Thermodynamic integration

Used to calculate the free energy difference 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 $\lambda =1$, $U_{\lambda }=U$

$\Delta A=A-A_{0}=\int _{0}^{1}d\lambda <{\frac {\partial U_{\lambda }}{\partial \lambda }}>_{\lambda }$

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