Speculation on Spatial Density and Acceleration

In a previous note (1), we argued that the classical statistical mechanics of a Maxwell-Boltzman gas is linked (in a canonical distribution sense) to an additive scalar( ei = kinetic energy of a particle) and leads to P(ei)= C exp(-ei/T) if one uses information loss considerations i.e P(a1)P(a2)=P(a3) where a is the additive scalar. We further argued that a quantum particle also satisfies this information loss equation, but with a being a vector such as momentum. This leads to exp(ipx) as a probability, although an unusual one.

     In (2) we argued that a classical equilibrium distribution based on an additive scalar which is linked by a conservation law in space i.e.  a(at x1) + B(x1) = a(at x2) + B(x2) necessarily introduces a probability factor of the form exp(-iB(x)/T), but leaves the exp(-ai/T) distribution unchanged. We suggested this be interpreted as a spatial density linked to acceleration of particles. In this note, we argue that if both the quantum exp(ipx) and the classical MB distribution both follow from the same P(a1)P(a2)=P(a3) condition, then the notion of density and acceleration should also be found in quantum mechanics, even for exp(ipx) which represents constant velocity.