Modification of Fermi-Dirac/Bose-Einstein Distributions Based on Pure State Entropy?

 In a previous note (1) we argued that for a wavefunction W(x,t)= Sum over n an exp(iEnt) Wn(x), the entropy for a particle in a box with infinite potential walls from (2) is:

 S/(-k)= Sum over n |anan| ln |anan|  ((1)) and  should be modified to: S/(-k)= Sum over n  |anan| ln|anan| + |anan| Sp(n,L) + |anan| Sx(L) ((2)) where Sp and Sx are the pure state momentum and spatial entropies i.e  Sp/-k = Integral dp  an(p)an(p) ln|an(p)an(p)|  and Sx/(-k) = Integral dx W*n(x)Wn(x) ln[Wn*Wn|. 

    In (2) the Clausius relationship:  dQ/T = dS is used where dQ = Sum over n  En d|anan| and T is temperature to find an expression for |anan|. In general this would be: |anan| = exp(-En/T) , but with ((2)) we find: |anan| = exp(-En/kT) exp(-1-B(n)) ((3)) where B(n)= Sp(n)+Sx. L portions form Sp and Sx cancel so there is no L dependence, L being the size of the box.

   Given the different form ((3)) we suggest that the Fermi-Dirac (FD) and Bose-Einstein (BE) distributions may change if one uses the grand partition function approach. For example, exp(-ni ei/kT) is replaced by: exp(-ni ei/kT) exp(-ni(1+B(n)). This leads to changes in the FD and BD distributions:   zexp(-Ei/kT)exp(-1-B(i) / [1 +/- exp(-u/kT) exp(-ei/kT) exp(-1-B(i))] where z=exp(u/kT), u being the chemical potential. In the high n limit B(n)--> constant (3) so the Maxwell-Boltzmann (MB) distribution remains unchanged. The MB distribution has been verified to high precision and so modifications related to 1+B(n) might have been noticed had B(n) remained a function of n.