Vortex pattern generated by flowing water
In
today’s scientific terminology, yet another relationship emerges between
chaos
The
theme of order has played an important role in the evolution of natural
sciences. Physical laws are a manifestation of this order, and the belief
that such laws exist for all systems, no matter how complex, has been an
abiding article of faith among scientists. The search for order has
inevitably led to an obsession with domains where it is seemingly absent,
and which consequently are intrinsically chaotic. The result is that the
word chaos has acquired a well-defined meaning in physics. Ironically
enough, certain systems which are chaotic by this definition have recently
been found to be not so chaotic after all.
The
first serious encounter with systems which seemed to be totally chaotic
occurred when physicists picked up enough courage to examine systems with
a very large
Although
the motion of a single gas molecule could be kept track of, it was clear
Fractal
In all chaos there is cosmos
In all disorder a secret order.
Carl Jung
Let
us develop each of these notions in somewhat greater detail. Consider
first a ball moving inside a deep depression. If the ball is acted on by
no force other than that of friction (and gravity) it will eventually come
to rest at the bottom of the depression. This location is, therefore, an
attractor for the motion which is, however, not chaotic. It is, in fact,
normally referred to as simply a fixed-point. If the ball is now acted
upon by a small periodic force, it will eventually settle into
oscillations of a given amplitude. Its periodic motion at this point
constitutes yet another attractor, named, Limit Cycle. For certain values
of the force and for certain shapes of the well in which the particle is
moving, the motion in general becomes irregular. It is at this point that
we are said to have entered the chaotic regime.
It
turns out that in a number of cases this irregular motion actually has
certain regularities, which are easiest described if we keep track not
just of positions but velocities as well. This can be very conveniently
done by moving into an imagined space which has additional axes on which
we plot movement (which is proportional to the velocities). This space is
referred to as Phase Space. Now, in the chaotic regime, the system moves
along a trajectory which is continuous but winds on itself in such a way
that if we take a cross-section at right angles to the trajectory, we get
a fractal, that is, a set of points which constitute a pattern which is
such that if we magnify any part of it, we recover the original pattern.
The subset of Phase Space traced out by this trajectory has come to be
known as a Strange Attractor.
Another
aspect of order relates to the scale of time and distance over which it is
observable. Physicists have defined the full range of scales to be their
domain-from the Universe in its entirety to the shortest sub-nuclear
distances. It is fascinating that laws have been discovered on essentially
all length and time scales. The statement of these laws is relatively
straight forward when degrees of freedom on different scales do not
interact with each other. Recently, a fair amount of progress has been
made on understanding situations where many length-scales simultaneously
come in to play. These situations are not unlike periods of revolutionary
turmoil when macro and micro-structures in society begin to interact in an
intimate way. In physics, it was the recognition that scale-invariance
(similar to the one we see in fractals) characterized some of these
situations, which allowed some progress to be made. The systems and
circumstances discussed above are by no means exhaustive that they are
illustrative of the scientist’s search for order in the very heart of
chaos. This search will undoubtedly create new paradigms for exploration
and will draw upon ideas from a wide spectrum of disciplines.
Deshdeep Sahdev
