2
1 Overview
motion often show large regions of chaos. Hard sphere gases are rigorously chaotic.
The foundations of statistical mechanics (and thermodynamics) are based on the
assumption that the underlying dynamics is chaotic.
The term quantum chaos refers to the signatures of classical chaos in the quantum
dynamics of particles whose classical limit is chaotic. The quantum signatures of
chaos appear wherever the classically chaotic regions have a size equal to ¯
h d in
phase space, where ¯
h is Planck’s constant and d is the number of degrees of freedom.
Signatures of chaos occur, for some parameter ranges, in most quantum systems and
determine if a quantum system can thermalize. The signatures of chaos can be used
to control quantum transitions. They can also destabilize quantum systems. In fact,
the deeper we look into the fundamental dynamics governing the world, the more
we see the profound impact of chaotic behavior.
The phase space of all conservative nonlinear systems with three or more degrees
of freedom (with a few exceptions) forms an Arnol’d web (Arnol’d 1963). An
Arnol’d web consists of a fractal set of resonances and chaos that fill the phase
space. Depending on the degree of development of the web, an initial condition
(one that is not known to infinite precision), may evolve deterministically for a long
but finite period of time, or may begin to exhibit random behavior after a short
time. The ubiquitous Arnol’d web, in conservative dynamical systems, provides the
mechanism to thermalize the world.
In this regard, one of the important discoveries in quantum physics in recent
years is that the information content of conservative quantum systems is extremized
(minimized) when the underlying classical system undergoes a transition to chaos.
The information content of the conservative quantum system approaches that of a
system whose dynamics is governed by a random Hamiltonian matrix that is chosen
to minimize information content.
In subsequent, sections we will first give a brief historical overview of the history
of conservative dynamics and chaos theory. Then we will describe the content of the
remaining chapters of this book.
1.2 Historical Overview
On April 28, 1686 the first of the three books that comprise Newton’s Principia was
formally presented to the Royal Society and, by July 1687 the complete first edition
(consisting of perhaps 300 copies) was published Newton (1686). The publication
of this work was probably the most important single event in the history of science
because it formulated the science of mechanics in terms of just three basic laws:
• A body maintains its state of rest or uniform velocity unless a net force acts on it.
• The time rate of change of momentum, p, is equal to the net force, F, acting on it.
• To every action there is an equal and opposite reaction.
In the Principia, Newton not only wrote the three laws but also gave a systematic
mathematical framework for exploring the implications of these laws. In addition,
1 Overview
motion often show large regions of chaos. Hard sphere gases are rigorously chaotic.
The foundations of statistical mechanics (and thermodynamics) are based on the
assumption that the underlying dynamics is chaotic.
The term quantum chaos refers to the signatures of classical chaos in the quantum
dynamics of particles whose classical limit is chaotic. The quantum signatures of
chaos appear wherever the classically chaotic regions have a size equal to ¯
h d in
phase space, where ¯
h is Planck’s constant and d is the number of degrees of freedom.
Signatures of chaos occur, for some parameter ranges, in most quantum systems and
determine if a quantum system can thermalize. The signatures of chaos can be used
to control quantum transitions. They can also destabilize quantum systems. In fact,
the deeper we look into the fundamental dynamics governing the world, the more
we see the profound impact of chaotic behavior.
The phase space of all conservative nonlinear systems with three or more degrees
of freedom (with a few exceptions) forms an Arnol’d web (Arnol’d 1963). An
Arnol’d web consists of a fractal set of resonances and chaos that fill the phase
space. Depending on the degree of development of the web, an initial condition
(one that is not known to infinite precision), may evolve deterministically for a long
but finite period of time, or may begin to exhibit random behavior after a short
time. The ubiquitous Arnol’d web, in conservative dynamical systems, provides the
mechanism to thermalize the world.
In this regard, one of the important discoveries in quantum physics in recent
years is that the information content of conservative quantum systems is extremized
(minimized) when the underlying classical system undergoes a transition to chaos.
The information content of the conservative quantum system approaches that of a
system whose dynamics is governed by a random Hamiltonian matrix that is chosen
to minimize information content.
In subsequent, sections we will first give a brief historical overview of the history
of conservative dynamics and chaos theory. Then we will describe the content of the
remaining chapters of this book.
1.2 Historical Overview
On April 28, 1686 the first of the three books that comprise Newton’s Principia was
formally presented to the Royal Society and, by July 1687 the complete first edition
(consisting of perhaps 300 copies) was published Newton (1686). The publication
of this work was probably the most important single event in the history of science
because it formulated the science of mechanics in terms of just three basic laws:
• A body maintains its state of rest or uniform velocity unless a net force acts on it.
• The time rate of change of momentum, p, is equal to the net force, F, acting on it.
• To every action there is an equal and opposite reaction.
In the Principia, Newton not only wrote the three laws but also gave a systematic
mathematical framework for exploring the implications of these laws. In addition,