Widom Scaling - Definitions

Definitions

The critical exponents and are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

, for

$\chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases}$

$c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases}$

where

measures the temperature relative to the critical point.

Near the critical point, Widom's scaling relation reads

where has an expansion

$f(t/|M|^{1/\beta})\approx 1+{\rm const}\times( t/|M|^{1/\beta})^\omega +\dots$ ,

with being Wegner's exponent governing the approach to scaling.

Read more about this topic: Widom Scaling

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