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5 Arnol’d Diffusion
Arnol’d diffusion. In systems with two DoF, KAM tori can block the diffusion
of trajectories throughout the phase space. That is no longer the case for systems
with three or more DoF. For non-integrable systems with three or more degrees
of freedom, trajectories will definitely diffuse throughout the phase space. The
question one must then ask is “how rapid is the diffusion”?
Because Arnol’d diffusion occurs in systems with three or more degrees of
freedom, it is difficult to visualize and difficult to analyze. However, there are a
few systems for which the Arnol’d web has been studied in some detail, and we
describe some of these systems in the following sections.
We start in Sect. 5.2 with an example of an Arnol’d web that forms when two
delta-kicked rotors (standard maps) are coupled (Kaneko and Bagley 1985). Kaneko
and Bagley were able to locate the resonance lines for this system and construct a
map of the dynamics that shows clearly how Arnol’d diffusion fundamentally alters
the nature of the dynamics from that described for a single standard map in Chap. 3.
An estimate of the distance in phase space that a trajectory can diffuse, in a given
amount of time, in a system with an Arnol’d web, was obtained by Nekhoroshev
(1971, 1977) and is discussed in Sect. 5.3. Each resonance line in the web is
surrounded by a resonance region and a stochastic web. For integrable systems,
that are rendered non-integrable by a small perturbation, Nekhoroshev showed that
diffusion occurs along the stochastic layer of the resonance line and is a slow process
(the Nekhoroshev regime). However, as the strength of the perturbation is increased,
the resonance regions along the lines can begin to overlap and global diffusion can
begin. This is called the Chirikov regime (Chirikov 1979). (See also (Lichtenberg
and Lieberman 1991) for additional discussions regarding Arnol’d diffusion.)
A beautiful graphical picture of the Arnol’d diffusion process was obtained by
Froeschle et al. (2000). Some of their results are described and shown in Sect. 5.4.
A real-world system that shows Arnol’d diffusion and is amenable to experiment
is a time-periodically driven optical lattice. Such systems have been studied in
experiments for the case of a periodically modulated optical lattice with one space
dimension (Steck et al. 2001). But periodically driven lattices with one space
dimension only have two DoF and do not exhibit Arnol’d diffusion. In Sect. 5.5,
we show that a classical periodically modulated optical lattice with two space
dimensions does contain a complete Arnold web and will show a global transition
to chaos (Boretz and Reichl 2016). In Chap. 10, we show that when the dynamics is
governed by quantum mechanics, it will also be affected by Arnol’d diffusion.
Probably the most studied classical system, that clearly shows the effects of
Arnol’d diffusion, is the solar system. In Sect. 5.6, we discuss what is known,
to date, about how the planets and other objects in the solar system are affected
by the Arnol’d web that permeates the solar system phase space. In the past, the
Arnol’d web was also a concern for high energy particle accelerators because it
could destabilize the particle beams. This issue is discussed in Sect. 5.7 and, finally
in Sect. 5.8, we make some concluding remarks.
5 Arnol’d Diffusion
Arnol’d diffusion. In systems with two DoF, KAM tori can block the diffusion
of trajectories throughout the phase space. That is no longer the case for systems
with three or more DoF. For non-integrable systems with three or more degrees
of freedom, trajectories will definitely diffuse throughout the phase space. The
question one must then ask is “how rapid is the diffusion”?
Because Arnol’d diffusion occurs in systems with three or more degrees of
freedom, it is difficult to visualize and difficult to analyze. However, there are a
few systems for which the Arnol’d web has been studied in some detail, and we
describe some of these systems in the following sections.
We start in Sect. 5.2 with an example of an Arnol’d web that forms when two
delta-kicked rotors (standard maps) are coupled (Kaneko and Bagley 1985). Kaneko
and Bagley were able to locate the resonance lines for this system and construct a
map of the dynamics that shows clearly how Arnol’d diffusion fundamentally alters
the nature of the dynamics from that described for a single standard map in Chap. 3.
An estimate of the distance in phase space that a trajectory can diffuse, in a given
amount of time, in a system with an Arnol’d web, was obtained by Nekhoroshev
(1971, 1977) and is discussed in Sect. 5.3. Each resonance line in the web is
surrounded by a resonance region and a stochastic web. For integrable systems,
that are rendered non-integrable by a small perturbation, Nekhoroshev showed that
diffusion occurs along the stochastic layer of the resonance line and is a slow process
(the Nekhoroshev regime). However, as the strength of the perturbation is increased,
the resonance regions along the lines can begin to overlap and global diffusion can
begin. This is called the Chirikov regime (Chirikov 1979). (See also (Lichtenberg
and Lieberman 1991) for additional discussions regarding Arnol’d diffusion.)
A beautiful graphical picture of the Arnol’d diffusion process was obtained by
Froeschle et al. (2000). Some of their results are described and shown in Sect. 5.4.
A real-world system that shows Arnol’d diffusion and is amenable to experiment
is a time-periodically driven optical lattice. Such systems have been studied in
experiments for the case of a periodically modulated optical lattice with one space
dimension (Steck et al. 2001). But periodically driven lattices with one space
dimension only have two DoF and do not exhibit Arnol’d diffusion. In Sect. 5.5,
we show that a classical periodically modulated optical lattice with two space
dimensions does contain a complete Arnold web and will show a global transition
to chaos (Boretz and Reichl 2016). In Chap. 10, we show that when the dynamics is
governed by quantum mechanics, it will also be affected by Arnol’d diffusion.
Probably the most studied classical system, that clearly shows the effects of
Arnol’d diffusion, is the solar system. In Sect. 5.6, we discuss what is known,
to date, about how the planets and other objects in the solar system are affected
by the Arnol’d web that permeates the solar system phase space. In the past, the
Arnol’d web was also a concern for high energy particle accelerators because it
could destabilize the particle beams. This issue is discussed in Sect. 5.7 and, finally
in Sect. 5.8, we make some concluding remarks.