Introduction to Quantum Chaos
Classical chaos:
Not a theory in the fundamental sense, a unifying collection of
concepts, in fact a dominant theme of classical mechanics that
was missed for about 200 years!
Non-linear dynamics, nearly always impossible to
obtain exact solutions analytically, computer
simulations are important (and have driven the field).
Quantum chaos:
Is there any? Short answer is no. It’s a misleading name!
Classical Hamiltonian chaos rests on the notion of trajectory,
and the fine structure of phase space; concepts that have no
place in quantum mechanics.
What is the nature of a quantum system for which the
corresponding classical system is chaotic?
Classical chaos
Example: double pendulum.
• Only one more degree of freedom than the
standard periodic pendulum.
• Unpredictable trajectory and sensitivity to initial
conditions.
• Complex non-predictable behavior from simple
systems: deterministic chaos.
Why are some systems chaotic? What is the
origin of chaotic behavior? Why the sensitivity to
initial conditions? Are chaotic systems in some
(statistical) way as predictable and controllable as
integrable systems.
Classical chaos
Origins of chaos theory can be found in the end of the
19th centuary with Poincare and others who tried to answer
the basic question: is the solar system stable?
3-body problem admits no
closed form solution.
Perturbation theory in general
is expected to diverge (though
is very successful when it can
be controlled).
In fact, signatures of
chaos can be found in the
solar system; e.g.
asteroid distribution.
Chaos in other contexts
Chaotic behavior is found in many
contexts which can be modeled
by a non-linear system of some
kind.
Since most of nature is non-linear,
chaos is everywhere. (However,
question: How to distinguish chaos
from noise?)
Quantum chaos
Quantum scarring.
Localization.
Universal forms of the distribution function of eigenvalue spacing.
Outline of the course
Classical chaos:
Lecture 1: Introduction, logistic map.
Lecture 2-3: Renormalization in the logistic map, kicked rotator
and the standard map, introduction to Hamiltonian chaos and
the KAM theorem.
Quantum chaos:
Rough order of topics; division into lectures not decided!
The kicked quantum rotator.
Quantum cat map.
Universal features of quantum chaos.
Gutzwiller trace formulae and quantum scarring.
Format of lectures: Generally most of the pictures on slides, and most of the
formulas on the blackboard.
Books, other course resources
Many books can be found in library, both quantum and classical
chaos: dig in!
Very good online book: http://chaosbook.org/ by one of the
pioneers of the field.
Popular science books abound, e.g. Ian Stewart “Does god
play dice”, and James Gleik “Chaos” (ignores all the Russian
contributions to non-linear dynamics and chaos!).
Myself: tend to be found in the office most weekday afternoons
during semester.
Completing the course
Two ways to do this:
1) Attend lectures and “problems classes”.
2) Attend lectures and do mini-project at the end
of the course (I prefer this one).
Logistic map
Many features of non-Hamiltonian chaos can be seen in this simple map
(and other similar one dimensional maps).
Why? Universality.
Period doubling
Intermittency
Crisis
Ergodicity
Strange attractors
Periodic/Aperiodic mix
Topological Cantor set
Time series of the logistic map
Superstable points; convergence to attractor very rapid
Critical slowing down near a bifurcation; convergence very slow
Time series of the logistic map
Superstable points; convergence to attractor very rapid
Critical slowing down near a bifurcation; convergence very slow.
Time series of the logistic map
Bifurcation to a two cycle attractor.
Time series of the logistic map
A series of period doublings of the attractor occur for increasing values
of r, the “biotic potential” as it is sometimes known for the logistic map.
Time series of the logistic map
At a critical value of r the dynamics become aperiodic. However, note that
initially close trajectories remain close. Dynamics not ergodic.
Time series of the logistic map
At r = 4 one finds aperiodic motion, in which initially close trajectories
exponentially diverge.
Liapunov exponents
Liapunov exponents
Self similarity, intermittency, “crisis”
The period doubling regime
• Derivative identical for all points in cycle.
• All points become unstable simultaneously; period doubling topology can
only be that shown above.
Period doubling in experiment
First experimental evidence was via convection currents in mercury, so
called Bernard experiment, many others followed e.g. nonlinear driven
RCL-oscillator, etc.
Period doubling constants (very) close to those of the logistic map!
Other one dimensional maps
Other one dimensional maps show a rather similar structure; LHS
is the sine map and RHS one that I made up randomly.
In fact the structure of the periodic cycles with r is universal.
Universality
2nd order phase transitions: details, i.e. interaction form, does not matter
near the phase transition.
Depend only on (e.g.) symmetry of order parameter, dimensionality,
range of interaction
Can be calculate from simple models that share these feature with
(complicated) reality.
Renormalisation
Change coordinates such that superstable
point is at origin
Renormalisation
Renormalisation
Renormalisation
Summary
Logistic map shows many features of chaotic driven dissipative systems:
• Fixed point analysis of 1d map.
• Period doubling regimes with both qualitative and quantitative universality.
• In the chaotic regime commingling of unstable periodic orbits and chaotic
orbits.
• Strange attractors: fractal structure (topological Cantor set), ergodic, but
nearby trajectories exponentially diverge.
Next week: Renormalization for the logistic map,
and start of the kicked rotator (shares many features
with Hamiltonian chaos).
Lorenz attractor