Editing Critical exponents
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
==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. | ||
Line 19: | Line 18: | ||
:<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 | 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. | ||
==Susceptibility exponent: <math>\gamma</math>== | ==Susceptibility exponent: <math>\gamma</math>== | ||
[[Susceptibility]] | [[Susceptibility]] | ||
Line 26: | Line 24: | ||
:<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 | 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. | ||
==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 | Theoretically one has <math>\nu = 0.63012(16)</math><ref name="Campostrini2002"> </ref> for the three dimensional Ising model, and <math>\nu = 0.67155(27)</math><ref name="Campostrini2001"> </ref> for the three-dimensional XY universality class. | ||
==Rushbrooke equality== | |||
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 | |||
:<math> | :<math>\alpha + 2\beta + \gamma =2</math>. | ||
Using the above-mentioned values one has: | |||
Using the above-mentioned values | |||
:<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math> | :<math>0.1096 + (2\times0.32653) + 1.2373 = 1.99996</math> | ||
==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. \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math> | :<math>\left. \right. C_p \sim \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math> | ||
where | where <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3. | ||
==Epsilon divergence== | ==Epsilon divergence== | ||
Line 73: | Line 52: | ||
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]] |