Introduction

Differential expression of the first principle of thermodynamics:

$$dE=TdS-pdV$$

$$dE=TdS-pdV$$

Boltzmann relation between entropy $S$$S$ and number of microstates at a given energy $g(E)$$g(E)$:

$$S=k\log(g(E))$$

$$S=k\log (g(E))$$

The above defines an extensive property of the system, i.e. if we have two subsystems at $S_{1},S_{2}$$S_{1}, S_{2}$ then $S=S_{1}+S_{2}=k\log(g_{1}g_{2})=k\log(g)$$S=S_{1}+S_{2}=k\log (g_{1}g_{2})=k\log (g)$.