Universality classes: Difference between revisions
(→Ising) |
Carl McBride (talk | contribs) m (→Ising: Added some internal links) |
||
| Line 27: | Line 27: | ||
==Directed percolation== | ==Directed percolation== | ||
==Ising== | ==Ising== | ||
The Hamiltonian of the Ising model is | The [[Hamiltonian]] of the [[Ising model]] is | ||
<math> | <math> | ||
| Line 36: | Line 36: | ||
where <math>S_i=\pm 1</math> and the summation runs over the lattice sites. | where <math>S_i=\pm 1</math> and the summation runs over the lattice sites. | ||
The order parameter is | The [[Order parameters | order parameter]] is | ||
<math> | <math> | ||
m=\sum_i S_i | m=\sum_i S_i | ||
</math> | </math> | ||
In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the critical exponents are | In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are | ||
<math> | <math> | ||
\alpha=0 | \alpha=0 | ||
</math> | </math> | ||
(In fact, the specific | (In fact, the [[Heat capacity |specific heat]] diverges logarithmically with the [[Critical points |critical temperature]]) | ||
<math> | <math> | ||
Revision as of 13:46, 20 July 2011
|
This article is a 'stub' page, it has no, or next to no, content. It is here at the moment to help form part of the structure of SklogWiki. If you add sufficient material to this article then please remove the {{Stub-general}} template from this page. |
| 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 d} | 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 n} | 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 \sigma} | name |
| 3-state Potts | |||
| Ashkin-Teller | |||
| Chiral | |||
| Directed percolation | |||
| Ising | |||
| Local linear interface | |||
| Mean-field | |||
| Molecular beam epitaxy | |||
| Random-field |
3-state Potts
Ashkin-Teller
Chiral
Directed percolation
Ising
The Hamiltonian of the Ising model is
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 H=\sum_{<i,j>}S_i S_j }
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 S_i=\pm 1}
and the summation runs over the lattice sites.
The order parameter is 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 m=\sum_i S_i }
In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the critical exponents are 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 \alpha=0 } (In fact, the specific heat diverges logarithmically with the critical temperature)
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 \beta=\frac{1}{8} }
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=\frac{7}{4} }
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 \delta=15 }