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

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

where

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

Isothermal integration

Ref. 1 Eq. 5:

$\frac{A({\rho }_2,T)}{N{k}_{B}T}=\frac{A({\rho }_1,T)}{N{k}_{B}T}+\underset{{\rho }_1}{\overset{{\rho }_2}{\int }}\frac{p(\rho )}{k_{B}T{\rho }^2}\mathrm{d}\rho$

Isobaric integration

Ref. 1 Eq. 6:

$\frac{G(T_2,p)}{N{k}_B{T}_2}=\frac{G(T_1,p)}{N{k}_B{T}_1}-\underset{T_1}{\overset{T_2}{\int }}\frac{H(T)}{N{k}_B{T}^2}\mathrm{d}T$

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

Isochoric integration

Ref. 1 Eq. 7:

$\frac{A(T_2,V)}{N{k}_B{T}_2}=\frac{A(T_1,V)}{N{k}_B{T}_1}-\underset{T_1}{\overset{T_2}{\int }}\frac{U(T)}{N{k}_B{T}^2}\mathrm{d}T$

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)