Universality classes: Difference between revisions
Carl McBride (talk | contribs) m (→Ising: Added some internal links) |
Carl McBride (talk | contribs) m (→Mean-field: Started Mean field section) |
||
| Line 61: | Line 61: | ||
==Local linear interface== | ==Local linear interface== | ||
==Mean-field== | ==Mean-field== | ||
The [[critical exponents]] of are derived as follows <ref>Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4 </ref>: | |||
====Heat capacity exponent: <math>\alpha</math>==== | |||
(final result: <math>\alpha=0</math>) | |||
====Magnetic order parameter exponent: <math>\beta</math>==== | |||
(final result: <math>\beta=1/2</math>) | |||
====Susceptibility exponent: <math>\gamma</math>==== | |||
(final result: <math>\gamma=1</math>) | |||
==Molecular beam epitaxy== | ==Molecular beam epitaxy== | ||
==See also== | ==See also== | ||
Revision as of 14:02, 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 \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 }
Local linear interface
Mean-field
The critical exponents of are derived as follows [1]:
Heat capacity exponent: 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}
(final result: 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} )
Magnetic order parameter exponent: 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}
(final result: 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=1/2} )
Susceptibility exponent: 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}
(final result: 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=1} )
Molecular beam epitaxy
See also
Random-field
- ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4