Stable manifold - Revision history

en>Wavelength: inserting 1 hyphen—wikt:finite-dimensional

2014-03-12T02:23:47Z

inserting 1 hyphen—wikt:finite-dimensional

← Older revision		Revision as of 04:23, 12 March 2014
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	The '~~''pair distribution function''' (PDF) describes~~ the ~~distribution of distances between pairs of particles contained within a given volume~~. ~~Mathematically, if '~~'a'' and '~~'b'' are two particles in~~ a ~~fluid, the PDF of~~ ''b'' with respect to ''a'', denoted by <math>g_{ab}(\vec{r})</math> is the probability of finding the particle ''b'' at the distance <math>\vec{r}</math> from ''a'', with ''a'' taken as the origin of coordinates.		Greetings. The writer's title is Phebe and she feels comfy when people use the complete title. For many years he's been operating as a meter reader and it's some thing he truly enjoy. For a while she's been in South Dakota. One of the very best issues in the world for him is to collect badges but he is struggling to find time for it.<br><br>My website ... [http://calvaryhill.net/xe/board_DsjO50/23648 home std test]

	~~== Overview ==~~
	~~The pair distribution function is used to describe the distribution of objects within a medium (for example, oranges in a crate or nitrogen molecules~~ in a gas cylinder). If the medium is homogeneous (i.e. every spatial location has identical properties), then there is an equal probability density for finding an object at any position <math>\vec{r}</math>:
	~~:<math>p(\vec{r})=1/V</math>,~~
	where <math>V</math> is the volume of the container. On the other hand, the likelihood of finding ''pairs of objects'' at given positions (i.e. the two-body probability density) is not uniform. For example, pairs of hard balls must be separated by at least the diameter of a ball. ~~The pair distribution function <math>g(\vec{r},\vec{r'})</math> is obtained by scaling the two-body probability density function by the total number of objects <math>N</math> and the size~~ of the ~~container:~~
	~~:<math>g(\vec{r}, \vec{r}') = p(\vec{r},\vec{r}') V^2 \frac{N-1}{N}</math>.~~
	~~In the common case where the number of objects~~ in the ~~container~~ is ~~large, this simplifies~~ to ~~give:~~
	~~:<math>g(\vec{r}, \vec{r}') \approx p(\vec{r},\vec{r}') V^2</math>.~~

	~~== Simple models and general properties ==~~
	~~The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:~~
	~~:<math>g(\vec{r})=1</math>,~~
	~~where <math>\vec{r}</math>~~ is ~~the separation between a pair of objects. However, this is inaccurate in the case of hard objects as discussed above, because~~ it ~~does not account for the minimum separation required between objects~~. ~~The hole-correction (HC) approximation provides a better model:~~
	~~:<math>g(r) =~~
	~~\begin{cases}~~
	~~0,&r<b,\\~~
	~~1,&r\geq{}b~~
	~~\end{cases},~~
	<~~/math~~>
	~~where~~ <~~math>b</math~~> ~~is the diameter of one of the objects~~.

	~~Although the HC approximation gives a reasonable description of sparsely packed objects, it breaks down for dense packing~~. This may be illustrated by considering a box completely filled by identical hard balls so that each ball touches its neighbours. In this case, every pair of balls in the box is separated by a distance of exactly <math>r=nb</math> where <math>n</math> is a positive whole number. ~~The pair distribution for a volume completely filled by hard spheres is therefore a set of~~ [~~[Dirac delta function]]s of the form~~:
	~~:<math>g(r)=\sum\limits_i\delta(r-ib)<~~/~~math>.~~

	~~Finally, it may be noted that a pair of objects which are separated by a large distance have no influence on each other's position (provided that the container is not completely filled). Therefore,~~
	~~:<math>\lim\limits_{r\to\infty}g(r) = 1<~~/~~math>~~.

	In general, a pair distribution function will take a form somewhere between the sparsely packed (HC approximation) and the densely packed (delta function) models, depending on the packing density <math>f</~~math>.~~

	~~== Radial pair distributions ==~~
	Of special practical importance is the radial pair distribution function, which is independent of orientation. It is a major descriptor for the atomic structure of amorphous materials (glasses, polymers) and liquids. The radial PDF can be calculated directly from physical measurements like [[light scattering]] or [[x-ray powder diffraction]] through the use of [[Fourier Transform]].

	~~In Statistical Mechanics the PDF is given by the expression:~~

	~~<math>~~
	~~g_{ab}(r) = \frac{1}{N_{a} N_b}\sum\limits_{i=1}^{N_a} \sum\limits_{j=1}^{N_b} \langle \delta( \vert \mathbf{r}_{ij} \vert -r)\rangle~~
	~~</math>~~

	~~The [http:~~//~~www.diffpy.org Diffpy] project is used to match crystal structures with PDF data derived from X-ray or neutron diffraction data.~~

	~~== See also ==~~

	* [[Radial distribution function]]
	* [[classical-map hypernetted-chain method]]

