Editing Computational implementation of integral equations
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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 | and a 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; | ||
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Picard iteration generates a solution of an initial value problem for an ordinary differential equation (ODE) using fixed-point iteration. | Picard iteration generates a solution of an initial value problem for an ordinary differential equation (ODE) using fixed-point iteration. | ||
Here are the four steps used to solve integral equations: | Here are the four steps used to solve integral equations: | ||
===Closure relation <math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>=== | ===1. Closure relation <math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>=== | ||
(Note: for linear fluids <math>\mu = \nu =0</math>) | (Note: for linear fluids <math>\mu = \nu =0</math>) | ||
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Thus | Thus | ||
:<math> | :<math>\gamma(12)=\gamma (r,x_1x_2,y,z_1z_2)</math>. | ||
====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 [[ | where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomial]] <math>P_\nu(cos \theta)</math> | ||
where <math>y_j</math> are the <math>\nu</math> roots of the [[ | 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]] | ||
<math>T_{\nu}(\ cos \chi)</math> | <math>T_{\nu}(\ cos \chi)</math> | ||
thus | thus | ||
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where | where | ||
:<math>\hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta)</math> | :<math>\hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta)</math> | ||
where <math>d_{s \mu}^m(\theta)</math> is the angular, <math>\theta</math>, part of the | where <math>d_{s \mu}^m(\theta)</math> is the angular, <math>\theta</math>, part of the | ||
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and | and | ||
:<math> | :<math>e_s(y)=\exp(is\phi)</math> | ||
:<math> | :<math>e_{\mu}(z)= \exp(i\mu \chi)</math> | ||
For the limits in the summations | For the limits in the summations | ||
:<math> | :<math>L_1= \max (s,\nu_1)</math> | ||
:<math> | :<math>L_2= \max (s,\nu_2)</math> | ||
The above equation constitutes a separable five-dimensional transform. To rapidly evaluate | The above equation constitutes a separable five-dimensional transform. To rapidly evaluate | ||
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Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math> | Use [[Gauss-Legendre quadrature]] for <math>x_1</math> and <math>x_2</math> | ||
Use [[Gauss-Chebyshev quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math> | Use [[Gauss-Chebyshev quadrature]] for <math>y</math>, <math>z_1</math> and <math>z_2</math> | ||
thus | |||
:<math>c_{mns}^{\mu \nu} (r) = w^3 | :<math>c_{mns}^{\mu \nu} (r) = w^3 | ||
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:<math>w=\frac{1}{NG}</math> | :<math>w=\frac{1}{NG}</math> | ||
===Perform FFT from Real to Fourier space <math>c_{mns}^{\mu \nu} (r) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k)</math>=== | ===Perform FFT from Real to Fourier space<math>c_{mns}^{\mu \nu} (r) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k)</math>==== | ||
This is non-trivial and is undertaken in three steps: | This is non-trivial and is undertaken in three steps: | ||
#Conversion from axial reference frame to spatial reference frame, ''i.e.'' | |||
:<math>c_{mns}^{\mu \nu} (r) \rightarrow c_{\mu \nu}^{mnl} (r)</math> | :<math>c_{mns}^{\mu \nu} (r) \rightarrow c_{\mu \nu}^{mnl} (r)</math> | ||
this is done using the Blum transformation | this is done using the Blum transformation \cite{JCP_1972_56_00303,JCP_1972_57_01862,JCP_1973_58_03295}: | ||
:<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left( | :<math>g_{\mu \nu}^{mnl}(r) = \sum_{s=-\min (m,n)}^{\min (m,n)} \left( | ||
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\right)g_{mns}^{\mu \nu} (r)</math> | \right)g_{mns}^{\mu \nu} (r)</math> | ||
#'''Fourier-Bessel Transforms''': <math>c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)</math> | |||
:<math>c_{\mu \nu}^{mnl} (r) \rightarrow \tilde{c}_{\mu \nu}^{mnl} (k)</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> | :<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 | (see Blum and Torruella Eq. 5.6 \cite{JCP_1972_56_00303} or Lado Eq. 39 \cite{MP_1982_47_0283}), | ||
where <math>J_l(x)</math> is a [[ | 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 | of Fourier transforms, see Eqs. 49a, 49b, 50 of Lado \cite{MP_1982_47_0283}. | ||
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. | ||
<math>g(q)=2\pi \int_0^\infty f(r) J_0(2 \pi qr)r ~{\rm d}r</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 | |||
''i.e.'' | |||
<math>\tilde{c}_{\mu \nu}^{mnl} (k) \rightarrow \tilde{c}_{mns}^{\mu \nu} (k) | |||
</math> | |||
this is done using the Blum transformation | this is done using the Blum transformation | ||
<math>g_{mns}^{\mu \nu} (r) | |||
= \sum_{l=|m-n|}^{m+n} \left( | = \sum_{l=|m-n|}^{m+n} \left( | ||
\begin{array}{ccc} | \begin{array}{ccc} | ||
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g_{\mu \nu}^{mnl}(r)</math> | g_{\mu \nu}^{mnl}(r)</math> | ||
==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)] | ||
==References== | ==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)] | #[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)] | ||
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#[http://dx.doi.org/10.1080/00268978200100222 F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics '''47''' pp. 313-317 (1982)] | #[http://dx.doi.org/10.1080/00268978200100222 F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics '''47''' pp. 313-317 (1982)] | ||
#[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)] | #[http://dx.doi.org/10.1080/00268978900101981 Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics '''68''' pp. 87-95 (1989)] | ||