copyright 2017 Jennifer Coopersmith
The beautiful hand-in-hand advance of physics and mathematics is well known. For example, there was the (uncomfortable) discovery, by the Ancient Greeks, that some measurements led to ‘irrational’ numbers; and the differential calculus enabled the definition of instantaneous speed, without the need for division by zero. I would like to bring to your attention one example from thermodynamics.
In a thermodynamic system we usually have many parameters, such as P, V, T, S, etc. We are then, of necessity, thrust into the world of partial differential equations (seeing how one parameter varies with respect to another while yet a third is held fixed). The Carnot Cycle was devised by a young French engineer, Sadi Carnot, in 1824, and was an idealized physical mock-up of a mathematical problem. During one section of the Cycle, P varies with V while T is held constant; and during another section (the ‘adiabatic section’) P varies with V while the entropy, S, is held constant. Thus, the partial differential equations can be solved. This was sheer genius, all the more remarkable when we remember that the Cycle is a fiction (it never happens exactly like that in practice) and, moreover, entropy (and energy) hadn’t even been discovered at this time.
In physics, we perform a mapping between physical things and numbers. We then perform various mathematical operations on these numbers, and then ‘translate’ the mathematical consequences back into physical things (we make physical predictions). Now here’s a curious question: we may wonder whether a mathematical thought-experiment must be capable of being carried out physically, albeit with some difficulty, if it is to have physical relevance. However, it turns out that this requirement is not a necessity. For example, in the Principle of Least Action, a mathematical test of ‘flatness’ is carried out, and this test is carried out in a ‘virtual space’ which sometimes implies conditions that cannot be physically realized. Nevertheless, the Principle of Least Action turns out to be one of the cornerstones of physics. Perhaps this finding is not so alarming if we remind ourselves of other more familiar scenarios. Consider the statistic ‘average family size’, useful in the social sciences. A family size of, say, 2.2 children can never be realized in practice, and yet ‘average family size’ may still be a very useful measure.