Internal Energy and Helmholtz Free Energy

Substituting Eqs. (32) and (31) into Eq. (25) we obtain that the internal energy $E$ can be computed from the partition function $Z$ as follows,

$$E = -\frac{\partial \text{ln} Z}{\partial \beta} \Bigg )_X.$$ (35)

Furthermore, substituting Eqs. (32) and (31) into Eq. (22) we obtain

$$S = -k \{ \hat{\rho}(-\beta \hat{H} - Z) \} = k \beta E + k Z.$$ (36)

In the next section we prove that the parameter $T \equiv (k \beta)^{-1}$ can be identified with the temperature of the ensemble. Therefore,

$$A = E - T S = -k T Z,$$ (37)

is the Helmholtz free energy, that according to Eq. (35) satisfies the following thermodynamic equation,

$$E = \frac{\partial(\beta A)}{\partial \beta} \Bigg )_X.$$ (38)