Magnetic trap (atoms): Difference between revisions
en>BobbyDavid No edit summary |
|||
Line 1: | Line 1: | ||
In [[statistical mechanics]] the '''Ornstein–Zernike equation''' (named after [[Leonard Ornstein]] and [[Frits Zernike]]) is an [[integral equation]] for defining the direct [[correlation function (statistical mechanics)|correlation function]]. It basically describes how the [[correlation]] between two molecules can be calculated. Its applications can mainly be found in fluid theory. | |||
== Derivation == | |||
The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to<ref>Frisch, H. & Lebowitz J.L. ''The Equilibrium Theory of Classical Fluids'' (New York: Benjamin, 1964)</ref> for the full derivation. | |||
It is convenient to define the [[total correlation function]]: | |||
: <math> h(r_{12})=g(r_{12})-1 \, </math> | |||
which is a measure for the "influence" of molecule 1 on molecule 2 at a distance <math>r_{12}</math> away with <math>g(r_{12})</math> as the [[radial distribution function]]. In 1914 Ornstein and Zernike proposed <ref>Ornstein, L. S. and Zernike, F. Accidental deviations of density and opalescence at the critical point of a single substance. Proc. Acad. Sci. Amsterdam 1914, 17, 793-806</ref> to split this influence into two contributions, a direct and indirect part. The direct contribution is ''defined'' to be given by the [[direct correlation function]], denoted <math>c(r_{12})</math>. The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as | |||
: <math> h(r_{12})=c(r_{12}) + \rho \int d \mathbf{r}_{3} c(r_{13})h(r_{23}) \, </math> | |||
which is called the '''Ornstein–Zernike equation'''. Its interest is that, by eliminating the indirect influence, <math>c(r)</math> is shorter-ranged than <math>h(r)</math> and can be more easily described. The OZ equation has the interesting property that if one multiplies the equation by <math>e^{i\mathbf{k \cdot r_{12}}}</math> with <math>\mathbf{r_{12}}\equiv |\mathbf{r}_{2}-\mathbf{r}_{1}|</math> and integrate with respect to <math>d \mathbf{r}_{1}</math> and <math>d \mathbf{r}_{2}</math> one obtains: | |||
: <math> \int d \mathbf{r}_{1} d \mathbf{r}_{2} h(r_{12})e^{i\mathbf{k \cdot r_{12}}}=\int d \mathbf{r}_{1} d \mathbf{r}_{2} c(r_{12})e^{i\mathbf{k \cdot r_{12}}} + \rho \int d \mathbf{r}_{1} d \mathbf{r}_{2} d \mathbf{r}_{3} c(r_{13})e^{i\mathbf{k \cdot r_{12}}}h(r_{23}). \, </math> | |||
If we then denote the [[Fourier transforms]] of <math>h(r)</math> and <math>c(r)</math> by <math>\hat{H}(\mathbf{k})</math> and <math>\hat{C}(\mathbf{k})</math> this rearranges to | |||
: <math> \hat{H}(\mathbf{k})=\hat{C}(\mathbf{k}) + \rho \hat{H}(\mathbf{k})\hat{C}(\mathbf{k}) \, </math> | |||
from which we obtain that | |||
: <math> \hat{C}(\mathbf{k})=\frac{\hat{H}(\mathbf{k})}{1 +\rho \hat{H}(\mathbf{k})} \,\,\,\,\,\,\, \hat{H}(\mathbf{k})=\frac{\hat{C}(\mathbf{k})}{1 -\rho \hat{C}(\mathbf{k})}. \, </math> | |||
One needs to solve for both <math>h(r)</math> and <math>c(r)</math> (or, equivalently, their Fourier transforms). This requires an additional equation, known as a [[Closure (mathematics)|closure]] relation. The Ornstein–Zernike equation can be formally seen as a definition of the direct correlation function <math>c(r)</math> in terms of the total correlation function <math>h(r)</math>. The details of the system under study (most notably, the shape of the interaction potential <math>u(r)</math>) are taken into account by the choice of the closure relation. Commonly used closures are the [[Percus–Yevick approximation]], well adapted for particles with an impenetrable core, and the [[hypernetted-chain equation]], widely used for "softer" potentials. | |||
More information can be found in.<ref>D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976</ref> | |||
== See also == | |||
