Sunday, February 7, 2010

Concepts of Order & Disorder III: Spontaneous Order

Figure 1(below): A large sunspot group shows spontaneous organization from a background of chaotic plasma.

We left off in the 2nd instalment of this series looking at the basis for how a separatrix - analogous to an equilibrium position or barrier between the two contrasting magnetic fields- can spontaneously form on the Sun if conditions are right. At the position where the competing fields are precisely neutralized lies the ‘neutral point’.[1] The actual matching sunspot region might appear as in Figure 1.

In fact, I took the above photograph using a Schmidt-Cassegrain telescope on Nov. 4, 1980. What it clearly shows is that self-organized and magnetically complex plasma systems can arise out of a background of chaos. Plasma itself is chaotic with ions, currents interacting in many random variations. But magnetic fields can order these.

An example closer to Earth is the aurora, such as I observed near Chena Hot Springs, Alaska in March of 2005. This particular aurora displayed two perfectly symmetrical parallel green “tubes”, arcing from north to south horizon. Did an "intelligent designer" craft two natural fluorescent tubes in the sky? Not at all. The inimitable procession to order (observed over two hours) was dictated by the (pre-existing) presence of the auroral oval around the pole and the polar electro-jet, after impinging electrons from the solar wind began to decelerate into the oval and form currents in sheets. These were then shaped by the ambient magnetic field of Earth into the two parallel tubes visible near Chena. Order out of chaos!

A much more elaborate bifurcation diagram is shown in Fig. 2 . One might correctly refer to it as depicting multiple bifurcations but each characterized by different periods. Thus, the sort of doubling in the first (left) portion of the diagram is similar to that shown in Fig. 2 of Part II. Beyond that, however, we now see at least two more additional doublings of stable solutions – each displaying bifurcation from the one preceding it. Beyond the obvious bifurcations lies a chaotic region, mostly grey. However, a few successive bands of ‘order’ emerge within it against the chaotic background bifurcation, associated with many complex physical processes, from polymer growth and collapse, to origin of solar flare conditions in solar coronal loops-arches. In Figure 2, the diagram is plotted with control parameter lambda along the horizontal axis, and some state variable zeta along the vertical.

More intriguing, if one magnifies portions of this chaotic domain, small regions displaying self-similarity appear. For example, exhibiting the sort of periodic splitting visible in the larger map. One may rightly conjecture that whatever systems reside within these bands, say persistent solar flare regions[2], or a replicating proto-cell, it emerged from chaos to exhibit self-organization.

Many other examples abound: a normally functioning cell suddenly becomes malignant; the molecules/particles of a liquid, initially with random arrangements, suddenly assume an orderly, lattice-type of structure when the liquid freezes (e.g. when water turns to ice); elementary atomic ‘magnets’ originally distributed randomly, suddenly oriented in the same basic direction - creating magnetism in a ferrous material. In each instance, the system has undergone a transition from a more disordered state, to a more ordered one. Bifurcation has occurred, setting the evolution of the system on a fundamentally different path from what it was earlier.

In the evolutionary sphere, specific combinations of amino acids probably contributed to system state change leading to a pre-biotic cell or protenoid.[3] One could view the transition from non-reproductive- non-growth to replicating-growing states as a ‘symmetry breaking’ in the organic molecules that yield a very primitive living cell.[4] Once formed, the cell possesses all the attributes of life including reproduction. At this stage, replication and further evolution can occur. In one particular simulation, using macromolecules with specified monomers of a given initial size, I obtained the results as shown in Fig. 3.

The tendency observed was for smaller length units to evolve to greater length polymers, as if the longer length had been preordained by selection. In the (Juliabrot fractal) model I used, entropy was expressed as a function of length and some partition function z. Longer lengths prevailed because the difference in free energy was heavily weighted in that direction with entropy taken into account.

Aside: In the model depicted in Fig. 3, an iterative mapping was set up with z’ (new value) = z^2 + k, where z = x + iy, and k = a +bi, with i= sqrt(-1). A conformal mapping was then performed where w = f(x + iy). The entropy S/k = l ln (zeta),for smaller lengths l << k. Also for S/k = (L+ l) ln (zeta) for larger.

The point is that this order emerges entirely within a fabric of random processes that are embedded within the physical laws of the cosmos. Supernatural and externally postulated agents are rendered totally redundant. We no longer require them in terms of necessary or sufficient explanations of anything, including the origin of the universe. We can leave them behind much as we've left behind the use of wasps to sting disease "humours" out of existence, having now replaced those wasps with vaccines.

[1] In actual sunspot regions one is much more likely to trace the path of the ‘neutral line’, obtained from solar magnetograms.

[2] Indeed, one of the best examples is the dynamic spectrum of a large solar flare in which differing radiofrequency bursts- called Type I, II, III, IV appear. Amidst the background of the spectrum, the burst regions define what can be regarded as domains of ‘order’.

[3] Protein-like polymers formed spontaneously by heating dry mixtures of amino acids at temperatures over 150° C. The products, also called thermal proteins, range in molecular weight from 4,000 to 10,000 daltons.

[4] This is considered in terms of using what are called spin glass models. See e.g. Bar-Yam, Yaneer: 1997, Dynamics of Complex Systems, Addison-Wesley, pp. 466-68. The point is, a critical polymerization threshold is crossed to weight the outcome in terms of more self organized systems. Hence, there is a definite bias underlying the way amino acids polymerize to form peptides and protenoids. The emergence of this bias is what I mean by ‘symmetry breaking’, i.e. displaced from the background of an equalized distribution or diffusion of products with wide ranging heterogeneity.

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