Thermodynamic integration

Thermodynamic integration is used to calculate the difference in the Helmholtz energy function between two states. The path must be continuous and reversible. One has a continuously variable energy function $U_{\lambda }$ such that $\lambda =0$, $U_{\lambda }=U_{0}$ and $\lambda =1$, $U_{\lambda }=U$

$\Delta A=A-A_{0}=\int _{0}^{1}d\lambda \left\langle {\frac {\partial U_{\lambda }}{\partial \lambda }}\right\rangle _{\lambda }$

where

$\left.U_{\lambda }\right.=(1-\lambda )U_{0}+\lambda U$.