Computational implementation of integral equations: Difference between revisions

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Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).
Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).
==Picard iteration==
==Picard iteration==
===Closure relation===
 
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:
~\\
===1. Closure relation===
1) {\bf Closure relation}: $\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)$\\
<math>\gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)</math>
(Note: for linear fluids $\mu = \nu =0$)\\
(Note: for linear fluids <math>\mu = \nu =0</math>)
~\\
 
i) Perform the summation
====Perform the summation====
\begin{equation}
 
g(12)=g({\bf r}_{12},\omega_1,\omega_2)=\sum_{mns\mu \nu} g_{mns}^{\mu \nu}({\bf r}_{12}) \Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2)
:<math>g(12)=g(r_{12},\omega_1,\omega_2)=\sum_{mns\mu \nu} g_{mns}^{\mu \nu}(r_{12}) \Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2)</math>
\end{equation}
 
where ${\bf r}_{12}$ is the separation between molecular centers and  
where <math>r_{12}</math> is the separation between molecular centers and  
$\omega_1,\omega_2$ the sets of Euler angles needed to specify the orientations of the two molecules, with
<math>\omega_1,\omega_2</math> the sets of [[Euler angles]] needed to specify the orientations of the two molecules, with
\begin{equation}
 
\Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2) = \sqrt{(2m+1)(2n+1)}  \mathcal{D}_{s \mu}^m (\omega_1)  \mathcal{D}_{\overline{s} \nu}^n (\omega_2)
:<math>\Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2) = \sqrt{(2m+1)(2n+1)}  \mathcal{D}_{s \mu}^m (\omega_1)  \mathcal{D}_{\overline{s} \nu}^n (\omega_2)</math>
\end{equation}
 
with $\overline{s} = -s$.\\
with <math>\overline{s} = -s</math>.
~\\
 
ii) Define the variables
====Define the variables====
\begin{equation}
 
x_1= \cos \theta_1
:<math>\left. x_1 \right.= \cos \theta_1</math>
\end{equation}
:<math>\left. x_2\right.= \cos \theta_2</math>
\begin{equation}
:<math>\left. z_1 \right.= \cos \chi_1</math>
x_2= \cos \theta_2
:<math>\left. z_2 \right.= \cos \chi_2</math>
\end{equation}
:<math>\left. y\right.= \cos \phi_{12}</math>
\begin{equation}
 
z_1 = \cos \chi_1
Thus <math>\gamma(12)=\gamma (r,x_1x_2,y,z_1z_2)</math>.
\end{equation}
 
\begin{equation}
====Evaluate====
z_2 = \cos \chi_2
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>
\end{equation}
where the <math>x_i</math> are the <math>\nu</math> roots of the [[Legendre polynomial]] <math>P_\nu(cos \theta)</math>
\begin{equation}
where <math>y_j</math> are the  <math>\nu</math> roots of the [[Chebyshev polynomial]] <math>T_{\nu}(\ cos \phi)</math>
y= \cos \phi_{12}
and where <math>z_{1_k},z_{2_k}</math> are the  <math>\nu</math> roots of the [[Chebyshev polynomial]]
\end{equation}
<math>T_{\nu}(\ cos \chi)</math>
Thus $\gamma(12)=\gamma (r,x_1x_2,y,z_1z_2)$.\\
~\\
iii) Evaluations of  $\gamma (12)$ are performed at the discrete points $x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}$\\
where the $x_i$ are the $\nu$ roots of the Legendre polynomial $P_\nu(cos \theta)$
~\\
where $y_j$ are the  $\nu$ roots of the Chebyshev polynomial $T_{\nu}(\ cos \phi)$\\
and where $z_{1_k},z_{2_k}$ are the  $\nu$ roots of the Chebyshev polynomial $T_{\nu}(\ cos \chi)$\\
~\\
thus
thus
\begin{equation}
 
