I believe things have gotten more than a little side-tracked by the discussion of the "N-body Problem", such that the more important issuess receive less attention. An example of such an issue would be the state of research within the physical/natural sciences and the life sciences, and in particular whether this research supports a view of reality that is ontologically reductionist and deterministic.
My hope is that if I can suitably address the "N-body" problem, I wouldn't do what I did above: devote at least half my response to that issue, and then having spent so much time on it, fail to suitably respond to the issues directly related to the possibility of conscious, self-determining systems. With that in mind:
Here is the initial request which began the N-body discussion component of this thread:
Legion, can you explain how functional processes happens in a simpler example? Such as the n-body gravitational problem?
(because reading thousands of words either way is getting tiresome, and I suspect it's not actually getting us anywhere useful.)
(the second statement after the request could not be more true, but is also rather ironic)
This above request was followed by further clarification about what was meant and why it was relevant:
Assuming that there is no radioactive atoms involved, all of a cell's workings are essentially an n-body problem in QED. (The two interatomic forces are too small to notice, and gravity is too large to notice.)
Specifically, I meant the example of modelling n particles obeying an inverse-square law between each other
Now that both the request and clarifications are organized in a single place, I hope to construct a single organized response.
(from
N-Body Problems And Models (World Scientific, 2004) by Donald Greenspan, p. xi under the section title "Problem Statement")
The above description, and indeed the N-body problem itself, is an extension of Newtonian physics and the problems Newton and other had understanding the orbits of the planets around the sun. However, it can be useful with objects in general:
Imagine that a remote control car strikes me at 30kmh. it might hurt, but it's nothing compared to the effect of a truck hitting me at the same speed. In other words, "force" is related to mass. However, for Newton, "mass" was a whole different ball game because he was dealing with planets. So for him, things like mass and velocity were complicated by the
amount of mass, because mass means gravitational force, and planets have enough mass for this to really matter.
However, changed around a bit, an"n-body" problem can becomes much more interesting,
even when n is 1: Imagine a pendulum hanging from center mass, perpendicular to but above the ground. Let's also say that we know the mass of the pendulum, it's center, how long it is, the angles we create (relative an imagined set of right angles on either side) when we raise it, and that we simply release it.
Knowing all this, applying an isolated system w/o friction, and armed with Newton's F=
ma, this "1-body" problem would appear easy on to model, i.e., calculate the future motion.
As it turns out,
this "one-body" problem has no (known) general solution method. It's the simplest type of nonlinear systems problem: a 1d oscillating system.
Notice that we haven't had to deal with thermodynamics, quantum processes, molecular reactions, fields, and most of basic phsyics. What happens when we do?
I mentioned in other posts the (thankfully increasingly addressed) problem within the life sciences when it comes to physics. The inadequacy of even the extremely complex, sophisticated,computational approaches used in molecular and chemical physics are typically inadequate. However, as usual, there are exceptions to the general rule: Dr. Hans Frauenfelder was a pioneer in the field of biological physics, and the volume
The Physics of Proteins: An Introduction to Biological Physics and Molecular Biophysics ( from the series
Biological and Medical Physics, Biomedical Engineering; Springer, 2010) was put together by the editors in an attempt not just for use by students & researchers, but also to "capture" Fauenfelder's ingenuity.
However, as is typical of such volumes, some material is work of other specialists to fill certain gaps. So chap. 3 is R. H. Austin's "Biomoecules, Spin Glasses, Glasses, and Solids", which begins by comparing the complexity of
"a few many-body systems", to biological systems. For exampe, Austin notes "Solids or glasses cannot be modified on an atomic or molecular scale at a particular point: modifications are either periodic or random. In contrast, a protein can be changed at any desired place at the molecular level: Through genetic engineering, the primary sequence is modified at the desired location, and this modification leads to the corresponding change in the protein." (p. 14). The various molecular dynamics Austin compares to biomolecular dynamics are qualitatively different.
Thus, for example, "the energy surface of a crystal, that is, its energy as a function of its conformation, is nondegenerate and has a single minimum ...
We can, in principle, describe the conformation of a system by giving the coordinates of all atoms. The energy hypersurface then is a function of all of these coordinates"
This does not mean these many-body systems are simple to reduce in order to understand the emergent structure through its constituents. One barrier is the phenomenon (aptly) termed "frustration":
"Frustration generally implies that there is no global ground energy state but rather a large number of nearly isoenergetic states separated by large energy barriers. In other words: imagine that you take a large interacting system and split it into two parts. Minimize the energy of the two separate parts. Now bring the two parts back together. A frustrated system will not be at the global energy minimum because of the interactions across the surface of the two systems.
There are a number of physical consequences that arise from frustration. The primary one that we wish to stress here is
the presence of a dense multitude of nearly isoenergetic states separated by a distribution of energy barriers between the states. This is due to the presence of
frustrated loops that cannot be easily broken by any simple symmetry transformation. This complexity inherently leads to distributions of relaxation times." (italics in original; emphasis added)
However, as complex as things like spin glass dynamics can be, they are nothing compared to biomolecules. Biomolecules "are complex many-body systems. Their size indicates that they lie at the border
between classical and quantum systems. Since motion is essential for their biological function,
collective phenomena play an important role. Moreover, we can expect that many of the features involve nonlinear processes.
Function, from storing information, energy, charge, and matter, to transport and catalysis, is an integral characteristic of biomolecules. The physics of biolomecules stands now where nuclear, particle, and condensed mattter physics were around 1930." (emphasis added).
We can further illustrate the difficulties biomolecular processes present by one found in the book: ligand binding to hemoglobin. Hill, who attempted to simply the process for modelling, "introduced in 1913 a hypothetical nonlinear equation that is
unphysical in that it assumes an n-body interaction...it implies that
n ligands
compete simultaneously for one binding site" (p. 87). Using the ol' n-body approach failed miserably. The model had no "physical interpretation" but rather was "a way to parameterize the data we dont understand in terms of the basic physics of proteins" (p. 89).
The reductionist approach (including the extended "n-body" of the early 20th century) which succeeded in the case of some nonbiological chemical reactions, stands in stark contrast with biomolecules, as "biomolecules provide a complex but highly organized environment that can affect the course of the reaction" (p. 125) rendering the reduction of the reaction process to its constitutents ineffective.
To return again to this:
Legion, can you explain how functional processes happens in a simpler example? Such as the n-body gravitational problem?
The answer is (again) "
no", but this time with an example of just such an attempt in the early 20th century, before it was widely recognized that the extension of 17th century methods presented problems in general, and were replaced or incorporated into the dynamical systems approach. Additionally, while the complex, nonlinear dynamics of molecules present more than there share of challenges, they are qualitatively different from the complexity inherent in biomolecular processes, structures, and functioning.
"n-body" isn't used not simply because it has been replaced by "many-body" problems. The pioneers of "chaos theory" incorporated equations of motion into their models to start with, and it was the suprising complexity of these equations to work even in "1-body" cases like the pendulum which rendered so thoroughly irrelevent the "n-body problem", even the extended form (which was not the "n-body gravitational problem").
