Critical exponents

2013-02-17T16:49:32Z

93.207.220.254: /* Magnetic order parameter exponent: \beta */

'''Critical exponents'''. Groups of critical exponents form [[universality classes]].
==Reduced distance: <math>\epsilon</math>==
<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.

:<math>\epsilon = \left| 1 -\frac{T}{T_c}\right|</math>

Note that this implies a certain symmetry when the [[Critical points|critical point]] is approached from either 'above' or 'below', which is not necessarily the case.
==Heat capacity exponent: <math>\alpha</math>==
The isochoric [[heat capacity]] is given by <math>C_v</math>

:<math>\left. C_v\right.=C_0 \epsilon^{-\alpha}</math>

Theoretically one has <math>\alpha = 0.1096(5)</math><ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E '''65''' 066127 (2002)]</ref> for the three dimensional [[Ising model]], and <math>\alpha = -0.0146(8)</math><ref name="Campostrini2001" >[http://dx.doi.org/10.1103/PhysRevB.63.214503 Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B '''63''' 214503 (2001)]</ref> for the three-dimensional XY [[Universality classes |universality class]].
Experimentally <math>\alpha = 0.1105^{+0.025}_{-0.027}</math><ref>[http://dx.doi.org/10.1103/PhysRevE.59.1795 A. Haupt and J. Straub "Evaluation of the isochoric heat capacity measurements at the critical isochore of SF6 performed during the German Spacelab Mission D-2", Physical Review E '''59''' pp. 1795-1802 (1999)]</ref>.

==Magnetic order parameter exponent: <math>\beta</math>==
The magnetic order parameter, <math>m</math> is given by

:<math>\left. m\right. = m_0 \epsilon^\beta</math>

Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.

==Susceptibility exponent: <math>\gamma</math>==
[[Susceptibility]]

:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math>

Theoretically one has <math>\gamma = 1.2373(2)</math><ref name="Campostrini2002"> </ref> for the [[Universality classes#Ising |three dimensional Ising model]], and <math>\gamma = 1.3177(5)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
==Correlation length==

:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math>

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==
====Fisher inequality====
The Fisher inequality (Eq. 5 <ref>[http://dx.doi.org/10.1103/PhysRev.180.594 Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review '''180''' pp. 594-600 (1969)]</ref>)

:<math>\gamma \le (2-\eta) \nu</math>
====Griffiths inequality====
The Griffiths inequality (Eq. 3 <ref>[http://dx.doi.org/10.1103/PhysRevLett.14.623 Robert B. Griffiths "Thermodynamic Inequality Near the Critical Point for Ferromagnets and Fluids", Physical Review Letters '''14''' 623-624 (1965)]</ref>):

:<math>(1+\delta)\beta \ge 2-\alpha'</math>
====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>

:<math>d\nu \ge 2-\alpha</math>
====Liberman inequality====
<ref>[http://dx.doi.org/10.1063/1.1726488 David A. Liberman "Another Relation Between Thermodynamic Functions Near the Critical Point of a Simple Fluid", Journal of Chemical Physics '''44''' 419-420 (1966)]</ref>
====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

:<math>\alpha' + 2\beta + \gamma' \ge 2</math>.

Using the above-mentioned values<ref name="Campostrini2002"> </ref> one has:

:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>
====Widom inequality====
The Widom inequality <ref>[http://dx.doi.org/10.1063/1.1726135 B. Widom "Degree of the Critical Isotherm", Journal of Chemical Physics '''41''' pp. 1633-1634 (1964)]</ref>

:<math>\gamma' \ge \beta(\delta -1)</math>
==Hyperscaling==
==Gamma divergence==
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form

:<math>\left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math>

where <math>\kappa_T</math> is the [[Compressibility#Isothermal compressibility | isothermal compressibility]]. <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state#Critical exponents | Van der Waals equation of state]], and is usually 1.2 to 1.3.

==Epsilon divergence==
When approaching the critical point along the critical isotherm the divergence is of the form

:<math>\left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}</math>

where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8.
==References==
<references/>
[[category: statistical mechanics]]

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Critical exponents