Single Particle Probabilities and Trial Runs in the Fermi-Dirac/Bose Einstein Cases

  If one tosses a die or coin, different outcomes occur with a probability p(i) where i is the outcome. This probability may be measured by performing trial runs. For N trial runs, where N is very large, one might expect Np(outcome i) results of an outcome i. In other words, p(outcome i) is both the probability and describes the trial run. Even in the Maxwell-Boltzmann case, one has a probability p(ei) where ei is kinetic energy (no V(x)) and reaction balance p(e1)p(e2) = p(e3)p(e4) represents trial runs. Thus the trial run is again characterized by p(ei) as ei’s contribution.

     It is, however, possible to have coupling of probabilities in certain physical situations even though p(outcome i) still exists. In such a case, the number of outcome i results for N large trials is not N p(outcome i). In other words, p(i) does not fully describe the trial run. There may be other physical rules in place. If elastic scattering of particles is the way trial runs occur, e1+e2 = e3 + e4 represents the salient information in the problem and even if there is nonindependence of p(outcome i) in the scattering, one should at least obtain a function Q(p(e1)) such that Q(p(e1)) = -e1/T. Q(p(ei)), however, need not be p(ei). 

   We consider the physical rule given by Pauli stating that only one particle may fit in an energy level (no spin). Thus there is either 0 or 1 particle in the level. This should inhibit scattering, so the trial run is no longer simply described by p(ei). Rather the probability to scatter into level ej is (1-n(ej)) and so one has:   n(e1)[1-n(e3)] n(e2)[1-n(e4)]= n(e3)[1-n(e1)]n(e4)[1-n(e2)] where n(ej) is the average occupancy, a number between 0 and 1. The opposite of only 0 or 1 particle in a state ei, is to have any possible number of particles in this state and an enhancement, not a hindrance given by 1+p(ej). This is only a guess, but experiment shows that bosons follow this rule.

    Given a definition of trial runs in terms of the inhibition and enhancement factors, one may do two things. First, since the trial rule seems to involve different numbers of particles in a state associated with a p(ei), one may consider sums of p(ei) (power number of particles) when describing the system statistically (i.e. grand canonical scheme). This approach gives rise to linear terms (a+bp(ei)) which may in turn be formulated in terms of a factorial/arrangement equation which may be maximized subject to the constraints Sum over i ei n(ei) and Sum over i n(ei). We note that this maximization is not simply in terms of all degrees of freedom in the system, but is specifically linked with the rules associated with the trial runs and the constraints. 

We note, however, that issues seem to arise with the factorial scheme in the boson case.

  Thus, overall we argue that a  single particle probability and its trial run scheme are key to establishing the statistical behaviour of a system. (We also note that single particle independent probability, which is an approximation, also affects the final results.)