Transcript Heuristics - Integrating Hierarchical Structures

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.
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Definition: Systematic Process:
Other things being equal!
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(1) The whole of a systematic process and its every
event can be accounted for by a single set of
correlations;
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(2) Any situation can be deduced from any other
without an explicit consideration of intervening
situations; and
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(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
predictionNis possible,
but on a step by step
1
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.
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
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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
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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.
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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
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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.]
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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:
γ1
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 equipotential.
 Take the result as the new set of boundary conditions.
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.