Does anyone know of any REALLY good derivations of the grand canonical partition function(T,V,μ) from the hamiltonian. I am using the graduate level thermodynamics book by tester and there appears to be some algebric manipulation that occurs going from the ensemble to the partition function once the hamiltonian is applied and a new λ term is defined which then becomes part of the grand function. Also I am refering to the partition function that includes the gibbs and phase space correction factors. I understand the specific partition function comes from the fermi-dirac distribution which I still need to follow up on and there is a good MIT open course ware video on youtube that goes into the fermi-dirac distribution. My ultimate goal is to go from manually deriving the boltzman distribution which is postulary and I have sketched out a good conceptual picture by drawing little molecules TO the van der waals equation of state. I could perhaps skip to the hemholtz macroscopic equation and go from there but I want to see the details of how these EOS are arived at. Then these equations of state are used to evalute things like joule thompson going from the combined first and second laws but there has always been an uneasy feeling of double dipping there since in theory the first and second laws could be derived from statistical mechanics?