*[[Percus–Yevick approximation]], a closure relation for solving the OZ equation | |||
*[[Hypernetted-chain equation]], a closure relation for solving the OZ equation | |||
== References == | |||
<references/> | |||
== External links == | |||
*[http://cbp.tnw.utwente.nl/PolymeerDictaat/node15.html The Ornstein–Zernike equation and integral equations] | |||
*[http://www4.ncsu.edu/~ctk/PAPERS/OZwavelet4.pdf Multilevel wavelet solver for the Ornstein–Zernike equation Abstract] | |||
*[http://www.iop.org/EJ/S/UNREG/lLP4nnFLwybFbk9aWg47cQ/article/0953-8984/12/38/101/c038l1.pdf Analytical solution of the Ornstein–Zernike equation for a multicomponent fluid] | |||
*[http://www.iop.org/EJ/article/0295-5075/54/4/475/6545.html The Ornstein–Zernike equation in the canonical ensemble] | |||
*[http://www.springerlink.com/content/g9mf9h70a2wugujl Ornstein–Zernike Theory for Finite-Range Ising Models Above T<sub>c</sub>] | |||
{{DEFAULTSORT:Ornstein-Zernike equation}} | |||
[[Category:Statistical mechanics]] | |||
[[Category:Integral equations]] |
Revision as of 00:18, 20 December 2013
In statistical mechanics the Ornstein–Zernike equation (named after Leonard Ornstein and Frits Zernike) is an integral equation for defining the direct correlation function. It basically describes how the correlation between two molecules can be calculated. Its applications can mainly be found in fluid theory.
Derivation
The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to[1] for the full derivation.
It is convenient to define the total correlation function:
which is a measure for the "influence" of molecule 1 on molecule 2 at a distance away with as the radial distribution function. In 1914 Ornstein and Zernike proposed [2] to split this influence into two contributions, a direct and indirect part. The direct contribution is defined to be given by the direct correlation function, denoted . The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as
which is called the Ornstein–Zernike equation. Its interest is that, by eliminating the indirect influence, is shorter-ranged than and can be more easily described. The OZ equation has the interesting property that if one multiplies the equation by with and integrate with respect to and one obtains:
If we then denote the Fourier transforms of and by and this rearranges to
from which we obtain that
One needs to solve for both and (or, equivalently, their Fourier transforms). This requires an additional equation, known as a closure relation. The Ornstein–Zernike equation can be formally seen as a definition of the direct correlation function in terms of the total correlation function . The details of the system under study (most notably, the shape of the interaction potential ) are taken into account by the choice of the closure relation. Commonly used closures are the Percus–Yevick approximation, well adapted for particles with an impenetrable core, and the hypernetted-chain equation, widely used for "softer" potentials. More information can be found in.[3]
See also
- Percus–Yevick approximation, a closure relation for solving the OZ equation
- Hypernetted-chain equation, a closure relation for solving the OZ equation
References
- ↑ Frisch, H. & Lebowitz J.L. The Equilibrium Theory of Classical Fluids (New York: Benjamin, 1964)
- ↑ Ornstein, L. S. and Zernike, F. Accidental deviations of density and opalescence at the critical point of a single substance. Proc. Acad. Sci. Amsterdam 1914, 17, 793-806
- ↑ D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976
External links
- The Ornstein–Zernike equation and integral equations
- Multilevel wavelet solver for the Ornstein–Zernike equation Abstract
- Analytical solution of the Ornstein–Zernike equation for a multicomponent fluid
- The Ornstein–Zernike equation in the canonical ensemble
- Ornstein–Zernike Theory for Finite-Range Ising Models Above Tc