Quantum Flow Versus Classical MaxEnt Balance

 Maximization of entropy subject to  a constraint Sum over i ei P(ei) = Eave may be used to determine the Maxwell-Boltzmann factor exp(-ei/T), where T is temperature. In the case of a potential V(x), one may maximize  -P(x) ln(P(x)) subject to the constraint  TV(x)P(x). Average potential energy is not an a priori known value, but we have argued in the past that maximization of entropy subject to a constraint, followed by a derivative leads to a balance equation.

 What about a classical mechanical bound problem? This is usually considered nonstatistical . We suggest it may be a statistical problem, but one for which maxent applies to flow, in particular the quantum bound problem. We have argued before that for a free particle, one may maximize Shannon's entropy  -P(t)ln(P(t)) subject to a periodic constraint i Et P(t) and -P(x)ln(P(x)) subject to i px P(x) to obtain exp(-iEt) and exp(-ipx). We argue these are associated with flows in time and space, and that there are problems in nature  based on such  flow and its interference.

   Classically, a bound system is a flow problem because a particle moves to the right and then left,  between turning points. This means the particle is confined or localized,, but exp(ipx) is not. exp(ipx) is two dimensional to account for motion in a certain direction, but both cos(px) and sin(px) exist everywhere in space. A bound system should not have one directional motion,on average, thus either cos(px) or sin(px) applies, i.e. a certain parity. How does one account for a probability value in all of space? We argue that in the flow exp(ipx) picture, one must consider a bound problem in equilibrium with particles trying to enter or escape. By creating an ensemble W(x)=Sumo v over p a(p)exp(ipx) and having W(x) tend to 0 at x= +-/infinite one may create such a flow picture. At first one may consider any a(p) values, but we argue that either a(p)=-a(p) or a(p)=a(-p) for all p in this note, i.e. there is no mixing of parity. We try to show this. We then argue that V(x) does not impede this flow statistics, but only imposes conditions on the average kinetic energy at x. Thus Newtonian mechanics deals only with this restriction and not with the underlying average of maxent flow probabilities exp(ipx).

   A classical statistical gas is not a flow problem, but a balance problem with or without V(x)..Interparticle collisions change the kinetic energy of a particle and hence break its exp(-iet) flow probability. The stochasticity is then not in the flow, but in the flow-breaking collisions

We suggest that this is linked to maximization of entropy subject to a constraint and deals with different probabilities than flow ones (exp(ipx) and exp(-iEt). A Rabi frequency problem on the other hand is a flow in time problem with the magnetic field not inhibiting this flow. Finally, there may be problems which at first appear to be a balance problem, but are really a flow one, like one dimensional photon reflection/refraction.