Thermodynamic integration - Revision history

Carl McBride: /* References */ Added a recent publication

2017-04-20T13:33:23Z

References: Added a recent publication

← Older revision		Revision as of 15:33, 20 April 2017
Line 28:		Line 28:
	*[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/10.1063/1.3023062 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)]		*[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)]
			*[http://dx.doi.org/10.1063/1.4979493 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)]


	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]

Carl McBride: /* References */ Added a recent publication

2015-06-23T10:08:38Z

References: Added a recent publication

← Older revision		Revision as of 12:08, 23 June 2015
Line 27:		Line 27:
	'''Related reading'''		'''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/10.1063/1.3023062 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]]		[[category:classical thermodynamics]]

Carl McBride at 09:21, 5 July 2011

2011-07-05T09:21:51Z

← Older revision		Revision as of 11:21, 5 July 2011
Line 1:		Line 1:
	'''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 '''must''' be ''continuous'' and ''reversible'' (Ref. 1 Eq. 3.5)		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>		:<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>

Carl McBride: /* References */

2010-10-06T09:50:31Z

References

← Older revision		Revision as of 11:50, 6 October 2010
Line 22:		Line 22:
	*[[Gibbs-Duhem integration]]		*[[Gibbs-Duhem integration]]
	==References==		==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/10.1103/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)		#[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/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]		'''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)]
	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]

Carl McBride: /* References */ Added a new reference concerning errors.

2008-12-05T11:33:03Z

References: Added a new reference concerning errors.

← Older revision		Revision as of 13:33, 5 December 2008
Line 24:		Line 24:
	#[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/10.1103/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)		#[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/10.1063/1.3023062 Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics '''129''' 214112 (2008)]
	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]

Carl McBride: Tidy up

2008-08-05T10:44:48Z

Tidy up

← Older revision		Revision as of 12:44, 5 August 2008
Line 1:		Line 1:
	'''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 must be ''continuous'' and ''reversible''.		The path '''must''' be ''continuous'' and ''reversible'' (Ref. 1 Eq. 3.5)
	~~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 - ~~A_0~~ = \~~int_0~~^~~1 d~~\lambda \left\langle \frac{\partial U_\lambda}{\partial \lambda} \right\rangle_{\lambda}~~</math>~~		:<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>

	~~where~~

	~~:<math>~~\~~left.U_\lambda\right.=(1-~~\lambda~~)U_0 + \lambda U~~</math>.
	==Isothermal integration==		==Isothermal integration==
	Ref. 1 Eq. 5:		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>		:<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:		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>		:<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>
Line 20:		Line 14:
	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:		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>		:<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>
Line 28:		Line 22:
	*[[Gibbs-Duhem integration]]		*[[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)		#[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]]

Carl McBride: Added some equations

2008-08-04T10:11:27Z

Added some equations

@@ Line 1: / Line 1: @@
-'''Thermodynamic integration''' is used to calculate the difference in the [[Helmholtz energy function]] between two states.
+'''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''.
 One has a  continuously variable energy function <math>U_\lambda</math> such that
@@ Line 9: / Line 9: @@
 :<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.
 ==See also==
 *[[Gibbs-Duhem integration]]
 ==References==
 [[category:classical thermodynamics]]

Carl McBride at 12:51, 12 February 2008

2008-02-12T12:51:18Z

← Older revision		Revision as of 14:51, 12 February 2008
Line 9:		Line 9:

	:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.		:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.
			==See also==
			*[[Gibbs-Duhem integration]]
	==References==		==References==
	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]

Carl McBride at 15:30, 29 January 2008

2008-01-29T15:30:42Z

← Older revision		Revision as of 17:30, 29 January 2008
Line 4:		Line 4:
	<math>\lambda=0</math>, <math>U_\lambda=U_0</math> and <math>\lambda=1</math>, <math>U_\lambda=U</math>		<math>\lambda=0</math>, <math>U_\lambda=U_0</math> and <math>\lambda=1</math>, <math>U_\lambda=U</math>

	:<math>\Delta A = A - A_0 = \int_0^1 d\lambda \langle\frac{\partial U_\lambda}{\partial \lambda}\rangle_{\lambda}</math>		:<math>\Delta A = A - A_0 = \int_0^1 d\lambda \left\langle \frac{\partial U_\lambda}{\partial \lambda} \right\rangle_{\lambda}</math>

	where		where

	:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.		:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.
			==References==
	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]

Nice and Tidy at 16:03, 25 May 2007

2007-05-25T16:03:23Z

← Older revision		Revision as of 18:03, 25 May 2007
Line 9:		Line 9:

	:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.		:<math>\left.U_\lambda\right.=(1-\lambda)U_0 + \lambda U</math>.
			[[category:classical thermodynamics]]