fbpx
Wikipedia

Ornstein–Zernike equation

In statistical mechanics the Ornstein–Zernike (OZ) equation is an integral equation introduced[1] by Leonard Ornstein and Frits Zernike that relates different correlation functions with each other. Together with a closure relation, it is used to compute the structure factor and thermodynamic state functions of amorphous matter like liquids or colloids.

Context edit

The OZ equation has practical importance as a foundation for approximations for computing the pair correlation function of molecules or ions in liquids, or of colloidal particles. The pair correlation function is related via Fourier transform to the static structure factor, which can be determined experimentally using X-ray diffraction or neutron diffraction.

The OZ equation relates the pair correlation function to the direct correlation function. The direct correlation function is only used in connection with the OZ equation, which can actually be seen as its definition.[2]

Besides the OZ equation, other methods for the computation of the pair correlation function include the virial expansion at low densities, and the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. Any of these methods must be combined with a physical approximation: truncation in the case of the virial expansion, a closure relation for OZ or BBGKY.

The equation edit

To keep notation simple, we only consider homogeneous fluids. Thus the pair correlation function only depends on distance, and therefore is also called the radial distribution function. It can be written

 

where the first equality comes from homogeneity, the second from isotropy, and the equivalences introduce new notation.

It is convenient to define the total correlation function as:

 

which expresses the influence of molecule 1 on molecule 2 at distance  . The OZ equation

 

splits this influence into two contributions, a direct and indirect one. The direct contribution defines the direct correlation function,   The indirect part is due to the influence of molecule 1 on a third, labeled molecule 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 molecule 3.

By eliminating the indirect influence,   is shorter-ranged than   and can be more easily modelled and approximated. The radius of   is determined by the radius of intermolecular forces, whereas the radius of   is of the order of the correlation length.[3]

Fourier transform edit

The integral in the OZ equation is a convolution. Therefore, the OZ equation can be resolved by Fourier transform. If we denote the Fourier transforms of   and   by   and  , respectively, and use the convolution theorem, we obtain

 

which yields

 

Closure relations edit

As both functions,   and  , are unknown, one needs an additional equation, known as a closure relation. While the OZ equation is purely formal, the closure must introduce some physically motivated approximation.

In the low-density limit, the pair correlation function is given by the Boltzmann factor,

 

with   and with the pair potential  .[4]

Closure relations for higher densities modify this simple relation in different ways. The best known closure approximations are:[5][6]

The latter two interpolate in different ways between the former two, and thereby achieve a satisfactory description of particles that have a hard core and attractive forces.

References edit

  1. ^ Ornstein, L.S.; Zernike, F. (1914). (PDF). Proceedings of the Royal Netherlands Academy of Arts and Sciences. 17: 793–806. Bibcode:1914KNAB...17..793. Archived from the original (PDF) on 2021-02-06. – Archived 24 Sep 2010 at the 'Digital Library' of the Dutch History of Science Web Center.
  2. ^ V I Kalikmanov: Statistical Physics of Fluids. Basic Concepts and Applications. Springer, Berlin 2001
  3. ^ Kalikmanov p 140
  4. ^ Kalikmanov p 137
  5. ^ Kalikmanov pp 140-141
  6. ^ McQuarrie, D.A. (May 2000) [1976]. Statistical Mechanics. University Science Books. p. 641. ISBN 9781891389153.

External links edit

  • "The Ornstein–Zernike equation and integral equations". cbp.tnw.utwente.nl.
  • "Multilevel wavelet solver for the Ornstein–Zernike equation" (PDF). ncsu.edu (Abstract).
  • "Analytical solution of the Ornstein–Zernike equation for a multicomponent fluid" (PDF). iop.org.
  • "The Ornstein–Zernike equation in the canonical ensemble". iop.org.

