Statistical Periodicity in Classical and Quantum Physics

 It is known in the literature that there exists a strong link between quantum mechanics (e.g. a particle in a box with an infinite potential at the walls) and standing sound waves as described in (1) (or even waves on a string). This relationship goes beyond the sinusoidal solutions and includes interference and other features discussed in (1). In (2), we suggest that periodicity in both classical and quantum solutions may be based on the statistical idea of maximum entropy. For example, a particle in a classical gas with no potential has equal probability to be at any point x. If one wishes to retain this idea of equal probability at each x, yet have momentum linked to x, then exp(ipx) is a solution which allows for this. It has a modulus of one and is also periodic in space which is the “next best thing” to being constant. In classical statistical mechanics (3), the Boltzmann transport equation allows one to find a density which describes sound in a medium. In this note, we examine the various assumptions used to derive this wave-equation and investigate whether one may anticipate this periodicity at a stage in the derivation prior to the sound wave equation being derived. We argue that a periodic solution may be part of a more general statistical idea of maximum entropy.