\gamma(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k})=
:<math>\gamma(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k})=
\sum_{\nu , \mu ,  s = -M }^M \sum_{m=L_2}^M \sum_{n=L_1}^M  
\sum_{\nu , \mu ,  s = -M }^M \sum_{m=L_2}^M \sum_{n=L_1}^M  
\gamma_{mns}^{\mu \nu} (r)
\gamma_{mns}^{\mu \nu} (r)
\hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i})
\hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i})
e_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k})
e_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k})</math>
\end{equation}
 
where
where
\begin{equation}
\begin{equation}

Revision as of 12:00, 30 May 2007

Integral equations are solved numerically. One has the Ornstein-Zernike relation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma (12)} and a closure relation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 (12)} (which incorporates the bridge function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(12)} ). The numerical solution is iterative;

  1. trial solution for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma (12)}
  2. calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 (12)}
  3. use the Ornstein-Zernike relation to generate a new Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma (12)} etc.

Note that the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 (12)} is local, i.e. the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 (12)} at a given point is given by the value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma (12)} at this point. However, the Ornstein-Zernike relation is non-local. The way to convert the Ornstein-Zernike relation into a local equation is to perform a (fast) Fourier transform (FFT). Note: convergence is poor for liquid densities. (See Ref.s 1 to 6).

Picard 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:

1. Closure relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_{mns}^{\mu \nu} (r) \rightarrow c_{mns}^{\mu \nu} (r)} (Note: for linear fluids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = \nu =0} )

Perform the summation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(12)=g(r_{12},\omega_1,\omega_2)=\sum_{mns\mu \nu} g_{mns}^{\mu \nu}(r_{12}) \Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{12}} is the separation between molecular centers and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_1,\omega_2} the sets of Euler angles needed to specify the orientations of the two molecules, with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi_{\mu \nu s}^{mn}(\omega_1,\omega_2) = \sqrt{(2m+1)(2n+1)} \mathcal{D}_{s \mu}^m (\omega_1) \mathcal{D}_{\overline{s} \nu}^n (\omega_2)}

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{s} = -s} .

Define the variables

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. x_1 \right.= \cos \theta_1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. x_2\right.= \cos \theta_2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. z_1 \right.= \cos \chi_1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. z_2 \right.= \cos \chi_2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. y\right.= \cos \phi_{12}}

Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(12)=\gamma (r,x_1x_2,y,z_1z_2)} .

Evaluate

Evaluations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma (12)} are performed at the discrete points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{i_1}x_{i_2},y_j,z_{k_1}z_{k_2}} where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} are the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} roots of the Legendre polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\nu(cos \theta)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j} are the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} roots of the Chebyshev polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{\nu}(\ cos \phi)} and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_{1_k},z_{2_k}} are the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} roots of the Chebyshev polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{\nu}(\ cos \chi)} thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k})= \sum_{\nu , \mu , s = -M }^M \sum_{m=L_2}^M \sum_{n=L_1}^M \gamma_{mns}^{\mu \nu} (r) \hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i}) e_s(j) e_{\mu} (z_{1_k}) e_{\nu} (z_{2_k})}

