Perturbative expansion

If the term $U'$ is small, then it is possible to use a perturbative approach to compute the integral in . Let us expand the quantity $\langle U' \rangle_\lambda$ near $\lambda = 0$:

$$\langle U' \rangle_\lambda = \langle U' \rangle_{\lambda = 0} + \... ...c{d \langle U' \rangle_\lambda}{d\lambda}\right )_{\lambda = 0} + o(\lambda^2).$$ (7.82)

We have:

$$\frac{d \langle U' \rangle_\lambda}{d\lambda} = \frac{d}{d \lambd... ...{{\bf r}\})}}{\int_V d^3\{{\bf r}\} e^{-\beta U_\lambda(\{{\bf r}\})}} \right),$$ (7.83)

and recalling that $\frac{\partial U_\lambda}{\partial \lambda} = U'$ we obtain:

$$\frac{d \langle U' \rangle_\lambda}{d\lambda} = -\beta \left [ \langle U'^2 \rangle_\lambda - \langle U' \rangle_\lambda^2 \right ],$$ (7.84)

and so

$$\langle U' \rangle_\lambda = \langle U' \rangle_{\lambda = 0} - \... ...gle_{\lambda = 0} - \langle U' \rangle_{\lambda = 0}^2 \right ] + o(\lambda^2).$$ (7.85)

In the case where it is possible to ignore the $o(\lambda^2)$ term, i.e. where $\langle U' \rangle_\lambda$ is well approximated by a linear function of $\lambda$, then the integral in  becomes:

$$F - F^h = \langle U' \rangle_{\lambda = 0} - \frac{1}{2} \beta \l... ...angle U'^2 \rangle_{\lambda = 0} - \langle U' \rangle_{\lambda = 0}^2 \right ].$$ (7.86)

Expression  is very powerful, because it only requires sampling the canonical ensemble generated by $U^h$, and computing the average value of $U'$ and of its fluctuations over the ensemble.