To be clear about the difference between
determinism and
chaos in physics it helps to
have the idea of a
dynamical
system for it is in these
terms that both chaos and determinism can be easily
defined.
A dynamical system is a system in motion
(a ball rolling, a star evolving, a sound wave
propagating I'm sure you get the idea). Its
motion (the systems)
is governed by a set of equations
(in the former examples these could be respectively
Newton's equations of motion ;
The diffusion equation
coupled to equations describing nuclear burning
and
hydrodynamic equations {stars are complicated};
hydrodynamic equations)
Just think of the state of the system as being
a position in some "state space" or some "phase space"
The system moves around according to the equations.
Determinism:
If you can use the equations of motion to calculate
where the system will be in the state space given
as information only where the system currently is then
the system is deterministic.
Determinism can fail in two ways, in
practice and in principle.
All systems we know fail practically
because we are unable to measure exactly where in the
state space a system is. This is because of errors in measurements and the
large numbers of degrees of freedom
involved in even moderately simple systems.
Think of trying to keep track of all of the particles of
air in a room in which you wanted to calculate
the speed of sound. Pierre LaPlace a
French mathematician once said that
if he knew the positions and velocities of all of the particles in
the universe he could know the future by using
Newtons laws of motion. In practice he could never
gather all of this information but determinism can also
fail in principle.
In Quantum theory the
uncertainty principle sets a limit
on the amount of information that we can have about any system irrespective of
our ability to measure the system.
Due to this principle the world is not deterministic.
(this is a point that many philosophers seem
to have allowed to slip them by in since 1923(4)?,
as it makes a discussion of free will
of academic interest only (but then again most philosophers
are academics and these are the questions that they should be interested in :) ))
Chaos is independent of determinism. The definition
of a Chaotic system is a system in which you
start with two positions in state space that are as
close together as you like
, as you apply the equations
of motion of the system the two paths in state
space diverge. They move away from each other.
Alternatively an stable system is one in which
adjacent points in state space remain together
as you iterate them in time with
the equations of motion.
One might call a system in which all points in state space
converge a pathologically stable system, however it is quite
common behavior as friction will tend to do this to a system,
but that leads one into a discussion of
thermodynamics.