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, i.e., the system must evolve through a succession of equilibrium states (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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

Related reading

• Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics 129 214112 (2008)
• Maria Concetta Abramo, Carlo Caccamo, Dino Costa, Paolo V. Giaquinta, Gianpietro Malescio, Gianmarco Munaò, and Santi Prestipino "On the determination of phase boundaries via thermodynamic integration across coexistence regions", Journal of Chemical Physics 142 214502 (2015)
• J. D. Doll, P. Dupuis, and P. Nyquist "Thermodynamic integration methods, infinite swapping, and the calculation of generalized averages", Journal of Chemical Physics 146 134111 (2017)