Monte Carlo in the microcanonical ensemble

2007-07-04T01:46:44Z

70.158.67.162:

== Integration of the kinetic degrees of freedom ==

Consider a system of <math> \left. N \right. </math> identical particles, with total energy <math> \left. H \right. </math> given by:

: <math> H(X^{3N},P^{3N}) = \sum_{i=1}^{3N} \frac{p_i^2}{2m} U \left( X^{3N} \right) </math>; (Eq.1)

where:

* <math> \left. X^{3N} \right. </math> represents the 3N Cartesian position coordinates of the particles
* <math> \left. P^{3N} \right. </math> stands for the the 3N momenta.

The first term on the right hand side of (Eq. 1) is the [[Kinetic energy |kinetic energy]], whereas the second term is
the [[Potential energy | potential energy]] (a function of the positional coordinates).

Now, let us consider the system in a [[Microcanonical ensemble |microcanonical ensemble]];
let <math> \left. E \right. </math> be the total energy of the system (constrained in this ensemble).

The probability, <math> \left. \Pi \right. </math> of a given position configuration <math> \left. X^{3N} \right. </math>, with potential energy
<math> U \left( X^{3N} \right) </math> can be written as:

: <math> \Pi \left( X^{3N}|E \right) \propto
\int d P^{3N} \delta \left[ K(P^{3N})
- \Delta E \right]
</math> ; (Eq. 2)
where:
* <math> \left. \delta(x) \right. </math> is the [[Dirac delta distribution|Dirac's delta function]]

* <math> \Delta E = E - U\left(X^{3N}\right) </math>.

The Integral in the right hand side of (Eq. 2) corresponds to the surface of a 3N-dimensional (<math> p_i; i=1,2,3,\cdots 3N </math>) hyper-sphere of radius
<math> r = \left. \sqrt{ 2 m \Delta E } \right. </math> ;
therefore:

:<math> \Pi \left( X^{3N}|E \right) \propto \left[ E- U(X^{3N}) \right]^{(3N-1)/2}
</math>.

See Ref. 1 for an application of Monte Carlo simulation using this ensemble.

[[Category: Monte Carlo]]

== References ==

#[http://dx.doi.org/10.1103/PhysRevE.64.042501 N. G. Almarza and E. Enciso "Critical behavior of ionic solids" Physical Review E 64, 042501 (2001) (4 pages) ]

Ornstein-Zernike relation

2007-07-04T01:43:59Z

70.158.67.162:

Notation:
*<math>g(r)</math> is the [[Pair distribution function | pair distribution function]].
*<math>\Phi(r)</math> is the [[Intermolecular pair potential | pair potential]] acting between pairs.
*<math>h(1,2)</math> is the [[Total correlation function | total correlation function]].
*<math>c(1,2)</math> is the [[Direct correlation function | direct correlation function]].
*<math>\gamma (r)</math> is the [[Indirect correlation function | indirect]] (or ''series'' or ''chain'') correlation function.
*<math>y(r_{12})</math> is the [[Cavity correlation function | cavity correlation function]].
*<math>B(r)</math> is the [[ bridge function]].
*<math>\omega(r)</math> is the [[Thermal potential | thermal potential]].
*<math>f(r)</math> is the [[Mayer f-function]].

The '''Ornstein-Zernike relation''' (OZ) integral equation is
:<math>h=h\left[c\right]</math>
where <math>h[c]</math> denotes a functional of <math>c</math>. This relation is exact.
This is complemented by the closure relation
:<math>c=c\left[h\right]</math>
Note that <math>h</math> depends on <math>c</math>, and <math>c</math> depends on <math>h</math>.
Because of this <math>h</math> must be determined [[self-consistently]].
This need for self-consistency is characteristic of all many-body problems.
(Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)
:<math>h(1,2) = c(1,2) \int \rho^{(1)}(3) c(1,3)h(3,2) d3</math>
If the system is both homogeneous and isotropic, the OZ relation becomes (Ref. 1Eq. 6)

:<math>\gamma (r) \equiv h(r) - c(r) = \rho \int h(r')~c(|r - r'|) dr'</math>

In words, this equation (Hansen and McDonald, section 5.2 p. 107)
``...describes the fact that the ''total'' correlation between particles 1 and 2, represented by <math>h(1,2)</math>,
is due in part to the ''direct'' correlation between 1 and 2, represented by <math>c(1,2)</math>, but also to the ''indirect'' correlation,
<math>\gamma (r)</math>, propagated via increasingly large numbers of intermediate particles."

Notice that this equation is basically a convolution, ''i.e.''
:<math>h \equiv c \rho h\otimes c </math>

(Note: the convolution operation written here as <math>\otimes</math> is more frequently written as <math>*</math>)
This can be seen by expanding the integral in terms of <math>h(r)</math>
(here truncated at the fourth iteration):

:<math>h(r) = c(r) \rho \int c(|r - r'|) c(r') dr'
\rho^2 \int \int c(|r - r'|) c(|r' - r''|) c(r'') dr''dr'
\rho^3 \int\int\int c(|r - r'|) c(|r' - r''|) c(|r'' - r'''|) c(r''') dr'''dr''dr'
\rho^4 \int \int\int\int c(|r - r'|) c(|r' - r''|) c(|r'' - r'''|) c(|r''' - r''''|) h(r'''') dr'''' dr'''dr''dr'</math>

''etc.''
Diagrammatically this expression can be written as (Ref. 2):

:[[Image:oz_diag.png]]

where the bold lines connecting root points denote <math>c</math> functions, the blobs denote <math>h</math> functions.
An arrow pointing from left to right indicates an uphill path from one root
point to another. An `uphill path' is a sequence of Mayer bonds passing through increasing
particle labels.
The OZ relation can be derived by performing a functional differentiation
of the grand canonical distribution function (HM check this).
==OZ equation in Fourier space==
The OZ equation may be written in Fourier space as (Eq. 5 in Ref. 3):

:<math>\hat{\gamma} = (I - \rho \hat{c})^{-1} \hat{c} \rho \hat{c}</math>

The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly
to:

:<math>\hat{\gamma} (k) = \frac{4 \pi}{k} \int_0^\infty r~\sin (kr) \gamma(r) dr</math>

:<math>\gamma (r) = \frac{1}{2 \pi^2 r} \int_0^\infty k~\sin (kr) \hat{\gamma}(r) dk</math>

Note:

:<math>\hat{h}(0) = \int h(r) dr</math>

:<math>\hat{c}(0) = \int c(r) dr</math>

==References==
#L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. '''17''' pp. 793- (1914)
#[http://dx.doi.org/10.1103/PhysRevA.45.816 James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A '''45''' pp. 816 - 824 (1992)]
#[http://dx.doi.org/10.1063/1.470724 Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics '''103''' pp. 2625-2633 (1995)]
#Hansen and MacDonald "Theory of Simple Liquids"

[[Category: Integral equations]]

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Monte Carlo in the microcanonical ensemble

Ornstein-Zernike relation