molecular details associated with changes in entropy. These ideas are closer to applicability in problems of making measures on the behavior of biological systems. Very generally, in the statistical mechanical context, an increase in entropy means a decrease in the order, which can be a quantitative observable reflecting a decrease in predictability and/or knowledge about the system. For example, we can locate the molecules of the gas more accurately when they are all on one side of the membrane-partitioned cylinder compared with the situation when the membrane is suddenly removed. This accompanying increase in ambiguity and decrease in knowledge in locating a set of gas particles reflects a statistical mechanical view of increases in entropy. Can anything general be said about the bounds on an increase in entropy? The statistical developments of the Yale mathematical physicist, Josiah Willard Gibbs (about 1875), consonant with the logical arguments of the Greek mathematician, Constantin Caratheodory (about 1910), conclude that the entropy increase goes to the maximum allowed by the constraints imposed by or upon the system. A change in likelihood as a probability is a characteristic way to quantify the entropy change, reflecting an alteration in knowledge or its reciprocal complement, uncertainty. The system’s entropic uncertainty said more colloquially,
and relevant to the Bell Syndrome’s women of my life, is its capacity for surprise.
A statistical mechanical approach to the total entropy of a bounded set of molecules in motion involves summing this property across all the participating molecules. We let N be the number of particles involved. As a problem in Newtonian mechanics, each of the N particles is represented in 6N dimensional phase space. That means that each point represents one of the N molecules in the three dimensions of location space plus three dimensions of motion space as its velocity, more specifically, the product of mass times velocity called momentum. This adds up to 6 dimensions of measurement. This so called phase space reconstruction of the molecules of a gas as individual particles are a daunting task, though fast computers and new algorithms are making computations from first principles more generally attainable. Those based on the first principles of short-range repulsion and long-range weak attraction among particles and the bumper-car collision
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