Universality classes: Difference between revisions
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| || || || || || ||Directed percolation | | || || || || || ||Directed percolation | ||
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| || || || 0 || <math>1/8</math> || <math>7/4</math> ||Ising | | || || || 0 || <math>1/8</math> || <math>7/4</math> || 2D Ising | ||
|- | |||
| || || || 0 || <math>1/8</math> || <math>7/4</math> || 3D Ising | |||
|- | |- | ||
| || || || || || ||Local linear interface | | || || || || || ||Local linear interface | ||
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</math> | </math> | ||
In three dimensions, the | In three dimensions, the critical exponents are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]] and [[Renormalisation group]] analysis provide accurate estimates: | ||
:<math> | |||
\nu=0.6298 (5) | |||
</math><ref name="Hasenbusch">[http://dx.doi.org/10.1103/PhysRevB.59.11471 M. Hasenbusch, K. Pinn and S. Vinti "Critical exponents of the three-dimensional Ising universality class from finite-size scaling with standard and improved actions", Physical Review B '''59''' pp. 11471-11483 (1999)]</ref> | |||
:<math> | |||
\alpha=0.108(5) | |||
</math> <ref name="Kolesik"> [http://dx.doi.org/10.1016/0378-4371(94)00302-A Miroslav Kolesik and Masuo Suzuki "Accurate estimates of 3D Ising critical exponents using the coherent-anomaly method", Physica A: Statistical and Theoretical Physics '''215''' pp. 138-151 (1995)]</ref> | |||
<math> | :<math> | ||
\ | \beta= 0.3269(6) | ||
</math> | </math> <ref name="Talapov">[http://dx.doi.org/10.1088/0305-4470/29/17/042 A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General '''29''' pp. 5727-5733 (1996)]</ref> | ||
<math> | :<math> | ||
\ | \gamma=1.237(4) | ||
</math> | </math><ref name="Kolesik"> </ref> | ||
<math> | :<math> | ||
\ | \delta=4.77(5) | ||
</math> | </math><ref name="Kolesik"> </ref> | ||
<math> | :<math> | ||
\ | \eta =0.0366(8) | ||
</math> | </math><ref name="Hasenbusch"> </ref> | ||
In four and higher dimensions, the critical exponents are | with a critical temperature of <math>K_c = J/k_BT_c = 0.2216544</math><ref name="Talapov"> </ref>. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections. | ||
==Local linear interface== | ==Local linear interface== | ||
==Mean-field== | ==Mean-field== | ||
Revision as of 11:11, 26 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 \beta} | 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} | 2D Ising | |||
| 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} | 3D 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
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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 }
In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations and Renormalisation group analysis provide accurate estimates:
- 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 \nu=0.6298 (5) } [1]
- 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=0.108(5) } [2]
- 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= 0.3269(6) } [3]
- 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=1.237(4) } [2]
- 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=4.77(5) } [2]
- 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 \eta =0.0366(8) } [1]
with a critical temperature of 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 K_c = J/k_BT_c = 0.2216544} [3]. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.
Local linear interface
Mean-field
The critical exponents of are derived as follows [4]:
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
References
- ↑ 1.0 1.1 M. Hasenbusch, K. Pinn and S. Vinti "Critical exponents of the three-dimensional Ising universality class from finite-size scaling with standard and improved actions", Physical Review B 59 pp. 11471-11483 (1999)
- ↑ 2.0 2.1 2.2 Miroslav Kolesik and Masuo Suzuki "Accurate estimates of 3D Ising critical exponents using the coherent-anomaly method", Physica A: Statistical and Theoretical Physics 215 pp. 138-151 (1995)
- ↑ 3.0 3.1 A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General 29 pp. 5727-5733 (1996)
- ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4