Thermodynamic integration

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 $\lambda =1$, $U_{\lambda }=U$

$\mathrm{\Delta }A=A-A_0=\underset{0}{\overset{1}{\int }}d\lambda ⟨\frac{\partial U_{\lambda }}{\partial \lambda }{⟩}_{\lambda }$

where

$U_{\lambda }=(1-\lambda )U_0+\lambda U$.