Two Kinds of Equilibria: Particle and Wave/Dynamic Based? Part II

  In Part I  we considered two kinds of statistical processes. One is based on a scalar additive which changes in stochastic processes e.g. kinetic energy in two body elastic collisions or the spatial shell in a 3D random walk problem. We argued that in such a case, one considers a1+a2= a3 (where ai is the distributive scalar) and a product probability scheme i.e  P(a1)P(a2) = P(a3) to find that P(ai) = C exp(-ai b) where b is a parameter which may be found if the average value of a is known. Furthermore, we argued that the distributive scalar process is identical to maximizing entropy subject to a constraint Sum over i ai P(ai). 

   The second process is based on a dynamic vector changing in the stochastic process. We used the problem of light reflecting and refracting (which we argued is similar to quantum mechanics for a pure bound state) and noted that the p (momentum) vector changes i.e. an incoming p along the x axis in the positive direction becomes -p on reflection or p2 on refraction in 1 dimension. We suggested that a dynamic probability is needed in such a case, namely an eigenfunction of the generator associated with the dynamic variable, e.g. x for p. The eigenfunction is then periodic. Using exp(ipx)s  (eigenfunctions of -id/dx) on may calculate average energy at x and fix weights i.e.  {Sum over p pp/2m a(p)exp(ipx)} / { Sum over p a(p)exp(ipx)} + V(x) = En, but there is no maximization of entropy.

   At first the additive scalar (distributive variable) and dynamic vector approaches seem to be very different (for example, one maximizes entropy and the other does not), but in this note we argue there are a number of similarities. We focus on these similarities and suggest these may perhaps be key features that unite the distributive scalar approach (classical statistical mechanics, level jumping  in quantum mechanics) with the dynamical vector approach (statistics of pure quantum bound state). In particular, both types of probabilities are linked with exponential functions although one is real and drops in value exp(-ei/T) and the other is complex and periodic exp(ipx) and P(a1)P(a2) =P(a3) in both cases. Also, both are eigenfunctions of generators. Thus we suggest that the similarities between the scalar additive approach (MB gas) and dynamical vector (quantum bound states) may be the key features of overall equilibrium type statistical mechanics.