Microcanonical ensemble

In such an ensemble the total energy ${\displaystyle E}$, instead of the temperature ${\displaystyle T}$, is fixed. If we do define[1] the density operator ${\displaystyle {\hat {\rho }}={\frac {e^{-}\beta {\hat {H}}}{Z}}}$, in this ensemble it takes the form: ${\displaystyle {\hat {\rho }}=\sum _{i\in E}{\frac {1}{g(E)}}\left|i\right\rangle \left\langle i\right|}$. The mean value of any operator ${\displaystyle {\hat {O}}}$ can be written as: ${\displaystyle \left\langle {\hat {O}}\right\rangle =Tr\left({\hat {O}}{\hat {\rho }}\right)}$. In this ensemble the entropy ${\displaystyle S}$ takes the role of thermodynamic potential.

1. This definition holds for any ensemble