Critical exponents: Difference between revisions

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, 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} 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

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. \chi \right. = \chi_0 \epsilon^{-\gamma}}

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 (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 = 0.67155(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 $\kappa _{T}$ 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]

[Campostrini2002-1] 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)

[Campostrini2001-2] 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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@@ Line 1: / Line 1: @@
-{{stub-general}}
+'''Critical exponents'''. Groups of critical exponents form [[universality classes]].
-{{Cleanup-rewrite}}
+==Reduced distance: <math>\epsilon</math>==
-[[Heat capacity |Specific heat]], ''C''
+<math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.
-:<math>\left. C\right.=C_0 \epsilon^{-\alpha}</math>
+:<math>\epsilon = \left| 1 -\frac{T}{T_c}\right|</math>
-Magnetic order parameter, ''m'',
+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>
-Correlation length
+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>
-where <math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e.
+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>\epsilon = \left| 1 -\frac{T}{T_c}\right|</math>
+:<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:
-Note that this implies a certain symmetry when the critical point is approached from either 'above' or 'below', which is not   necessarily the case.
+:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math>
-Rushbrooke equality
+====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>\alpha + 2\beta + \gamma =2</math>
+:<math>\gamma' \ge \beta(\delta -1)</math>
-====Gamma divergence====
+==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. 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==
 When approaching the critical point along the critical isotherm the divergence is of the form

Critical exponents: Difference between revisions

Latest revision as of 17:49, 17 February 2013

Contents

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]

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]

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]

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]

Correlation length[edit]

Inequalities[edit]

Fisher inequality[edit]

Griffiths inequality[edit]

Josephson inequality[edit]

Liberman inequality[edit]

Rushbrooke inequality[edit]

Widom inequality[edit]

Hyperscaling[edit]

Gamma divergence[edit]

Epsilon divergence[edit]

References[edit]

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Critical exponents: Difference between revisions

Latest revision as of 17:49, 17 February 2013

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]

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]

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]

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]

Correlation length[edit]

Inequalities[edit]

Fisher inequality[edit]

Griffiths inequality[edit]

Josephson inequality[edit]

Liberman inequality[edit]

Rushbrooke inequality[edit]

Widom inequality[edit]

Hyperscaling[edit]

Gamma divergence[edit]

Epsilon divergence[edit]

References[edit]

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