians uniform scaling

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. 

Figure 2. Uniform (1) and nonuniform (2-6) scaling paths of electronic charges in the complementary subsystems of M = (i4 B). The subsystem external potentials of the scaled hamiltonians for the starting and end points of each path are also listed, with the upper (lower) entries corresponding to ti (B).

The third coupling-constant hamiltonian to be considered [see Eq.(37)], again with all electron-electron repulsions scaled uniformly in accordance with the global factor 0 < K 2 (path 1 in Fig. 2),... 

Because of the uniform procedure used to derive the Umiting forms, the solutions obtained from them can be expected to bracket the D = Z solutions. In fact, simple dimensional interpolation (linear interpolation in 1/i between dimensional limits evaluated in the above scaling) allows one to obtain rough estimates of i = 3 energies for this and a variety of other simple systems, as shown in Fig. 1. It is important to note that it is only because of the use of a uniform scaling that this is true. For example, if one used units of (I —1)/2 Bohr radii for the D limit and ( >—1)2/4 Bohr radii for the D— oo limit [12], one would obtain Eq. (5) without the factors of and in firont of the cavity radius, and the solutions obtained from these hamiltonians would not be useful for quantitative calculations. 

---