148
5 Arnol’d Diffusion
In Fig. 5.9b, we show a strobe plot with the same initial conditions on (p x , x), but
now with the full Hamiltonian in Eq. (5.17) and initial values p y = 0.5, y(0) = π ,
but with b = 0. For b = 0, there is no coupling between the motions in the x
and y degrees of freedom and the system evolves as two uncoupled 2 DoF systems.
Fig. 5.9a, b are essentially indistinguishable.
In Fig. 5.9c and in Fig. 5.9d, we show the same strobe plots (same initial
conditions) but now with b = 0.002 and b = 0.02, respectively, so all degrees of
freedom are coupled. All four plots in Fig. 5.9 are run for 2000 periods of oscillation.
Weak diffusion across the KAM tori appears to have occurred in Fig. 5.9c. However,
in Fig. 5.9d, diffusion dominates the dynamics. This is consistent with our estimate
that the transition between the Nekhoroshev and Chirikov regimes occurs for
0.009≤b≤0.02. In Fig. 5.9, we see weak diffusion for b = 0.001 and more rapid
diffusion for b = 0.01.
In Fig. 5.9b, where b = 0, it is clear that the energy H (t) cannot undergo large
excursions. If the initial conditions lie on a KAM torus, the energy will undergo
small regular oscillations. If they lie in the chaotic region, the energy H (t) can
undergo small apparently random oscillations, but they are blocked by KAM tori
from large excursions in energy. However, if b =0 then, from Fig. 5.9c, d, it is clear
that the energy H (t) might undergo large energy oscillations because the KAM tori
no longer isolate regions of the phase space from one another. It is also clear that
the diffusion is faster for larger values of b.
It is useful to look at the behavior of the average energy of the system for
various parameter regimes. In Fig. 5.10, we plot the energy H (t) for four different
values of the lattice coupling strength b = 0.002, b = 0.02, b = 0.2, and
b = 2.0. In all four cases, the initial energy is E = 30 and the trajectory is
run for a time t = 2000. For each value of b, we have obtained energy plots for
ten different initial conditions on the energy surface. In Fig. 5.10, we show one
realization (out of ten) for each value of b. As we can see from the plots, the
energy fluctuates. We obtain the average energy for each plot and then average the
energy of the ten plots for each value of b. We find the following average energies:
(b, E av ) = (0.002, 35.8), (0.02, 46.0), (0.2, 79.0), (2.0, 186.3). This change in
the behavior of the average energy appears to be consistent with our estimates for
the transition between the Nekhoroshev and Chirikov regimes. The Chirikov regime
requires resonance overlap which, for low values of b, will occur in local regions
of the phase space because of the different sizes and locations of resonances. The
behavior of the average energy indicates that large scale diffusion in energy, due to
resonance overlap, begins to occur for b > 0.2 and steadily grows as b increases.
The quantum mechanical version of this time-periodically driven optical lattice
was also studied in Boretz and Reichl (2016). They found that the Arnol’d web
gave rise to Floquet eigenstates of the driven system that consisted of large numbers
of entangled energy eigenstates of the static lattice, and that the average energy of
atoms in the lattice increased dramatically in parameter regimes where the classical
system showed large fluctuations in the energy. Some of their results, regarding
Arnol’d diffusion on the quantum lattice, are described in Chap. 10.
5 Arnol’d Diffusion
In Fig. 5.9b, we show a strobe plot with the same initial conditions on (p x , x), but
now with the full Hamiltonian in Eq. (5.17) and initial values p y = 0.5, y(0) = π ,
but with b = 0. For b = 0, there is no coupling between the motions in the x
and y degrees of freedom and the system evolves as two uncoupled 2 DoF systems.
Fig. 5.9a, b are essentially indistinguishable.
In Fig. 5.9c and in Fig. 5.9d, we show the same strobe plots (same initial
conditions) but now with b = 0.002 and b = 0.02, respectively, so all degrees of
freedom are coupled. All four plots in Fig. 5.9 are run for 2000 periods of oscillation.
Weak diffusion across the KAM tori appears to have occurred in Fig. 5.9c. However,
in Fig. 5.9d, diffusion dominates the dynamics. This is consistent with our estimate
that the transition between the Nekhoroshev and Chirikov regimes occurs for
0.009≤b≤0.02. In Fig. 5.9, we see weak diffusion for b = 0.001 and more rapid
diffusion for b = 0.01.
In Fig. 5.9b, where b = 0, it is clear that the energy H (t) cannot undergo large
excursions. If the initial conditions lie on a KAM torus, the energy will undergo
small regular oscillations. If they lie in the chaotic region, the energy H (t) can
undergo small apparently random oscillations, but they are blocked by KAM tori
from large excursions in energy. However, if b =0 then, from Fig. 5.9c, d, it is clear
that the energy H (t) might undergo large energy oscillations because the KAM tori
no longer isolate regions of the phase space from one another. It is also clear that
the diffusion is faster for larger values of b.
It is useful to look at the behavior of the average energy of the system for
various parameter regimes. In Fig. 5.10, we plot the energy H (t) for four different
values of the lattice coupling strength b = 0.002, b = 0.02, b = 0.2, and
b = 2.0. In all four cases, the initial energy is E = 30 and the trajectory is
run for a time t = 2000. For each value of b, we have obtained energy plots for
ten different initial conditions on the energy surface. In Fig. 5.10, we show one
realization (out of ten) for each value of b. As we can see from the plots, the
energy fluctuates. We obtain the average energy for each plot and then average the
energy of the ten plots for each value of b. We find the following average energies:
(b, E av ) = (0.002, 35.8), (0.02, 46.0), (0.2, 79.0), (2.0, 186.3). This change in
the behavior of the average energy appears to be consistent with our estimates for
the transition between the Nekhoroshev and Chirikov regimes. The Chirikov regime
requires resonance overlap which, for low values of b, will occur in local regions
of the phase space because of the different sizes and locations of resonances. The
behavior of the average energy indicates that large scale diffusion in energy, due to
resonance overlap, begins to occur for b > 0.2 and steadily grows as b increases.
The quantum mechanical version of this time-periodically driven optical lattice
was also studied in Boretz and Reichl (2016). They found that the Arnol’d web
gave rise to Floquet eigenstates of the driven system that consisted of large numbers
of entangled energy eigenstates of the static lattice, and that the average energy of
atoms in the lattice increased dramatically in parameter regimes where the classical
system showed large fluctuations in the energy. Some of their results, regarding
Arnol’d diffusion on the quantum lattice, are described in Chap. 10.