Critical exponents

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Critical exponents. Groups of critical exponents form universality classes.

Reduced distance: \epsilon[edit]

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

\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: \alpha[edit]

The isochoric heat capacity is given by C_v

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

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

Magnetic order parameter exponent: \beta[edit]

The magnetic order parameter, m is given by

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

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

Susceptibility exponent: \gamma[edit]

Susceptibility

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

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

Correlation length[edit]

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

Theoretically one has \nu = 0.63012(16)[1] for the three dimensional Ising model, and \nu = 0.67155(27)[2] for the three-dimensional XY universality class.

Inequalities[edit]

Fisher inequality[edit]

The Fisher inequality (Eq. 5 [4])

\gamma \le (2-\eta) \nu

Griffiths inequality[edit]

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

(1+\delta)\beta \ge 2-\alpha'

Josephson inequality[edit]

The Josephson inequality [6][7][8]

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

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

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

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

Widom inequality[edit]

The Widom inequality [12]

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

Hyperscaling[edit]

Gamma divergence[edit]

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

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

where \kappa_T is the isothermal compressibility. \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

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

where \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) Cite error: Invalid <ref> tag; name "Campostrini2002" defined multiple times with different content Cite error: Invalid <ref> tag; name "Campostrini2002" defined multiple times with different content Cite error: Invalid <ref> tag; name "Campostrini2002" defined multiple times with different content Cite error: Invalid <ref> tag; name "Campostrini2002" defined multiple times with different content
  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) Cite error: Invalid <ref> tag; name "Campostrini2001" defined multiple times with different content Cite error: Invalid <ref> tag; name "Campostrini2001" defined multiple times with different content Cite error: Invalid <ref> tag; name "Campostrini2001" defined multiple times with different content
  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)