ornstein, zernike, equation, statistical, mechanics, ornstein, zernike, equation, integral, equation, introduced, leonard, ornstein, frits, zernike, that, relates, different, correlation, functions, with, each, other, together, with, closure, relation, used, c. In statistical mechanics the Ornstein Zernike OZ equation is an integral equation introduced 1 by Leonard Ornstein and Frits Zernike that relates different correlation functions with each other Together with a closure relation it is used to compute the structure factor and thermodynamic state functions of amorphous matter like liquids or colloids Contents 1 Context 2 The equation 3 Fourier transform 4 Closure relations 5 References 6 External linksContext editThe OZ equation has practical importance as a foundation for approximations for computing the pair correlation function of molecules or ions in liquids or of colloidal particles The pair correlation function is related via Fourier transform to the static structure factor which can be determined experimentally using X ray diffraction or neutron diffraction The OZ equation relates the pair correlation function to the direct correlation function The direct correlation function is only used in connection with the OZ equation which can actually be seen as its definition 2 Besides the OZ equation other methods for the computation of the pair correlation function include the virial expansion at low densities and the Bogoliubov Born Green Kirkwood Yvon BBGKY hierarchy Any of these methods must be combined with a physical approximation truncation in the case of the virial expansion a closure relation for OZ or BBGKY The equation editTo keep notation simple we only consider homogeneous fluids Thus the pair correlation function only depends on distance and therefore is also called the radial distribution function It can be written g r 1 r 2 g r 1 r 2 g r 12 g r 12 g r 12 g 12 displaystyle g mathbf r 1 mathbf r 2 g mathbf r 1 mathbf r 2 equiv g mathbf r 12 g mathbf r 12 equiv g r 12 equiv g 12 nbsp where the first equality comes from homogeneity the second from isotropy and the equivalences introduce new notation It is convenient to define the total correlation function as h 12 g 12 1 displaystyle h 12 equiv g 12 1 nbsp which expresses the influence of molecule 1 on molecule 2 at distance r 12 displaystyle r 12 nbsp The OZ equation h 12 c 12 r d 3 r 3 c 13 h 32 displaystyle h 12 c 12 rho int text d 3 mathbf r 3 c 13 h 32 nbsp splits this influence into two contributions a direct and indirect one The direct contribution defines the direct correlation function c r displaystyle c r nbsp The indirect part is due to the influence of molecule 1 on a third labeled molecule 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 molecule 3 By eliminating the indirect influence c r displaystyle c r nbsp is shorter ranged than h r displaystyle h r nbsp and can be more easily modelled and approximated The radius of c r displaystyle c r nbsp is determined by the radius of intermolecular forces whereas the radius of h r displaystyle h r nbsp is of the order of the correlation length 3 Fourier transform editThe integral in the OZ equation is a convolution Therefore the OZ equation can be resolved by Fourier transform If we denote the Fourier transforms of h r displaystyle h mathbf r nbsp and c r displaystyle c mathbf r nbsp by h k displaystyle hat h mathbf k nbsp and c k displaystyle hat c mathbf k nbsp respectively and use the convolution theorem we obtain h k c k r h k c k displaystyle hat h mathbf k hat c mathbf k rho hat h mathbf k hat c mathbf k nbsp which yields c k h k 1 r h k and h k c k 1 r c k displaystyle hat c mathbf k frac hat h mathbf k 1 rho hat h mathbf k qquad text and qquad hat h mathbf k frac hat c mathbf k 1 rho hat c mathbf k nbsp Closure relations editAs both functions h displaystyle h nbsp and c displaystyle c nbsp are unknown one needs an additional equation known as a closure relation While the OZ equation is purely formal the closure must introduce some physically motivated approximation In the low density limit the pair correlation function is given by the Boltzmann factor g 12 e b u 12 r 0 displaystyle g 12 text e beta u 12 quad rho to 0 nbsp with b 1 k B T displaystyle beta 1 k text B T nbsp and with the pair potential u r displaystyle u r nbsp 4 Closure relations for higher densities modify this simple relation in different ways The best known closure approximations are 5 6 The Percus Yevick approximation for particles with impenetrable hard core the hypernetted chain approximation for particles with soft cores and attractive potential tails the mean spherical approximation the Rogers Young approximation The latter two interpolate in different ways between the former two and thereby achieve a satisfactory description of particles that have a hard core and attractive forces References edit Ornstein L S Zernike F 1914 Accidental deviations of density and opalescence at the critical point of a single substance PDF Proceedings of the Royal Netherlands Academy of Arts and Sciences 17 793 806 Bibcode 1914KNAB 17 793 Archived from the original PDF on 2021 02 06 Archived 24 Sep 2010 at the Digital Library of the Dutch History of Science Web Center V I Kalikmanov Statistical Physics of Fluids Basic Concepts and Applications Springer Berlin 2001 Kalikmanov p 140 Kalikmanov p 137 Kalikmanov pp 140 141 McQuarrie D A May 2000 1976 Statistical Mechanics University Science Books p 641 ISBN 9781891389153 External links edit The Ornstein Zernike equation and integral equations cbp tnw utwente nl Multilevel wavelet solver for the Ornstein Zernike equation PDF ncsu edu Abstract Analytical solution of the Ornstein Zernike equation for a multicomponent fluid PDF iop org The Ornstein Zernike equation in the canonical ensemble iop org Retrieved from https en wikipedia org w index php title Ornstein Zernike equation amp oldid 1214006130, wikipedia, wiki, book, books, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.