We continue now examining further examples of statistical mechanical bifurcations, specifically a series of systems that achieved a nominal or temporary state of order out of a disordered and highly chaotic background.
Some investigations- using computer simulations, disclose interesting properties associated with what we term ‘strange attractors’. For example, two merging regions of plasma that can interact over a large distance to form one single magnetic configuration. Leaving out all the details of the computational plasma physics, a strange attractor for such a budding, complex sunspot field might appear as depicted in Fig. 1a, with the line axes representing spatial (x, y, z) coordinates.
A sketch of the merging fields, as they might appear confined to two dimensions, including field lines and directions as well as the separatrix, is shown in Fig. 1b. The separatrix is analogous to an equilibrium position or barrier between the two contrasting fields. The region with counterclockwise oriented field lines on the left might be called ‘basin of attraction 1’ and the counterpart on the right ‘basin of attraction 2’. At the position where the competing fields are precisely neutralized lies the ‘neutral point’.[1]
The actual matching sunspot region might appear as in Fig. 2 (which I photographed using a Cassegrain telescope in November, 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.
A much more elaborate diagram, associated with many generically complex physical processes, from polymer growth and collapse, to origin of solar flare conditions and filaments near coronal loops-arches is plotted with control parameter 'lambda' (Greek letter) along the horizontal axis, and some state variable 'Zeta' along the vertical, is shown in Fig. 3.
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 the basic bifurcation diagram of the previous blog for this topic ('Examples of Statistical Mechanical Bifurcations' (1)) .
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. More intriguing, if one magnifies portions of this chaotic domain, small regions displaying self-similarity appear. For example, exhibiting the sort of period 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 become 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. 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 summarized in Fig. 4. 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, Zeta. Longer lengths prevailed because the difference in free energy was heavily weighted in that direction with entropy taken into account.
In the model depicted, an iterative mapping was set up with z’ (new value) = z^2 + k, where z = x + iy, and k = a +bi, with i= [(-1)]^1/2. A conformal mapping was then performed where w = f(x + iy). The entropy S/k = l ln (z), obtained for smaller lengths (l << L) and S/k = (L+ l) ln (z) for larger. In general, -d F/ d(kT) = S/k where (dF) is the free energy difference. We can write, therefore: F2 – F1 = - kT[(L + l)ln (z)- lln (z)] = -kT L ln (z), which is the free energy difference between polymers of length L and those of lengths L + l.
This sort of phenomenon, trivialized here in a simulated depiction, can apply in a real world sense to the alpha-beta transition in fibrous proteins or the helix random coil transition evident in solutions of nucleic acids and proteins. I suspect it can also apply at some level to a polymerization process leading to formation of protenoids which themselves are the likely precursors of coacervate microspheres, the most logical original life on Earth.
What does all this mean? Only that one needn’t postulate a supernatural agent to account for the apparent "order" of the cosmos. (And let us please bear in mind this order is not large in any way or pervasive, but confined to perhaps 0.000000000002% of the matter component of the universe, which itself is only 4% of the whole!) Whatever limited order emerges, therefore, can be explained entirely within the material-physical system, in terms of fields (electric, gravitational, magnetic) as well as matter per se.
[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.
Some investigations- using computer simulations, disclose interesting properties associated with what we term ‘strange attractors’. For example, two merging regions of plasma that can interact over a large distance to form one single magnetic configuration. Leaving out all the details of the computational plasma physics, a strange attractor for such a budding, complex sunspot field might appear as depicted in Fig. 1a, with the line axes representing spatial (x, y, z) coordinates.
A sketch of the merging fields, as they might appear confined to two dimensions, including field lines and directions as well as the separatrix, is shown in Fig. 1b. The separatrix is analogous to an equilibrium position or barrier between the two contrasting fields. The region with counterclockwise oriented field lines on the left might be called ‘basin of attraction 1’ and the counterpart on the right ‘basin of attraction 2’. At the position where the competing fields are precisely neutralized lies the ‘neutral point’.[1]
The actual matching sunspot region might appear as in Fig. 2 (which I photographed using a Cassegrain telescope in November, 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.
A much more elaborate diagram, associated with many generically complex physical processes, from polymer growth and collapse, to origin of solar flare conditions and filaments near coronal loops-arches is plotted with control parameter 'lambda' (Greek letter) along the horizontal axis, and some state variable 'Zeta' along the vertical, is shown in Fig. 3.
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 the basic bifurcation diagram of the previous blog for this topic ('Examples of Statistical Mechanical Bifurcations' (1)) .
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. More intriguing, if one magnifies portions of this chaotic domain, small regions displaying self-similarity appear. For example, exhibiting the sort of period 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 become 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. 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 summarized in Fig. 4. 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, Zeta. Longer lengths prevailed because the difference in free energy was heavily weighted in that direction with entropy taken into account.
In the model depicted, an iterative mapping was set up with z’ (new value) = z^2 + k, where z = x + iy, and k = a +bi, with i= [(-1)]^1/2. A conformal mapping was then performed where w = f(x + iy). The entropy S/k = l ln (z), obtained for smaller lengths (l << L) and S/k = (L+ l) ln (z) for larger. In general, -d F/ d(kT) = S/k where (dF) is the free energy difference. We can write, therefore: F2 – F1 = - kT[(L + l)ln (z)- lln (z)] = -kT L ln (z), which is the free energy difference between polymers of length L and those of lengths L + l.
This sort of phenomenon, trivialized here in a simulated depiction, can apply in a real world sense to the alpha-beta transition in fibrous proteins or the helix random coil transition evident in solutions of nucleic acids and proteins. I suspect it can also apply at some level to a polymerization process leading to formation of protenoids which themselves are the likely precursors of coacervate microspheres, the most logical original life on Earth.
What does all this mean? Only that one needn’t postulate a supernatural agent to account for the apparent "order" of the cosmos. (And let us please bear in mind this order is not large in any way or pervasive, but confined to perhaps 0.000000000002% of the matter component of the universe, which itself is only 4% of the whole!) Whatever limited order emerges, therefore, can be explained entirely within the material-physical system, in terms of fields (electric, gravitational, magnetic) as well as matter per se.
[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.
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