Computational implementation of integral equations - Revision history

Carl McBride at 14:57, 31 January 2008

2008-01-31T14:57:23Z

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 *[http://dx.doi.org/10.1016/0010-4655(70)90034-2  Taro Tamura  "Angular momentum coupling coefficients", Computer Physics Communications  '''1''' pp.  337-342 (1970)]
 *[http://dx.doi.org/10.1016/0010-4655(71)90030-0 J. G. Wills  "On the evaluation of angular momentum coupling coefficients", omputer Physics Communications  '''2''' pp. 381-382 (1971)]
 ==References==
 #[http://dx.doi.org/10.1080/00268977900102861 M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics '''38''' pp. 1781-1794 (1979)]

Carl McBride: /* Fourier-Bessel Transforms */

2008-01-31T14:51:10Z

Fourier-Bessel Transforms

← Older revision		Revision as of 16:51, 31 January 2008
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	(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3),		(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3),
	where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.		where <math>J_l(x)</math> is a [[Bessel functions \|Bessel function]] of order <math>l</math>.
	`step-down' operations can be performed by way of sin and cos operations		`step-down' operations can be performed by way of sin and cos operations
	of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado Ref. 3.		of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado Ref. 3.
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	:<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>		:<math>f(r)=2\pi \int_0^\infty g(q) J_0(2 \pi qr)q ~{\rm d}q</math>


	====Conversion from the spatial reference frame back to the axial reference frame====		====Conversion from the spatial reference frame back to the axial reference frame====

Carl McBride: /* Evaluate */

2008-01-31T14:49:03Z

Evaluate

← Older revision		Revision as of 16:49, 31 January 2008
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	Evaluations of <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>		Evaluations of <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>
	where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomials \|Legendre polynomial]] <math>P_\nu(cos \theta)</math>		where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomials \|Legendre polynomial]] <math>P_\nu(cos \theta)</math>
	where <math>y_j</math> are the <math>\nu</math> roots of the [[Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>		where <math>y_j</math> are the <math>\nu</math> roots of the [[Chebyshev polynomials \|Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
	and where <math>z_{1_k},z_{2_k}</math> are the <math>\nu</math> roots of the [[Chebyshev polynomial]]		and where <math>z_{1_k},z_{2_k}</math> are the <math>\nu</math> roots of the Chebyshev polynomial
	<math>T_{\nu}(\ cos \chi)</math>		<math>T_{\nu}(\ cos \chi)</math>
	thus		thus

Nice and Tidy: /* Ornstein-Zernike relation $\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)$ */

2007-07-10T15:50:21Z

Ornstein-Zernike relation <math>\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)</math>

← Older revision		Revision as of 17:50, 10 July 2007
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	For molecular fluids (see Eq. 19 of Lado Ref. 3)		For molecular fluids (see Eq. 19 of Lado Ref. 3)

			:<math>\tilde{{\mathbf S}}_{m}(k) = (-1)^{m}\rho \left[{\mathbf I} - (-1)^{m} \rho \tilde{\mathbf C}_{m}(k) \right]^{-1} \tilde{\mathbf C}_{m}(k)\tilde{\mathbf C}_{m}(k)</math>

	:<math>\tilde{~~S}_{m}(k) = (-1)^~~{m}\~~rho \left[I - (-1)^{m~~} ~~\rho \tilde C_{m~~}~~(k) \right]^{-1} \tilde C_{m}(k)\tilde C_~~{m}(k)		where <math>\tilde{{\mathbf S}}_{m}(k)</math> and <math>\tilde{\mathbf C}_{m}(k)</math> are matrices
	</math>
	~~where~~ <math>\tilde {S}_~~{m}(k)</math> and <math>\tilde C_~~{m}(k)</math> are matrices
	with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.		with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.

	For mixtures of simple fluids (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):		For mixtures of simple fluids (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):

	:<math>\tilde \Gamma (k) = D \left[ I - D \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math>		:<math>\tilde{\Gamma}(k) = {\mathbf D} \left[{\mathbf I} - {\mathbf D} \tilde{\mathbf C}(k)\right]^{-1} \tilde{\mathbf C}(k)\tilde{\mathbf C}(k)</math>

	===Conversion back from Fourier space to Real space===		===Conversion back from Fourier space to Real space===

Carl McBride: /* Evaluate */

2007-05-30T17:06:18Z

Evaluate

← Older revision		Revision as of 19:06, 30 May 2007
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	====Evaluate====		====Evaluate====
	Evaluations of <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>		Evaluations of <math>\gamma (12)</math> are performed at the discrete points <math>x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}</math>
	where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomial]] <math>P_\nu(cos \theta)</math>		where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomials \|Legendre polynomial]] <math>P_\nu(cos \theta)</math>
	where <math>y_j</math> are the <math>\nu</math> roots of the [[Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>		where <math>y_j</math> are the <math>\nu</math> roots of the [[Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
	and where <math>z_{1_k},z_{2_k}</math> are the <math>\nu</math> roots of the [[Chebyshev polynomial]]		and where <math>z_{1_k},z_{2_k}</math> are the <math>\nu</math> roots of the [[Chebyshev polynomial]]

