Chapter 1
Overview
Abstract The dynamical evolution of conservative chaotic systems is not deterministic because it is not possible to follow their dynamical evolution with any precision
for more than a short time. In quantum systems, signatures of chaos emerge when
classical chaos occupies phase space volumes greater than d ( is Planck’s constant
and d is number of degrees of freedom).
The phase space of most conservative nonlinear systems with three or more
degrees of freedom (with a few exceptions) is permeated by an Arnol’d web
consisting of a fractal set of nonlinear resonances that fill the phase space. The
Arnol’d web is the source of the chaos that causes thermalization of both classical
and quantum systems.
The publication of Newton’s Principia in 1686 and the success and power of
Newton’s laws led to the huge growth in science that we see today. Belief that
Newtonian mechanics is deterministic was shaken by the work of Poincaré who
showed that perturbation expansions must diverge due to nonlinear resonances,
making it impossible to make long-time predictions. When chaos manifests itself
in quantum systems, the information content of a quantum system is extremized
(minimized). In this book, we examine in more detail the mechanisms by which
chaos emerges in conservative classical and quantum systems.
Keywords Classical chaos · Quantum manifestations of chaos · Determinism ·
History of dynamics · Symmetries · Perturbation theory divergence · KAM tori ·
Nonlinear resonance · Renormalization theory · Random matrix theory · Path
integrals
1.1 Introduction
The existence of chaos in a conservative classical system means that the dynamical
evolution of the system is no longer deterministic. Most classical systems with three
or more degrees of freedom are either fully chaotic or have fractal regions of chaos
distributed throughout the phase space. For example, the solar system appears to
have regions where planetary motion is chaotic. Classical models of molecular
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_1
1
Overview
Abstract The dynamical evolution of conservative chaotic systems is not deterministic because it is not possible to follow their dynamical evolution with any precision
for more than a short time. In quantum systems, signatures of chaos emerge when
classical chaos occupies phase space volumes greater than d ( is Planck’s constant
and d is number of degrees of freedom).
The phase space of most conservative nonlinear systems with three or more
degrees of freedom (with a few exceptions) is permeated by an Arnol’d web
consisting of a fractal set of nonlinear resonances that fill the phase space. The
Arnol’d web is the source of the chaos that causes thermalization of both classical
and quantum systems.
The publication of Newton’s Principia in 1686 and the success and power of
Newton’s laws led to the huge growth in science that we see today. Belief that
Newtonian mechanics is deterministic was shaken by the work of Poincaré who
showed that perturbation expansions must diverge due to nonlinear resonances,
making it impossible to make long-time predictions. When chaos manifests itself
in quantum systems, the information content of a quantum system is extremized
(minimized). In this book, we examine in more detail the mechanisms by which
chaos emerges in conservative classical and quantum systems.
Keywords Classical chaos · Quantum manifestations of chaos · Determinism ·
History of dynamics · Symmetries · Perturbation theory divergence · KAM tori ·
Nonlinear resonance · Renormalization theory · Random matrix theory · Path
integrals
1.1 Introduction
The existence of chaos in a conservative classical system means that the dynamical
evolution of the system is no longer deterministic. Most classical systems with three
or more degrees of freedom are either fully chaotic or have fractal regions of chaos
distributed throughout the phase space. For example, the solar system appears to
have regions where planetary motion is chaotic. Classical models of molecular
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_1
1