Determinism and Bias in Information for Fermions and Bosons?

Information theory, which applies to compressing messages, suggests that ln(probability) is information. Shannon’s entropy is -1 times the average information i.e. -Sum over i P(i)ln(P(i)). For a classical gas in the canonical approximation, this is of the form of an average conserved quantity e.g Eave= Sum over i ei P(ei). 

   In deterministic classical mechanics one may characterize an elastic two body collision by e1+e2 = e3 + e4. In a Maxwell-Boltzmann (MB) gas these same deterministic reactions occur, but there is also a static probability P(ei). The question seems to be whether P(ei) brings any new “physical” information into a scattering reaction. One already knows that a particle contributes energy ei. Is there something else brought in? If the answer is no, then using the product of probabilities P(e1)P(e2) and taking ln leads to ln(P(e1)) + ln(P(e2)) suggests that ln(P(e1))= A (e1-B) where A and B are constant. Thus information considerations link probability P(ei) to a deterministic quantity ei. One does not have to consider reaction balance or the maximization of entropy subject to a constraint. One simply finds the appropriate probability  Pappropriate(e1) which describes an overall change to the state e1. For the MB case, P(appropriate)(ei) = P(ei) (MB) which gives a static description of the number of particles with ei. 

    In other cases, however, there may be bias. Consider quantum scattering of fermions or bosons. If the collisions are elastic, one again has the classical deterministic e1+e2=e3+e4 expression. There is also a static P(ei) (Fermi-Dirac) for fermions and P(ei) Bose-Einstein for bosons. One is, however, interested in collisions which create and maintain the equilibrium situation. If there is bias in these collisions, then Pappropriate(ei) may differ from P(ei). The information e1 is the full contribution of particle 1 to the collision. Thus one also needs a full probability describing probability movement out of state e1. P(e1) is proportional only to the probability flow out of e1 due to scattering, but there is a “bias” factor namely 1-P(e1) which prevents another particle from entering state e1. This biases a reaction. As a result Pappropriate(e1) = P(e1)/ (1-P(e1)) for a fermion. This may be seen from a reaction balance equation:P(e1)(1-P(e3)) P(e2)(1-P(e4)) = P(e3)(1-P(e1))P(e4)(1-P(e2)) for fermions. For bosons there is an enhancement. One may obtain product probabilities by forming P(ei)/(1-P(ei)). This is the full contribution in terms of probabilities of particle i to a collision and so its ln should equal A(e-B)i which is its full information contribution to the reaction. If the two are not equal, it means there is more information (excluding parameters such as T temperature and u the chemical potential) than ei in the problem.