Universality classes: Difference between revisions
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| <math>d</math> || <math>n</math> || <math>\sigma</math> || | | <math>d</math> || <math>n</math> || <math>\sigma</math> || <math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || class | ||
|- | |- | ||
| || || || 3-state Potts | | || || || || || || 3-state Potts | ||
|- | |- | ||
| || || ||Ashkin-Teller | | || || || || || ||Ashkin-Teller | ||
|- | |- | ||
| || || ||Chiral | | || || || || || ||Chiral | ||
|- | |- | ||
| || || ||Directed percolation | | || || || || || ||Directed percolation | ||
|- | |- | ||
| || || | | || || || 0 || <math>1/8</math> || <math>7/4</math> ||Ising | ||
|- | |- | ||
| || || ||Local linear interface | | || || || || || ||Local linear interface | ||
|- | |- | ||
| || || ||Mean-field | | || || ||0 || <math>1/2</math> || 1 ||Mean-field | ||
|- | |- | ||
| || || ||Molecular beam epitaxy | | || || || || || ||Molecular beam epitaxy | ||
|- | |- | ||
| || || ||Random-field | | || || || || || ||Random-field | ||
|} | |} | ||
==3-state Potts== | ==3-state Potts== | ||
Revision as of 14:05, 20 July 2011
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| 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} | 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} | 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} | class | |
| 3-state Potts | ||||||
| Ashkin-Teller | ||||||
| Chiral | ||||||
| Directed percolation | ||||||
| 0 | 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 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 7/4} | Ising | |||
| Local linear interface | ||||||
| 0 | 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 1/2} | 1 | Mean-field | |||
| Molecular beam epitaxy | ||||||
| Random-field |
3-state Potts
Ashkin-Teller
Chiral
Directed percolation
Ising
The Hamiltonian of the Ising model is
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} }
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:
(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
References
- ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4