Critical exponents: Difference between revisions

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'''Critical exponents'''. Groups of critical exponents form [[universality classes]].
==Reduced distance: <math>\epsilon</math>==
==Reduced distance: <math>\epsilon</math>==
<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.
<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.
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:<math>\left. m\right. = m_0 \epsilon^\beta</math>
:<math>\left. m\right. = m_0 \epsilon^\beta</math>


Theoretically one has <math>\beta =0.32653(10)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model,  and <math>\beta = 0.3485(2)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class.
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 exponent: <math>\gamma</math>==
[[Susceptibility]]  
[[Susceptibility]]  
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:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math>
:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math>


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


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


Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref>  for the three dimensional Ising model,  and <math>\nu = .671 55(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.
==Rushbrooke equality==
==Inequalities==
The Rushbrooke equality <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> , proposed by 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
====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 =2</math>.
:<math>\alpha' + 2\beta + \gamma'  \ge 2</math>.


Using the above-mentioned values 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 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==
==Gamma divergence==
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form


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


where <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3.
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==
==Epsilon divergence==
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where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8.
where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8.
==See also==
*[[Universality classes]]
==References==
==References==
<references/>
<references/>
[[category: statistical mechanics]]
[[category: statistical mechanics]]

Latest revision as of 17:49, 17 February 2013

Critical exponents. Groups of critical exponents form universality classes.

Reduced distance: 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 \epsilon} [edit]

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 \epsilon} is the reduced distance from the critical temperature, i.e.

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 \epsilon = \left| 1 -\frac{T}{T_c}\right|}

Note that this implies a certain symmetry when the critical point is approached from either 'above' or 'below', which is not necessarily the case.

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} [edit]

The isochoric heat capacity is given by 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 C_v}

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 \left. C_v\right.=C_0 \epsilon^{-\alpha}}

Theoretically one has 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 = 0.1096(5)} [1] for the three dimensional Ising model, and 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 = -0.0146(8)} [2] for the three-dimensional XY universality class. Experimentally 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 = 0.1105^{+0.025}_{-0.027}} [3].

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} [edit]

The magnetic order parameter, is given by

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 \left. m\right. = m_0 \epsilon^\beta}

Theoretically one has 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 =0.32653(10)} [1] for the three dimensional Ising model, and 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 = 0.3485(2)} [2] for the three-dimensional XY universality class.

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} [edit]

Susceptibility

Theoretically one has 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 = 1.2373(2)} [1] for the three dimensional Ising model, and 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 = 1.3177(5)} [2] for the three-dimensional XY universality class.

Correlation length[edit]

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 \left. \xi \right.= \xi_0 \epsilon^{-\nu}}

Theoretically one has 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 = 0.63012(16)} [1] for the three dimensional Ising model, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \nu =0.67155(27)} [2] for the three-dimensional XY universality class.

Inequalities[edit]

Fisher inequality[edit]

The Fisher inequality (Eq. 5 [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 \gamma \le (2-\eta) \nu}

Griffiths inequality[edit]

The Griffiths inequality (Eq. 3 [5]):

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+\delta)\beta \ge 2-\alpha'}

Josephson inequality[edit]

The Josephson inequality [6][7][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 d\nu \ge 2-\alpha}

Liberman inequality[edit]

[9]

Rushbrooke inequality[edit]

The Rushbrooke inequality (Eq. 2 [10]), based on the work of Essam and Fisher (Eq. 38 [11]) is given by

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' + 2\beta + \gamma' \ge 2} .

Using the above-mentioned values[1] one has:

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 0.1096 + (2\times0.32653) + 1.2373 = 1.99996}

Widom inequality[edit]

The Widom inequality [12]

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' \ge \beta(\delta -1)}

Hyperscaling[edit]

Gamma divergence[edit]

When approaching the critical point along the critical isochore (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 T > T_c} ) the divergence is of the form

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 \left. \right. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}}

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 \kappa_T} is the isothermal compressibility. 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} is 1.0 for the Van der Waals equation of state, and is usually 1.2 to 1.3.

Epsilon divergence[edit]

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

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 \left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}}

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 \epsilon} is 2/3 for the Van der Waals equation of state, and is usually 0.75 to 0.8.

References[edit]

  1. ↑ 1.0 1.1 1.2 1.3 1.4 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)
  2. ↑ 2.0 2.1 2.2 2.3 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)
  3. ↑ 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)
  4. ↑ Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review 180 pp. 594-600 (1969)
  5. ↑ Robert B. Griffiths "Thermodynamic Inequality Near the Critical Point for Ferromagnets and Fluids", Physical Review Letters 14 623-624 (1965)
  6. ↑ B. D. Josephson "Inequality for the specific heat: I. Derivation", Proceedings of the Physical Society 92 pp. 269-275 (1967)
  7. ↑ B. D. Josephson "Inequality for the specific heat: II. Application to critical phenomena", Proceedings of the Physical Society 92 pp. 276-284 (1967)
  8. ↑ Alan D. Sokal "Rigorous proof of the high-temperature Josephson inequality for critical exponents", Journal of Statistical Physics 25 pp. 51-56 (1981)
  9. ↑ David A. Liberman "Another Relation Between Thermodynamic Functions Near the Critical Point of a Simple Fluid", Journal of Chemical Physics 44 419-420 (1966)
  10. ↑ G. S. Rushbrooke "On the Thermodynamics of the Critical Region for the Ising Problem", Journal of Chemical Physics 39, 842-843 (1963)
  11. ↑ 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)
  12. ↑ B. Widom "Degree of the Critical Isotherm", Journal of Chemical Physics 41 pp. 1633-1634 (1964)
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