Editing Critical exponents
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Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class. | Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class. | ||
==Inequalities== | ==Inequalities== | ||
====Fisher | ====Fisher ==== | ||
====Josephson inequality==== | ====Josephson inequality==== | ||
The Josephson inequality <ref>[http://dx.doi.org/10.1088/0370-1328/92/2/301 B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society '''92''' pp. 269-275 (1967)]</ref><ref>[http://dx.doi.org/10.1088/0370-1328/92/2/302 B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society '''92''' pp. 276-284 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01008478 Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics '''25''' pp. 51-56 (1981)]</ref> | The Josephson inequality <ref>[http://dx.doi.org/10.1088/0370-1328/92/2/301 B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society '''92''' pp. 269-275 (1967)]</ref><ref>[http://dx.doi.org/10.1088/0370-1328/92/2/302 B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society '''92''' pp. 276-284 (1967)]</ref><ref>[http://dx.doi.org/10.1007/BF01008478 Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics '''25''' pp. 51-56 (1981)]</ref> | ||
:<math>d\nu \ge 2-\alpha</math> | :<math>d\nu \ge 2-\alpha</math> | ||
====Rushbrooke inequality==== | ====Rushbrooke inequality==== | ||
The Rushbrooke inequality (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref>), based on the work of Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by | The Rushbrooke inequality (Eq. 2 <ref>[http://dx.doi.org/10.1063/1.1734338 G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)]</ref>), based on the work of Essam and Fisher (Eq. 38 <ref>[http://dx.doi.org/10.1063/1.1733766 John W. Essam and Michael E. Fisher "Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point", Journal of Chemical Physics 38, 802-812 (1963)]</ref>) is given by | ||
:<math>\alpha | :<math>\alpha + 2\beta + \gamma \ge 2</math>. | ||
Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has: | Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has: | ||
:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math> | :<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math> | ||
====Widom | ====Widom relation==== | ||
==Hyperscaling== | ==Hyperscaling== | ||
==Gamma divergence== | ==Gamma divergence== |