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)
\[\Delta A = A(\lambda) - A(\lambda_0) = \int_{\lambda_0}^{\lambda} \left\langle \frac{\partial U(\mathbf{r},\lambda)}{\partial \lambda} \right\rangle_{\lambda} ~\mathrm{d}\lambda\]
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[edit] Isothermal integration
At constant temperature (Ref. 2 Eq. 5):
\[\frac{A(\rho_2,T)}{Nk_BT} = \frac{A(\rho_1,T)}{Nk_BT} + \int_{\rho_1}^{\rho_2} \frac{p(\rho)}{k_B T \rho^2} ~\mathrm{d}\rho \]
[edit] Isobaric integration
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.
[edit] Isochoric integration
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.
[edit] See also
[edit] References
- J. A. Barker and D. Henderson "What is "liquid"? Understanding the states of matter ", Reviews of Modern Physics 48 pp. 587 - 671 (1976)
- 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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