RE: More on "Entropy"

[David Van Wie <dvw@hamachi.epr.com>]

Tom Weinstein wrote:

>We used this formulation of entropy in Statistical Mechanics.  It's
>especially useful in Quantum Thermo where you can actually enumerate all
>of the states instead of relying on probabilistic arguments.

Sure, this formulation can be used.  As a pedagogic tool for explaining what 
a theory is all about, many formulations are discussed as if they have 
application in real world situations.  Of course (for pedagogic reasons), 
these discussions focus on systems in which there is a definition, and 
typically a well-behaved mathematical model, for all of the significant 
states.  Some instructors believe this will assist students in appreciating 
the concepts of statistical mechanics and quantum thermodynamics.

To build a working apparatus (or software systems, as we are discussing 
here), the designer is typically faced with the breakdown of well-behaved 
mathematical models.  Everything from degenerate states to the "baked in" 
uncertainty of certain states tends to undermine the mathematical 
foundations of a theorist's constructions.  Of course the theoretical models 
are absolutely critical, but the designer must always caution themselves 
against drawing inferences without measurements and clearly stated 
rationales that speak to these physical realities that lead to mathematical 
weaknesses.  Ultimately, the probabilistic nature of such systems may be 
"moved around," but not removed from the model!

Since the real world of actual measurements interferes with essentially 
everything we claim to "know" about quantities such as entropy, the real 
danger is assigning an independent "meaning" to these constructs.  Why? 
 Because these quantities do not exist independently, they only exist with 
respect to our predictive models of a system's behavior.  So these models do 
not really "enumerate" anything about states, but rather restate the 
probability assumptions of the model in the form of a "working equation."

In addition, drawing inferences as to the behavior of systems based on 
common mathematical form is simply inviting trouble, even at the theoretical 
level.  Mathematical models are not the real world, and the superficial 
mathematical consistency between say, the functional form of a resonance in 
a quantum well and a marble in a bowl, does not mean that the marble gives 
any special insight into the nature of the quantum well.  In fact, beyond 
the curiosity of similar equations, the most important information is in the 
distinctions and clarifications (emanating from theory) between the systems 
from a practical, apparatus building, real world perspective (as contrasted 
with the "everything is just a special case of X" perspective).

This danger is also present in designs for sources of entropy to seed RNGs 
for random data or to create uniformly distributed keys.  Well designed 
models will avoid rephrasing assumptions as conclusions, and will explicitly 
address the mathematical weaknesses upon which the theoretical arguments in 
support of the model are ultimately based.

dvw