Minimal Functional Forms in Statisical Mechanics Part 3

In Part 1, we argued that one may use the notion of rotational invariance to suggest that a probability for a certain velocity should be a function of the simplest invariant namely v dot v. We then used exp(-a v dot v) as a probability (using conservation of energy) and suggested extending this idea to space, i.e. proposing a probability exp(-a r dot r) or exp(-a x*x) in one dimension. Here, we investigate this situation in more detail. In particular, exp(-a x*x) corresponds to a random walk in one dimension which may be solved in terms of factorials and Stirling’s approximation followed by a Taylor series. It does not seem that rotational invariance is required at all. Neither is there any mention of reflection symmetry which is the 1-dimensional analogue.. We suggest, however, that rotational invariance is a key feature as the random walk problem in one dimension (steps to the right and left) is a subset of the problem of motion in two dimensions (or even three). As a result, we argue that P(x) = P(feven(x)). Then: P(feven(x)) P(feven(y)) P((feven(x)) = P( feven(r)) ((1)) ((1)) shows that it is in fact the simplest function demonstrating rotational invariance which solves this problem, i.e. feven(x) = x*x as advertised in Part 1 without much explanation.