Time Reversal Balance and the Effect of Flux in Statistical and Quantum Mechanics

In a previous note (1), we argued that the Maxwell-Boltzmann (MB) distribution exp(-p*p/2mT) and similar forms such as exp(-e1/T) and exp(-V(x)/T) were based on reactions of the form P(p1)P(p3)=P(p2)P(p4) or P(p1)V(1 to 2) = P(p2)V(2 to 1) ((1)) where p1, momentum, is being scattered into p2. Taking the natural log of this expression and comparing it to a conservation of kinetic energy (or energy levels in other cases) leads to the form exp(-p*p/2mT). This approach was applied to static statistical mechanics with a potential V(x), statistical mechanics with time independent flow, quantum mechanics with a temperature and fixed energy levels and the T=0 ground state quantum oscillator. It was argued that the MB type distribution breaks down for other bound state cases because there are two velocities in such cases. The first is the vrms(x) velocity where .5mvrms(x)*vrms(x) is the classical kinetic energy and u(x)=(1/W(x)) d/dx W(x) where W(x) is the bound state wavefunction. In this note, we try to give a specific calculation involving velocity (or flux) for the statistical mechanical case to show how it complements a reaction balance ((1)) as long as the flux velocity obeys Newtonian mechanics. For the ground state quantum oscillator, this is the case for the flux 1/W d/dx W, but not for other levels or potentials. Thus, this lack of Newton’s law in the flux 1/W d/dx W (which is responsible for d/dx d(x), where d(x) is the spatial density) seems to lead to the breakdown of the reaction balance ((1)) and the MB distribution in quantum mechanical bound states, leading to a new kind of statistical requirement for quantum bound states