1.3 Plan of the Book
7
Probably the most widely studied systems, as regards to the transition to chaos,
are systems driven by time-periodic external fields. With a time-periodic force,
one can cause a nonlinear system, with only one degree of freedom, to undergo
a transition to chaos. In such systems, energy is not conserved but due to a discrete
time translation invariance, the Floquet energy (quasi-energy) is conserved. Thus
all the techniques used in energy-conserving systems can be applied to these driven
systems.
1.3 Plan of the Book
The goal of this book is to provide a thorough grounding in classical and quantum
chaos theory, with the focus on topics that impact current and future research topics.
Chapters 2–5 provide a description of processes underlying chaotic classical
dynamics in conservative systems. Chapter 2 lays the foundations of the relevant
classical mechanics for understanding chaos, and focuses on those aspects of chaotic
behavior that will be used throughout the remainder of the book.
Chapter 3 deals primarily with systems that have two degrees of freedom. For
these systems it is possible to visualize the processes that lead to the onset of chaos,
because one can construct area preserving maps to follow the process. In Chap. 3,
we also focus on the fractal nature of structures in the phase space that lead to global
chaos, as parameters of the system are changed.
Chapter 4 deals with classical scattering processes and the fractal nature of
scattering dynamics, when the scatterer is chaotic or partially chaotic. Finally,
Chap. 5 focuses on the Arnold web that exists in nonlinear, non-integrable systems
with three of more degrees of freedom. The Arnol’d web provides the mechanism
for the global transition to chaos in systems with three or more degrees of freedom.
The remaining chapters of the book, Chaps. 6–10, examine the quantum manifestations of chaos. We start in Chap. 6 with a discussion of random matrix theory,
as applied to conservative Hamiltonian systems. Random matrix theory is based
on the assumption that the matrix elements of a hermitian or unitary matrix are
independent random variables. This implies certain behaviors of the eigenvalues and
eigenvectors of such systems that have been observed in quantum systems whose
classical counterpart is chaotic. As we also show in Chap. 6, a conservative quantum
system that shows random matrix-like behavior has become thermalized.
In Chap. 7, we discuss the behavior of some bounded quantum systems whose
classical counterparts undergo a transition to chaos. The Schrödinger equation for
these systems is linear. Nonlinearities appear in the Hamiltonian. We consider
chaotic billiards, spin systems, and small molecules which are anharmonic oscillators. A number of the results we describe have been realized in microwave cavity
experiments.
The connection between the quantum manifestations of chaos and random matrix
theory was first observed in nuclear scattering experiments. In Chap. 8, we describe
the theory originally developed by Wigner and Eisenbud (W-E) 1947 that allowed
7
Probably the most widely studied systems, as regards to the transition to chaos,
are systems driven by time-periodic external fields. With a time-periodic force,
one can cause a nonlinear system, with only one degree of freedom, to undergo
a transition to chaos. In such systems, energy is not conserved but due to a discrete
time translation invariance, the Floquet energy (quasi-energy) is conserved. Thus
all the techniques used in energy-conserving systems can be applied to these driven
systems.
1.3 Plan of the Book
The goal of this book is to provide a thorough grounding in classical and quantum
chaos theory, with the focus on topics that impact current and future research topics.
Chapters 2–5 provide a description of processes underlying chaotic classical
dynamics in conservative systems. Chapter 2 lays the foundations of the relevant
classical mechanics for understanding chaos, and focuses on those aspects of chaotic
behavior that will be used throughout the remainder of the book.
Chapter 3 deals primarily with systems that have two degrees of freedom. For
these systems it is possible to visualize the processes that lead to the onset of chaos,
because one can construct area preserving maps to follow the process. In Chap. 3,
we also focus on the fractal nature of structures in the phase space that lead to global
chaos, as parameters of the system are changed.
Chapter 4 deals with classical scattering processes and the fractal nature of
scattering dynamics, when the scatterer is chaotic or partially chaotic. Finally,
Chap. 5 focuses on the Arnold web that exists in nonlinear, non-integrable systems
with three of more degrees of freedom. The Arnol’d web provides the mechanism
for the global transition to chaos in systems with three or more degrees of freedom.
The remaining chapters of the book, Chaps. 6–10, examine the quantum manifestations of chaos. We start in Chap. 6 with a discussion of random matrix theory,
as applied to conservative Hamiltonian systems. Random matrix theory is based
on the assumption that the matrix elements of a hermitian or unitary matrix are
independent random variables. This implies certain behaviors of the eigenvalues and
eigenvectors of such systems that have been observed in quantum systems whose
classical counterpart is chaotic. As we also show in Chap. 6, a conservative quantum
system that shows random matrix-like behavior has become thermalized.
In Chap. 7, we discuss the behavior of some bounded quantum systems whose
classical counterparts undergo a transition to chaos. The Schrödinger equation for
these systems is linear. Nonlinearities appear in the Hamiltonian. We consider
chaotic billiards, spin systems, and small molecules which are anharmonic oscillators. A number of the results we describe have been realized in microwave cavity
experiments.
The connection between the quantum manifestations of chaos and random matrix
theory was first observed in nuclear scattering experiments. In Chap. 8, we describe
the theory originally developed by Wigner and Eisenbud (W-E) 1947 that allowed