Editing Critical points

{{Cleanup-rewrite}}
Critical points are singularities in the [[partition function]].
In the critical point vicinity  (Ref. 1 Eq. 17a)

:<math> \left.\frac{\partial P}{\partial n}\right\vert_{T}   \simeq 0</math> 

and

:<math>n \int_0^{\infty} c(r) ~4 \pi r^2 ~{\rm d}r \simeq  1</math>

For a review of the critical region see the work of Michael E. Fisher (Ref. 2).

 ... Turning now to the question of specific heats, it has long been known
 that real gases exhibit a large ``anomalous" specific-heat maximum
 above <math>T_c</math> which lies near the critical isochore and which is not expected on classical theory..." (Ref. 3)

also

 ... measurements (Ref 4) of <math>C_V(T)</math> for argon along the critical isochore suggest strongly that
 <math>C_V(T) \rightarrow \infty ~{\rm as} ~ T  \rightarrow  T_c \pm</math>. Such a result is again inconsistent with classical theory."

Thus in the vicinity of the liquid-vapour critical point, both the [[Compressibility | isothermal compressibility]] 
and the [[heat capacity]] at constant pressure diverge to infinity.
==Critical exponents==
[[Heat capacity |Specific heat]], ''C''

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

Magnetic order parameter, ''m'',

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

[[Susceptibility]] 

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

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.

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

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

:<math>\alpha + 2\beta + \gamma =2</math>
===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>

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===
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==
#[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics '''49''' pp. 1495-1504 (1983)]
#[http://dx.doi.org/10.1063/1.1704197  Michael E. Fisher "Correlation Functions and the Critical Region of Simple Fluids", Journal of Mathematical Physics '''5''' pp. 944-962 (1964)]
#[http://dx.doi.org/10.1016/S0031-8914(58)80093-2   A. Michels, J.M. Levelt and G.J. Wolkers "Thermodynamic properties of argon at temperatures between 0°C and −140°C and at densities up to 640 amagat (pressures up to 1050 atm.)", Physica '''24''' pp. 769-794 (1958)]
# M. I. Bagatskii and A. V. Voronel and B. G. Gusak "", Journal of Experimental and Theoretical Physics '''16''' pp. 517- (1963)
#[http://dx.doi.org/10.1103/PhysRevA.2.1047 Robert B. Griffiths and John C. Wheeler "Critical Points in Multicomponent Systems", Physical Review A '''2''' 1047 - 1064 (1970)]
#[http://dx.doi.org/10.1103/RevModPhys.46.597 Michael E. Fisher "The renormalization group in the theory of critical behavior", Reviews of Modern Physics '''46''' pp. 597 - 616 (1974)]
#[http://dx.doi.org/10.1146/annurev.pc.37.100186.001201  J. V. Sengers and  J. M. H. Levelt Sengers "Thermodynamic Behavior of Fluids Near the Critical Point", Annual Review of Physical Chemistry '''37''' pp. 189-222 (1986)]
[[category: statistical mechanics]]
[[category:classical thermodynamics]]