INTEGRATING HIERARCHICAL STRUCTURES

There are several computational and theoretical approaches to chemical dynamics in the literature.

So far, most (all?) fail to be applicable to laboratory time scales. It would be very useful to make some progress toward solving that problem.

Also, “hierarchies” and “multi-scales” are commonly discussed.

We need to know how these terms fit our problem, in a precise way. We need, therefore, a fresh look, a controlling context, so that we can know what type of result we should be looking for, before pushing ahead into detailed calculations.

Definition: Systematic Process:

Other things being equal!

(1) The whole of a systematic process and its every event can be accounted for by a single set of correlations;

(2) Any situation can be deduced from any other

without an explicit consideration of intervening situations; and

(3) The empirical investigation of such processes is marked not only by a notable facility in ascertaining and checking abundant and significant data but also by reaching a stage where all data fall into a single perspective, sweeping deductions become possible, and subsequent exact predictions regularly are fulfilled.

Examples of systematic and non-systematic processes

Planetary motion is approximately systematic. Related to planetary - a well engineered pendulum. Note that if an arbitrarily small periodic forcing term is

introduced into even the simple ODE for an ideal pendulum, the system becomes chaotic, and therefore non-systematic.

Molecules have vibrational energy frequencies. It follows that we cannot reasonably expect individual

or aggregates of molecular dynamics to be systematic.

Consequently, molecular dynamics will normatively require stochastic methods.

More Examples of Non-systematic processes

For N = 1, 2, 3, … billiard balls on a pool table, by all accounts, constitute a non-systematic process.

Note that the definition allows for the possibility that a non-systematic process be deducible in all of its events. In the pool table example, accurate prediction is possible, but on a step by step basis.

One may calculate as many steps as one pleases, but there is no way to sweep forward across the time variable to deduce locations and momenta at arbitrary later times.

1N

How to Study Non-Systematic Process

We suppose well-defined and identifiable individual events.

But, in a non-systematic process we cannot make long term predictions.

What is left? What do scientists do with processes such as: sequences of flips of a coin; the weather; genetic combinations in progeny; birth and death rates in a population, etc, etc?

Even if we cannot predict long term results for individuals, we can still count events!

In other words, statistical method is the natural way to investigate non-systematic processes.

Statistical Method

Does not seek to define events, but seeks ideal relative frequencies of already defined events; ignores differences, as long as they are random differences; and seeks instead to determine prevailing or central trends in sufficiently large sample sets data.

Statistical method by definition of non-systematic process must always allow for random differences.

Corollary: Statistical method does not apply directly to an individual event, and cannot be verified in any single event, but seeks to determine/analyze ideal distributions that can be verified in representative samples.

HIERACHIES AND COINCIDENTAL AGGREGATES IN MATHEMATICS

Divisor classes: Evens and Odds; clock arithmetic; days of the week arithmetic. Within each context, there is a pattern that governs and takes hold of arithmetic potential, and pulls it into to the patterns of the particular division class.

There is no breaking of rules of arithmetic. Rather, those rules are used within the specified context. Moreover, the patterns that emerge are not derivable from arithmetic alone, but are enforced by the rules of the division class. So, with respect to the rules of arithmetic, the patterns of the division class are merely “coincidental”.

Using logical operations alone, there is no transition from one system to the other.

Group structure explains patterns

It can be discovered that division classes can be combined, associated to each other, one can be a “subset” of another, and so on. With respect to the rules of the various division classes, however, this regular behavior between and among division classes can only be regarded as mere patterns of happy coincidence. There are unities that are evident, but are not explained merely by the rules of arithmetic, and the rules of division classes.

However, examining the various division classes, one may reach the definition of “finite group” (commutative). Within the context of finite group theory, we can work out the entire theory of homomorphisms between groups, Cartesian products, and so on, and end up with the complete theory of finite abelian groups. Within this context, what in division classes were mere happy patterns of coincidence, become explained patterns.

Two orders of logical discourse

Division class rules are not broken, but are subsumed within the higher context of group theory.

Using logical operations alone, there is no transition from one system to the other, neither upwards to groups, nor downwards to division classes.

Same type of transition in empirical domain

In an analogous way, the regular behavior of subatomic elements are mere happy patterns of coincidence from the perspective of physical laws.

The physical laws of subatomic elements do not provide the systematic unification and explanation of the patterns of atomic combinations and chemical processes. This control becomes possible through and within the higher context established through the discovery of the periodic table.

So, again, we have coincidental aggregates from a lower perspective, but explanation and unification within the higher context.

And there is no transition between the orders of discourse, on logic alone.

Multi-scales

This leads to multi-scales: Note that multi-scale in this sense is not necessarily “larger” or “smaller” in time. But, the measurement scales will in most cases be different. For, the higher principle organizes a coincidental aggregate of lower events, which has its own measurement scale. Therefore, the aggregate of events could conceivably require a larger scale.

The normative distinction (magnitudes aside) regards the fact that that measurements for chemical events proper belong to a different logical order from the lower structures and processes.

Scope of a multi-scale approach

Using a knowledge-based approach for a chemical reaction:

One may use physical laws to compute/ determine probabilities of physical events.

Calculations will necessarily be stochastic and under tight assumptions of “other things being equal”.

In particular, physical computations require that time scales be based on physical laws.

Physical time scale is not chemical time scale.

Hierarchical probabilities

Under controlled circumstances, and given chemical configurations, there will be probabilities for physical events.

