Editing Thermodynamic integration

'''Thermodynamic integration''' is used to calculate the difference in the [[Helmholtz energy function]], <math>A</math>, 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)

:<math>\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</math>
==Isothermal integration==
At constant [[temperature]] (Ref. 2 Eq. 5):

:<math>\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 </math>
==Isobaric integration==
At constant [[pressure]] (Ref. 2 Eq. 6):

:<math>\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 </math>

where <math>G</math> is the [[Gibbs energy function]] and <math>H</math> is the [[enthalpy]].
==Isochoric integration==
At constant volume (Ref. 2 Eq. 7):

:<math>\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 </math>

where <math>U</math> is the [[internal energy]].
==See also==
*[[Gibbs-Duhem integration]]
==References==
<references/>
#[http://dx.doi.org/10.1103/RevModPhys.48.587      J. A. Barker and D. Henderson "What is "liquid"? Understanding the states of matter ", Reviews of Modern Physics '''48''' pp. 587 - 671 (1976)]
#[http://dx.doi.org/10.1088/0953-8984/20/15/153101  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'''
*[http://dx.doi.org/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]
*[http://dx.doi.org/10.1063/1.4921884 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)]

[[category:classical thermodynamics]]