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Sunday, October 3, 2010

Putting on my Top Hat

I ran across an intriguing abstract the other day that ties into Thom's topological catastrophe surfaces.  But alas I cannot get hold of the whole article. 

First Things First
Rene Thom developed catastrophe theory in algebraic topology many years ago.  Like so-called "chaos" theory, it is somewhat mis-named.  In French, apparently, "catastrophe" means only a sudden, discontinuous change, as when a stretched rubber band stops growing longer and snaps; or when a waterfall changes from laminar flow to turbulent flow.
The catastrophe surface is a manifold in response (or "state") space that consists of all the equilibrium points of the response variables over all values of the parameter variables.  The are the "attractors" toward which a system will move in systems governed by a potential function; and thus they function as a type of Aristotelian "final cause."  



By Thom's theory, there are only seven canonical "shapes" that the surface of equilibrium points can take on.  (All others consist of a combination of these.)  For example: when there are two phase variables [parameters] and one state variable [response] in a system governed by a potential function, the set of all equilibria of Y over all values of X1 and X2 forms a surface called a "cusp" as pictured to the left. 

For certain parameter values (X1,X2) there are two distinct equilibrium points.  The set of such parameters is called the "bifurcation set."  In the picture, this is the funnel shape on the "floor" of the space.  This is "parameter space" or the "control surface." 

When the system is outside the bifurcation set, it has only one possible value and will proceed rapidly toward that value.  Hence, the term "attractor."  Suppose the system state is on the upper sheet forward-left, about where the differential sign is written. 

Now suppose that the parameters change so that the system state moves to the left along the surface.  The parameters enter the bifurcation set and the system suddenly acquires a second equilibrium state.  It stays however on the upper sheet where the "twist" or "pleat" is. 

Then, when the parameters continue to move left, the system leaves the bifurcation set and suddenly there is only one equilibrium point -- and it's on the lower fold of the sheet!  The system rapidly snaps to the lower sheet like dropping off a cliff.  (All the points between are unstable, so the system does not linger.)  Thus, there is a "quantum" leap in the system state. 

Higher level catastrophes, like three Xs and two Ys, too highly dimensioned to draw.  But this model has been used to model flight-or-fight reactions, economic crashes, Pournelle's two-dimensional political scale, corrosion of metal, and a variety of different things. 

Which brings us to....

The Fifth Element. 

Here is the Abstract I ran across recently: 

Using Arnold's Classification Theorem applied to a four-dimensional manifold, it is shown that there is only a finite number of ways in which energy can discontinuously change state. It is demonstrated that each of these energy flow pathways can be associated with a distinct elementary particle. The theory not only shows how the formation of particles from the stress-energy present in the space-time manifold can be predicted from first principles, but also that there must exist five fundamental forces in a universe in which discontinuous energy transitions are possible. Finally, the existence of a new, as yet undiscovered particle is predicted, which is associated with this new fifth force.
"Use of catastrophe theory to obtain a fundamental understanding of elementary particle stability,"
International Journal of Theoretical Physics, Volume 25, Number 7, 711-715


The "four-dimensional manifold" in question is the space-time manifold and "a universe in which discontinuous energy transitions are possible" is one in which quantum mechanics works.  That would seem to mean our own. 

The prediction is that there is another force that goes along with the nicely symmetric group:

.............................Weak......................Strong
Long Distance:  Gravity....................Electromagnetism
Short Distance:  Weak force............Nuclear [Strong force]
One wonders how a fifth force fits in?  There is grave skepticism regarding the existence of "dark energy," and it may or may not be the one predicted by the topology of the universe.  Don't know without the full article -- and probably not even then!  

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