The critical exponents ${\displaystyle \alpha ,\alpha ',\beta ,\gamma ,\gamma '}$  and ${\displaystyle \delta }$  are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows

${\displaystyle M(t,0)\simeq (-t)^{\beta }}$ , for ${\displaystyle t\uparrow 0}$ 

${\displaystyle M(0,H)\simeq |H|^{1/\delta }\mathrm {sign} (H)}$ , for ${\displaystyle H\rightarrow 0}$ 

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

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

where

${\displaystyle t\equiv {\frac {T-T_{c}}{T_{c}}}}$  measures the temperature relative to the critical point.

Near the critical point, Widom's scaling relation reads

${\displaystyle H(t)\simeq M|M|^{\delta -1}f(t/|M|^{1/\beta })}$ .

where ${\displaystyle f}$  has an expansion

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

with ${\displaystyle \omega }$  being Wegner's exponent governing the approach to scaling.

The scaling hypothesis is that near the critical point, the free energy ${\displaystyle f(t,H)}$ , in ${\displaystyle d}$  dimensions, can be written as the sum of a slowly varying regular part ${\displaystyle f_{r}}$  and a singular part ${\displaystyle f_{s}}$ , with the singular part being a scaling function, i.e., a homogeneous function, so that

${\displaystyle f_{s}(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}f_{s}(t,H)\,}$ 

Then taking the partial derivative with respect to H and the form of M(t,H) gives

${\displaystyle \lambda ^{q}M(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}M(t,H)\,}$ 

Setting ${\displaystyle H=0}$  and ${\displaystyle \lambda =(-t)^{-1/p}}$  in the preceding equation yields

${\displaystyle M(t,0)=(-t)^{\frac {d-q}{p}}M(-1,0),}$  for ${\displaystyle t\uparrow 0}$ 

Comparing this with the definition of ${\displaystyle \beta }$  yields its value,

${\displaystyle \beta ={\frac {d-q}{p}}\equiv {\frac {\nu }{2}}(d-2+\eta ).}$ 

Similarly, putting ${\displaystyle t=0}$  and ${\displaystyle \lambda =H^{-1/q}}$  into the scaling relation for M yields

${\displaystyle \delta ={\frac {q}{d-q}}\equiv {\frac {d+2-\eta }{d-2+\eta }}.}$ 

Hence

${\displaystyle {\frac {q}{p}}={\frac {\nu }{2}}(d+2-\eta ),~{\frac {1}{p}}=\nu .}$ 

Applying the expression for the isothermal susceptibility ${\displaystyle \chi _{T}}$  in terms of M to the scaling relation yields

${\displaystyle \lambda ^{2q}\chi _{T}(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}\chi _{T}(t,H)\,}$ 

Setting H=0 and ${\displaystyle \lambda =(t)^{-1/p}}$  for ${\displaystyle t\downarrow 0}$  (resp. ${\displaystyle \lambda =(-t)^{-1/p}}$  for ${\displaystyle t\uparrow 0}$ ) yields

${\displaystyle \gamma =\gamma '={\frac {2q-d}{p}}\,}$ 

Similarly for the expression for specific heat ${\displaystyle c_{H}}$  in terms of M to the scaling relation yields

${\displaystyle \lambda ^{2p}c_{H}(\lambda ^{p}t,\lambda ^{q}H)=\lambda ^{d}c_{H}(t,H)\,}$ 

Taking H=0 and ${\displaystyle \lambda =(t)^{-1/p}}$  for ${\displaystyle t\downarrow 0}$  (or ${\displaystyle \lambda =(-t)^{-1/p}}$  for ${\displaystyle t\uparrow 0)}$  yields

${\displaystyle \alpha =\alpha '=2-{\frac {d}{p}}=2-\nu d}$ 

As a consequence of Widom scaling, not all critical exponents are independent but they can be parameterized by two numbers ${\displaystyle p,q\in \mathbb {R} }$  with the relations expressed as

${\displaystyle \alpha =\alpha '=2-\nu d,}$ 

${\displaystyle \gamma =\gamma '=\beta (\delta -1)=\nu (2-\eta ).}$ 

The relations are experimentally well verified for magnetic systems and fluids.

• H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
• H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online)