It was not until the 19th Century that solid evidence against this deterministic veiw was found. First of all, forces other than gravity (namely the electric and magnetic forces) had been discovered. Clearly there was more to understanding all of the forces acting on a particle than just understanding gravity. Further, Henri Poincare had shown that the behavior of even a small number of particles acting only under the force of gravity cannot be easily predicted. Poincare found that the motion of three or more particles under gravity can be quite complicated and although the motion is, in theory, exactly determined by Newton's Laws and the force of gravity, in practice it cannot be predicted over long times. The problem is that the motion of such a system depends very sensitively on the initial positions and velocities of the particles involved. If one could specify these initial conditions with infinite accuracy then one could predict all of the future motion of the system (provided one had an infinite amount of time in which to do the calculation). In practice, though, one cannot know these initial conditions to infinite accuracy and it turns out that even very tiny errors in these values will lead to inaccurate prediction of the motion of the system after relatively short times.
More recently it has been found that many systems behave this way. With accurate initial conditions one can accurately predict the future motion for short times, but one can never predict the long-time behavior of the system. A system that has this property of unpredictability is referred to as chaotic. Chaos is defined mathematically as a situation in which the distance between two particles that start off in close to the same place, with clost to the same velocity, increases exponentially with time. This property is referred to as "sensitive dependence on initial conditions" and it basically means that particles with nearly identical initial conditions can have drastically different motion as time goes on. This is the essence of chaos.
One way to distinguish between regular (non-chaotic) behavior and chaotic motion is by looking at a diagram called a Poincare surface of section (SOS). A SOS is a plot that shows the position (the x-coordinate of the plot) and velocity (the y-coordinate) of a particle at various times. This space that treats velocity as a coordinate along with position is called phase space. If the system has more than one spatial dimension then one must make a slice through the phase space (which has twice as many dimensions as the number of spatial dimensions: i.e. a 2D system has a 4-dimensional phase space) in order to create a 2D plot that can be printed on a page. In 1D systems that have forces that are periodic in time, one simply plots the position and velocity after each period. This type of SOS is called a strobe plot. Examples of strobe plots for a system with a mixed phase space (displaying both regular and chaotic behavior) are shown below. It is easy to distinguish between the regions of regular motion and the regions of chaotic motion. Regular motion appears as well-defined lines running through the phase space. Regions of chaotic motion, on the other hand, appear as a random jumble of points. Note that J denotes the velocity (or momentum) of the particle in the 1D system and the Greek letter Theta denotes the position.
The ellipse-shaped feature near J=20 in the plots is called a resonance. In a system where the forces are
periodic in time, a resonance is a region of phase space where the natural
period of motion in the system without the periodic forces is nearly equal
to some integer multiple of the period of the forces. These resonances
play and important role in the development of chaos in a system. In this
system, the parameter denoted by the Greek letter epsilon controls the
strength of the periodic forces. As the forces become stronger (epsilon is
increased) the system becomes more chaotic as can be seen in the strobe
plots shown here. This occurs because the system has several resonances.
In figure (a) there is a chain of three elliptical "islands" visible near
J=5. This is another resonance, in addition to the larger resonance near
J=16. As the forces become stronger these resonances get larger. Wherever
two resonances overlap a region of chaotic motion will be created. Thus,
as the strength of the periodic forces is increased all of the resonances
grow, overlap, and produce a large region of chaotic motion as seen in
figure (c). This is essentially how chaos comes into being in systems like
this. For other kinds of systems (e.g. systems with more than one spatial
dimension, or systems without periodic forces) the details are different
but the basic phenomenon is the same.
Now that you've been introduced to chaos in classical (i.e. Newtonian)
systems, it is time to take a look at the strange world of Quantum
Mechanics.
[Go To "What is Quantum Mechanics?"]
OR
[Go To Quantum Chaos Home]