Given representative samples of events from the physical submanifold, there will be probabilities for the chemical events.

Put these two sets of probabilities together. Note: So far in the literature, there is some

evidence for this possibility, but not yet done in a controlled and clearly defined way.

These ideas lead to the following:

NORMATIVE INTEGRAL STATISTICAL STRCUTURES AND CONDITIONAL PROBABILITIES.

OVERVIEW Chemical probabilities Physical probabilities Combine chemical and physical probabilities

via conditional probabilities. Prob(Chem event given conditions C) =

Prob(Chem event given physical conditions) X Prob (physical event given C)

For the chemical probabilities

Make use of results “like” Gibson and Bruck, 2000. Efficient exact stochastic simulation of chemical systems ….”. There may be other similar articles. Caution though, there are some (fixable) errors in this article.

For the physical probabilities

So far in the literature, computations are bound to physical non-lab time scales. Also, in many cases the probabilities are not verifiable and therefore not admissible. [See also Nölting’s reference (Ch. 1 of his book) to known results that using only the laws of physics and supposing a random walk leads to inapplicable time scale results.]

For the physical probabilities, we need to lift to chemical lab time and whatever else, we need to get verifiable probabilities.

What follows in the next few pages is a specific strategy for possibly making this work at the physical level.

If not this idea, then something “like” it. That is, we will need to get lab time verifiable results and fit them together in the integral-hierarchical structures.

One way to possibly reach lab time statistical results: Bridge potential lines and use computational and/or Monte Carlo techniques?

Use “knowledge-based approach”, that is, invoke what we can from the higher organizing chemical structure:

To assume that the physical configuration is suitable to make the chemical reaction possible. One might assume, for example, that certain “hot spots” are suitably aligned.

As we have discussed, construct a pseudo-potential function (or functions? or stochastically time varying potentials?), under the hypothesis that other things remain approximately equal. This type of hyp. is crucial.

Now Partition the space using equal potential curves. Use an iterative technique as follows:

A possible computation scheme?

Start at equi-potential curve C1

This determines a set of boundary conditions. Recall, that using physics, we can only get

probabilities from computations in the lower order. For physical quantities, use super computing, parallel

computing, etc and possibly Monte Carlo simulation (quantum techniques? -- comments later), or some such, to compute the outcomes of short term highest probability. This will give a vector V1 (or possibly a family of equally probable vectors). Select one and propagate vector to the next curve C2 of equi-potential.

Take the result as the new set of boundary conditions.

1γ

Now Iterate

Between potential lines, computations are confined to non-chemical time scales.

Computations in the literature seem to be within this non-verifiable domain.

However, if propagated (joined together by the artifice of linking outcomes of highest probability), we could obtain “high probability outcomes trajectories”, in lab chemical time scales.

Scheme Cont’d

The high probability outcomes trajectory would be of the form V1V2 V3 …Vn.

Note: Without some approach that reaches beyond or across physics time scale boundaries to chemical time scales, the physical computations will necessarily be restricted to non-verifiable time scales of order (1/10)12 or so.

Therefore, if this particular scheme is not going to work, we need something “like” it. That is, use some other partition that produces chemical lab time scale.

Quantum – Chemistry?

Note that quantum chemistry techniques have been found to be not practical for analysis of complex molecules and their dynamics.

(There are several references that point to this difficulty. Two are: A. Neumaier. 1997. SIAM Review article: Molecular modeling of proteins…, SIAM Rev., 39, 1997, 407 – 460. See also M. Gibson and J. Bruck. 2000. Efficient exact stochastic simulation of chemical systems …., J. Phys. Chem., 104, 1876 – 1889, p. 1877. Many others.)

Why is Quantum Chemistry not useful for this sort of problem?

Quantum chemistry is actually quantum physics applied to the atomic constellations of chemical elements. So, we using a lower order logic to study higher order system.

Quantum physics looks to energy states rather than geometric dynamics as such. There are, therefore, many degrees of freedom and a vast range of possibilities to be accounted for.

On present showing, the limited application of quantum chemistry to complex molecular interactions is also because, as the chemical process proceeds in chemical time, at the physics level, “things do not remain equal”, and so quantum predictions need to be too broad for practical applications to be possible for lab time.

Appendix: An ODE can be used as the basis of a statistical) non-deterministic theory

See, for example, B. Nölting. 1999 and 2006. Protein Folding Kinetics, Biophysical Methods, 2nd ed., Springer, Berlin. ] (see, e.g., pp. 146 – 147) See photo on next slide. But, reasoning also possible to explain why ODE can be viable for statistical processes.

Photo from, (Nölting. pp. 146 – 147)

The energy minimization hypothesis for protein folding and other chemical reactions

It has been conjectured by some that a chemical reaction always occurs in a way that minimizes the geometric-physical energy functional.

But, on present showing, for a chemical reaction, one would need to look to the entire chemical process/scheme. In other words, it is feasible that local geometric efficiency could be sacrificed in order to achieve the higher chemical end - the products of a total chemical reaction.

Within the context of chemistry, this local sacrifice conceivably could be chemically more energy efficient, when one looks to the entire chemical process.

For an example from biology, a horse crossing a gravity gradient sacrifices local geometric energy in order to succeed in the higher order energy efficiency that better ensures its biological survival.

Corollary

Part 1. The energy hypothesis becomes reasonable when the chemical reaction is sufficiently limited, in the sense that physical law dominates.

Part 2. Otherwise, more information is needed.