where \begin{equation} \hat{d}_{s \mu}^m (x) = (2m+1)^{1/2} d_{s \mu}^m(\theta) \end{equation} where $d_{s \mu}^m(\theta)$ is the angular, $\theta$, part of the rotation matrix $\mathcal{D}_{s \mu}^m (\omega)$,\\ and \begin{equation} e_s(y)=\exp(is\phi) \end{equation} \begin{equation} e_{\mu}(z)= \exp(i\mu \chi) \end{equation} For the limits in the summations \begin{equation} \begin{equation} L_1= \max (s,\nu_1) \end{equation} \begin{equation} L_2= \max (s,\nu_2) \end{equation} The above equation constitutes a separable five-dimensional transform. To rapidly evaluate this expression it is broken down into five one-dimensional transforms: \begin{equation} \gamma_{l_2m}^{n_1n_2}(r,x_{1_i})=\sum_{l_1=L_1}^M \gamma_{l_1 l_2 m}^{n_1 n_2}(r) \hat{d}_{m n_1}^{l_1} (x_{1_i}) \end{equation} \begin{equation} \gamma_{m}^{n_1n_2}(r,x_{1_i},x_{2_i})=\sum_{l_2=L_2}^M \gamma_{l_2 m}^{n_1 n_2}(r,x_{1_i}) \hat{d}_{\overline{m} n_2}^{l_2} (x_{2_i}) \end{equation} \begin{equation} \gamma^{n_1n_2}(r,x_{1_i},x_{2_i},j)=\sum_{m=-M}^M \gamma_{m}^{n_1 n_2}(r,x_{1_i},x_{2_i}) e_m(j) \end{equation} \begin{equation} \gamma^{n_2}(r,x_{1_i},x_{2_i},z_{1_k})=\sum_{n_1=-M}^M \gamma^{n_1 n_2}(r,x_{1_i},x_{2_i},j) e_{n_1}(z_{1_k}) \end{equation} \begin{equation} \gamma(r,x_{1_i},x_{2_i},z_{1_k},z_{2_k})=\sum_{n_2=-M}^M \gamma^{n_2}(r,x_{1_i},x_{2_i},j,z_{1_k}) e_{n_2}(z_{2_k}) \end{equation} Operations involving the $e_m(y)$ and $e_n(z)$ basis functions are performed in complex arithmetic. The sum of these operations is asymptotically smaller than the previous expression and thus constitutes a ``fast separable transform". $NG$ and $M$ are parameters; $NG$ is the number of nodes in the Gauss integration, and $M$ the the max index in the truncated rotational invariants expansion.\\ ~\\ iv) Integrate over angles $c_2(12)$:\\ ~\\ Use Gauss-Legendre quadrature for $x_1$ and $x_2$\\ Use Gauss-Chebyshev quadrature for $y$, $z_1$ and $z_2$\\ thus \begin{equation} c_{mns}^{\mu \nu} (r) = w^3 \sum_{x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}=1}^{NG} w_{i_1}w_{i_2}c_2(r,x_{1_i},x_{2_i},j,z_{1_k},z_{2_k}) \hat{d}_{s \mu}^m (x_{1_i}) \hat{d}_{\overline{s} \nu}^n (x_{2_i}) e_{\overline{s}}(j) e_{\overline{\mu}} (z_{1_k}) e_{\overline{\nu}} (z_{2_k}) \end{equation} where the Gauss-Legendre quadrature weights are given by \begin{equation} w_i= \frac{1}{(1-x_i^2)}[P_{NG}^{'} (x_i)]^2 \end{equation} while the Gauss-Chebyshev quadrature has the constant weight \begin{equation} w=\frac{1}{NG} \end{equation}

Perform FFT from Real to Fourier space

Ng acceleration

References

  1. M. J. Gillan "A new method of solving the liquid structure integral equations" Molecular Physics 38 pp. 1781-1794 (1979)
  2. Stanislav Labík, Anatol Malijevský and Petr Voncaronka "A rapidly convergent method of solving the OZ equation", Molecular Physics 56 pp. 709-715 (1985)
  3. F. Lado "Integral equations for fluids of linear molecules I. General formulation", Molecular Physics 47 pp. 283-298 (1982)
  4. F. Lado "Integral equations for fluids of linear molecules II. Hard dumbell solutions", Molecular Physics 47 pp. 299-311 (1982)
  5. F. Lado "Integral equations for fluids of linear molecules III. Orientational ordering", Molecular Physics 47 pp. 313-317 (1982)
  6. Enrique Lomba "An efficient procedure for solving the reference hypernetted chain equation (RHNC) for simple fluids" Molecular Physics 68 pp. 87-95 (1989)
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