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 $\lambda =1$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle U_{\lambda }=U}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Delta A=A-A_{0}=\int _{0}^{1}d\lambda \langle {\frac {\partial U_{\lambda }}{\partial \lambda }}\rangle _{\lambda }}

where

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.U_{\lambda }\right.=(1-\lambda )U_{0}+\lambda U} .