Carl McBride at 16:26, 30 May 2007

2007-05-30T16:26:35Z

← Older revision		Revision as of 18:26, 30 May 2007
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	Integral equations are solved numerically.		Integral equations are solved numerically.
	One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>		One has the [[Ornstein-Zernike relation]], <math>\gamma (12)</math>
	and a [[~~Closures~~ \| closure relation]], <math>c_2 (12)</math> (which		and a [[closure relations \| closure relation]], <math>c_2 (12)</math> (which
	incorporates the [[bridge function]] <math>B(12)</math>).		incorporates the [[bridge function]] <math>B(12)</math>).
	The numerical solution is iterative;		The numerical solution is iterative;

Carl McBride: /* Ornstein-Zernike relation $\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)$ */

2007-05-30T15:02:58Z

Ornstein-Zernike relation <math>\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)</math>

← Older revision		Revision as of 17:02, 30 May 2007
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	where <math>\tilde {S}_{m}(k)</math> and <math>\tilde C_{m}(k)</math> are matrices		where <math>\tilde {S}_{m}(k)</math> and <math>\tilde C_{m}(k)</math> are matrices
	with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.		with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.
	~~For mixtures of simple fluids (see Ref. 10 and the thesis of Juan Antonio Anta pp. 107--109):~~

	~~<math>\tilde \Gamma~~ (~~k) = D \left[ I~~ - ~~D \tilde C(k)\right]^{~~-~~1} \tilde C(k)\tilde C(k~~)~~</math>~~		For mixtures of simple fluids (see Ref. 10 Juan Antonio Anta PhD thesis pp. 107--109):

			:<math>\tilde \Gamma (k) = D \left[ I - D \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math>

	===Conversion back from Fourier space to Real space===		===Conversion back from Fourier space to Real space===

Carl McBride: /* Conversion back from Fourier space to Real space */

2007-05-30T15:01:37Z

Conversion back from Fourier space to Real space

Carl McBride at 14:09, 30 May 2007

2007-05-30T14:09:52Z

Carl McBride at 14:05, 30 May 2007

2007-05-30T14:05:32Z

← Older revision		Revision as of 16:05, 30 May 2007
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	:<math>\tilde{c}_{\mu \nu}^{mnl} (k; l_1 l_2 l n_1 n_2) = 4\pi i^l \int_0^{\infty} c_{\mu \nu}^{mnl} (r; l_1 l_2 l n_1 n_2) J_l (kr) ~r^2 {\rm d}r</math>		:<math>\tilde{c}_{\mu \nu}^{mnl} (k; l_1 l_2 l n_1 n_2) = 4\pi i^l \int_0^{\infty} c_{\mu \nu}^{mnl} (r; l_1 l_2 l n_1 n_2) J_l (kr) ~r^2 {\rm d}r</math>

	(see Blum and Torruella Eq. 5.6 ~~\cite{JCP_1972_56_00303}~~ or Lado Eq. 39 ~~\cite{MP_1982_47_0283}~~),		(see Blum and Torruella Eq. 5.6 in Ref. 7 or Lado Eq. 39 in Ref. 3),
	where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.		where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.
	`step-down' operations can be performed by way of sin and cos operations		`step-down' operations can be performed by way of sin and cos operations

← Older revision		Revision as of 17:01, 30 May 2007
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	===Conversion back from Fourier space to Real space===		===Conversion back from Fourier space to Real space===
	<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>		:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>
	(basically the inverse of step 2).		(basically the inverse of step 2).
	====Axial reference frame to spatial reference frame====		====Axial reference frame to spatial reference frame====
	:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>		:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
	====Inverse Fourier-Bessel transform====		====Inverse Fourier-Bessel transform====
	:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)</math> `Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is		:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)</math>
			'Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is
	:<math>\gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k</math>		:<math>\gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k</math>
	====Change from spatial reference frame back to axial reference frame====		====Change from spatial reference frame back to axial reference frame====
	:<math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>.		:<math>\gamma^{mnl}_{\mu \nu} (r) \rightarrow \gamma_{mns}^{\mu \nu} (r)</math>.

