Thermodynamic integration

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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[edit]

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

\frac{A(\rho_2,T)}{Nk_BT} = \frac{A(\rho_1,T)}{Nk_BT} + \int_{\rho_1}^{\rho_2} \frac{p(\rho)}{k_B T \rho^2} ~\mathrm{d}\rho

Isobaric integration[edit]

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

\frac{G(T_2,p)}{Nk_BT_2} = \frac{G(T_1,p)}{Nk_BT_1} - \int_{T_1}^{T_2} \frac{H(T)}{Nk_BT^2} ~\mathrm{d}T

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

Isochoric integration[edit]

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

\frac{A(T_2,V)}{Nk_BT_2} = \frac{A(T_1,V)}{Nk_BT_1}  - \int_{T_1}^{T_2} \frac{U(T)}{Nk_BT^2} ~\mathrm{d}T

where U is the internal energy.

See also[edit]


  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)

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