	~~[[Category:Statistical mechanics]]~~
	~~[[Category:Condensed matter physics]~~]

en>Solomonfromfinland: added Category:Manifolds using HotCat

2013-04-27T03:36:58Z

added Category:Manifolds using HotCat

← Older revision		Revision as of 05:36, 27 April 2013
Line 1:		Line 1:
	The ~~author~~'~~s title is Christy~~. ~~My spouse doesn~~'~~t like it~~ the ~~way I do but what I really like performing~~ is ~~caving but I don~~'~~t have~~ the ~~time recently~~. ~~My wife and I live~~ in ~~Mississippi and I love~~ every ~~day living right here. My day job~~ is an ~~info officer but I've currently applied~~ for ~~another~~ 1.<br><br>~~My weblog~~ :: are ~~psychics real~~ ([~~http~~://~~ltreme~~.~~com~~/~~index~~.~~php?do~~=/~~profile-127790~~/~~info~~/ ~~ltreme~~.~~com~~])		The '''pair distribution function''' (PDF) describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if ''a'' and ''b'' are two particles in a fluid, the PDF of ''b'' with respect to ''a'', denoted by <math>g_{ab}(\vec{r})</math> is the probability of finding the particle ''b'' at the distance <math>\vec{r}</math> from ''a'', with ''a'' taken as the origin of coordinates.

			== Overview ==
			The pair distribution function is used to describe the distribution of objects within a medium (for example, oranges in a crate or nitrogen molecules in a gas cylinder). If the medium is homogeneous (i.e. every spatial location has identical properties), then there is an equal probability density for finding an object at any position <math>\vec{r}</math>:
			:<math>p(\vec{r})=1/V</math>,
			where <math>V</math> is the volume of the container. On the other hand, the likelihood of finding ''pairs of objects'' at given positions (i.e. the two-body probability density) is not uniform. For example, pairs of hard balls must be separated by at least the diameter of a ball. The pair distribution function <math>g(\vec{r},\vec{r'})</math> is obtained by scaling the two-body probability density function by the total number of objects <math>N</math> and the size of the container:
			:<math>g(\vec{r}, \vec{r}') = p(\vec{r},\vec{r}') V^2 \frac{N-1}{N}</math>.
			In the common case where the number of objects in the container is large, this simplifies to give:
			:<math>g(\vec{r}, \vec{r}') \approx p(\vec{r},\vec{r}') V^2</math>.

			== Simple models and general properties ==
			The simplest possible pair distribution function assumes that all object locations are mutually independent, giving:
			:<math>g(\vec{r})=1</math>,
			where <math>\vec{r}</math> is the separation between a pair of objects. However, this is inaccurate in the case of hard objects as discussed above, because it does not account for the minimum separation required between objects. The hole-correction (HC) approximation provides a better model:
			:<math>g(r) =
			\begin{cases}
			0,&r<b,\\
			1,&r\geq{}b
			\end{cases},
			</math>
			where <math>b</math> is the diameter of one of the objects.

			Although the HC approximation gives a reasonable description of sparsely packed objects, it breaks down for dense packing. This may be illustrated by considering a box completely filled by identical hard balls so that each ball touches its neighbours. In this case, every pair of balls in the box is separated by a distance of exactly <math>r=nb</math> where <math>n</math> is a positive whole number. The pair distribution for a volume completely filled by hard spheres is therefore a set of [[Dirac delta function]]s of the form:
			:<math>g(r)=\sum\limits_i\delta(r-ib)</math>.

			Finally, it may be noted that a pair of objects which are separated by a large distance have no influence on each other's position (provided that the container is not completely filled). Therefore,
			:<math>\lim\limits_{r\to\infty}g(r) = 1</math>.

			In general, a pair distribution function will take a form somewhere between the sparsely packed (HC approximation) and the densely packed (delta function) models, depending on the packing density <math>f</math>.

			== Radial pair distributions ==
			Of special practical importance is the radial pair distribution function, which is independent of orientation. It is a major descriptor for the atomic structure of amorphous materials (glasses, polymers) and liquids. The radial PDF can be calculated directly from physical measurements like [[light scattering]] or [[x-ray powder diffraction]] through the use of [[Fourier Transform]].

			In Statistical Mechanics the PDF is given by the expression:

			<math>
			g_{ab}(r) = \frac{1}{N_{a} N_b}\sum\limits_{i=1}^{N_a} \sum\limits_{j=1}^{N_b} \langle \delta( \vert \mathbf{r}_{ij} \vert -r)\rangle
			</math>

			The [http://www.diffpy.org Diffpy] project is used to match crystal structures with PDF data derived from X-ray or neutron diffraction data.

			== See also ==

			* [[Radial distribution function]]
			* [[classical-map hypernetted-chain method]]

			[[Category:Statistical mechanics]]
			[[Category:Condensed matter physics]]

en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-07T01:38:11Z

ISBNs (Build KC)

New page

The author's title is Christy. My spouse doesn't like it the way I do but what I really like performing is caving but I don't have the time recently. My wife and I live in Mississippi and I love every day living right here. My day job is an info officer but I've currently applied for another 1.<br><br>My weblog :: are psychics real ([http://ltreme.com/index.php?do=/profile-127790/info/ ltreme.com])