	==Ng acceleration==		==Ng acceleration==
	*[http://dx.doi.org/10.1063/1.1682399 Kin-Chue Ng "Hypernetted chain solutions for the classical one-component plasma up to Gamma=7000", Journal of Chemical Physics '''61''' pp. 2680-2689 (1974)]		*[http://dx.doi.org/10.1063/1.1682399 Kin-Chue Ng "Hypernetted chain solutions for the classical one-component plasma up to Gamma=7000", Journal of Chemical Physics '''61''' pp. 2680-2689 (1974)]

← Older revision		Revision as of 16:09, 30 May 2007
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	where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.		where <math>J_l(x)</math> is a [[Bessel function]] of order <math>l</math>.
	`step-down' operations can be performed by way of sin and cos operations		`step-down' operations can be performed by way of sin and cos operations
	of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado ~~\cite{MP_1982_47_0283}~~.		of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado Ref. 3.
	The Fourier-Bessel transform is also known as a '''Hankel transform'''.		The Fourier-Bessel transform is also known as a '''Hankel transform'''.
	It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.		It is equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel.
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	:<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho \tilde{c}_2 (k)}</math>		:<math>\tilde{\gamma}(k)= \frac{\rho \tilde{c}_2 (k)^2}{1- \rho \tilde{c}_2 (k)}</math>

	For molecular fluids (see Eq. 19 of Lado ~~\cite{MP_1982_47_0283}~~)		For molecular fluids (see Eq. 19 of Lado Ref. 3)


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	where <math>\tilde {S}_{m}(k)</math> and <math>\tilde C_{m}(k)</math> are matrices		where <math>\tilde {S}_{m}(k)</math> and <math>\tilde C_{m}(k)</math> are matrices
	with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.		with elements <math>\tilde S_{l_1 l_2 m}(k), \tilde{C}_{l_1 l_2 m}(k), l_1,l_2 \geq m</math>.
	For mixtures of simple fluids (see ~~\cite{JCP_1988_88_07715}~~ and the thesis of Juan Antonio Anta pp. 107--109):		For mixtures of simple fluids (see Ref. 10 and the thesis of Juan Antonio Anta pp. 107--109):

	<math>\tilde \Gamma (k) = D \left[ I - D \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math>		<math>\tilde \Gamma (k) = D \left[ I - D \tilde C(k)\right]^{-1} \tilde C(k)\tilde C(k)</math>
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	:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>		:<math>\tilde{\gamma}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}^{mnl}_{\mu \nu} (k)</math>
	====Inverse Fourier-Bessel transform====		====Inverse Fourier-Bessel transform====
	:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)</math> `Step-up' operations are given by Eq. 53 of ~~\cite{MP_1982_47_0283}~~. The inverse Hankel transform is		:<math>\tilde{\gamma}^{mnl}_{\mu \nu} (k) \rightarrow \gamma^{mnl}_{\mu \nu} (r)</math> `Step-up' operations are given by Eq. 53 of Ref. 3. The inverse Hankel transform is
	:<math>\gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k</math>		:<math>\gamma(r;l_1 l_2 l n_1 n_2)= \frac{1}{2 \pi^2 i^l} \int_0^\infty \tilde{\gamma}(k;l_1 l_2 l n_1 n_2) J_l (kr) ~k^2 {\rm d}k</math>
	====Change from spatial reference frame back to axial reference frame====		====Change from spatial reference frame back to axial reference frame====
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	#[http://dx.doi.org/10.1063/1.1678503 L. Blum "Invariant Expansion. II. The Ornstein-Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model", Journal of Chemical Physics '''57''' pp. 1862-1869 (1972)]		#[http://dx.doi.org/10.1063/1.1678503 L. Blum "Invariant Expansion. II. The Ornstein-Zernike Equation for Nonspherical Molecules and an Extended Solution to the Mean Spherical Model", Journal of Chemical Physics '''57''' pp. 1862-1869 (1972)]
	#[http://dx.doi.org/10.1063/1.1679655 L. Blum "Invariant expansion III: The general solution of the mean spherical model for neutral spheres with electostatic interactions", Journal of Chemical Physics '''58''' pp. 3295-3303 (1973)]		#[http://dx.doi.org/10.1063/1.1679655 L. Blum "Invariant expansion III: The general solution of the mean spherical model for neutral spheres with electostatic interactions", Journal of Chemical Physics '''58''' pp. 3295-3303 (1973)]
			#[http://dx.doi.org/10.1063/1.454286 P. G. Kusalik and G. N. Patey " On the molecular theory of aqueous electrolyte solutions. I. The solution of the RHNC approximation for models at finite concentration", Journal of Chemical Physics '''88''' pp. 7715-7738 (1988)]
	[[category: integral equations]]		[[category: integral equations]]

Computational implementation of integral equations - Revision history

Carl McBride at 14:57, 31 January 2008

Carl McBride: /* Fourier-Bessel Transforms */

Carl McBride: /* Evaluate */

Nice and Tidy: /* Ornstein-Zernike relation \tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k) */

Carl McBride: /* Evaluate */

Carl McBride at 16:26, 30 May 2007

Carl McBride: /* Ornstein-Zernike relation \tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k) */

Carl McBride: /* Conversion back from Fourier space to Real space */

Carl McBride at 14:09, 30 May 2007

Carl McBride at 14:05, 30 May 2007

Nice and Tidy: /* Ornstein-Zernike relation $\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)$ */

Carl McBride: /* Ornstein-Zernike relation $\tilde{c}_{mns}^{\mu \nu} (k) \rightarrow \tilde{\gamma}_{mns}^{\mu \nu} (k)$ */