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 (Ref. 1 Eq. 3.5)

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

Isothermal integration

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

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

Isobaric integration

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

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

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

Isochoric integration

At constant volume (Ref. 2 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. J. A. Barker and D. Henderson "What is "liquid"? Understanding the states of matter ", Reviews of Modern Physics 48 pp. 587 - 671 (1976)
2. 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)