Boltzmann statistics and the canonical partition function a derivation

1 Boltzmann statistics and the canonical partition function a derivation 

Our study of the NVT ensemble begins by treating a large heat reservoir thermally coupled to a smaller system using the NVE approach. The energy of the heat reservoir is denoted Er and the energy of the smaller subsystem, El. The composite system is assumed closed and the total energy is fixed Er + ) = E0 = constant. The composite system is assumed to be a closed NVE system and the subsystem is assumed to have constant N and V. 

In addition, from Equation (12.6) we know that In 2r is equal to f3Er + a, where the constant a does not depend on the energy of the system. From this relationship we have 

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