Editing Thermodynamic integration
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'''Thermodynamic integration''' is used to calculate the difference in the [[Helmholtz energy function]], <math>A</math>, between two states. | '''Thermodynamic integration''' is used to calculate the difference in the [[Helmholtz energy function]], <math>A</math>, between two states. | ||
The path | The path must be ''continuous'' and ''reversible''. | ||
One has a continuously variable energy function <math>U_\lambda</math> such that | |||
<math>\lambda=0</math>, <math>U_\lambda=U_0</math> and <math>\lambda=1</math>, <math>U_\lambda=U</math> | |||
:<math>\Delta A = A | :<math>\Delta A = A - A_0 = \int_0^1 d\lambda \left\langle \frac{\partial U_\lambda}{\partial \lambda} \right\rangle_{\lambda}</math> | ||
where | |||
:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>. | |||
==Isothermal integration== | ==Isothermal integration== | ||
Ref. 1 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> | :<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== | ==Isobaric integration== | ||
Ref. 1 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> | :<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> | ||
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where <math>G</math> is the [[Gibbs energy function]] and <math>H</math> is the [[enthalpy]]. | where <math>G</math> is the [[Gibbs energy function]] and <math>H</math> is the [[enthalpy]]. | ||
==Isochoric integration== | ==Isochoric integration== | ||
Ref. 1 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> | :<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> | ||
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*[[Gibbs-Duhem integration]] | *[[Gibbs-Duhem integration]] | ||
==References== | ==References== | ||
#[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) | #[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) | ||
[[category:classical thermodynamics]] | [[category:classical thermodynamics]] |