Thermodynamic integration

Thermodynamic integration is used to calculate the difference in the Helmholtz energy function, $A$, 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

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

Isothermal integration

Ref. 1 Eq. 5:

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

Isobaric integration

Ref. 1 Eq. 6:

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

where $G$ is the Gibbs energy function and $H$ is the enthalpy.

Isochoric integration

Ref. 1 Eq. 7:

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

where $U$ is the internal energy.

See also

• Gibbs-Duhem integration

References

1. C. Vega, E. Sanz, J. L. F. Abascal and E. G. Noya "Determination of phase diagrams via computer simulation: methodology and applications to water, electrolytes and proteins", Journal of Physics: Condensed Matter 20 153101 (2008) (section 4)