# Thermodynamic integration

Thermodynamic integration is used to calculate the difference in the Helmholtz energy function, ${\displaystyle A}$, between two states. The path must be continuous and reversible, i.e., the system must evolve through a succession of equilibrium states (Ref. 1 Eq. 3.5)

${\displaystyle \Delta A=A(\lambda )-A(\lambda _{0})=\int _{\lambda _{0}}^{\lambda }\left\langle {\frac {\partial U(\mathbf {r} ,\lambda )}{\partial \lambda }}\right\rangle _{\lambda }~\mathrm {d} \lambda }$

## Isothermal integration

At constant temperature (Ref. 2 Eq. 5):

${\displaystyle {\frac {A(\rho _{2},T)}{Nk_{B}T}}={\frac {A(\rho _{1},T)}{Nk_{B}T}}+\int _{\rho _{1}}^{\rho _{2}}{\frac {p(\rho )}{k_{B}T\rho ^{2}}}~\mathrm {d} \rho }$

## Isobaric integration

At constant pressure (Ref. 2 Eq. 6):

${\displaystyle {\frac {G(T_{2},p)}{Nk_{B}T_{2}}}={\frac {G(T_{1},p)}{Nk_{B}T_{1}}}-\int _{T_{1}}^{T_{2}}{\frac {H(T)}{Nk_{B}T^{2}}}~\mathrm {d} T}$

where ${\displaystyle G}$ is the Gibbs energy function and ${\displaystyle H}$ is the enthalpy.

## Isochoric integration

At constant volume (Ref. 2 Eq. 7):

${\displaystyle {\frac {A(T_{2},V)}{Nk_{B}T_{2}}}={\frac {A(T_{1},V)}{Nk_{B}T_{1}}}-\int _{T_{1}}^{T_{2}}{\frac {U(T)}{Nk_{B}T^{2}}}~\mathrm {d} T}$

where ${\displaystyle U}$ is the internal energy.