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Definitions

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

M ( t , 0 ) ≃ ( − t ) β {\displaystyle M(t,0)\simeq (-t)^{\beta }} , for t ↑ 0 {\displaystyle t\uparrow 0} M ( 0 , H ) ≃ | H | 1 / δ s i g n ( H ) {\displaystyle M(0,H)\simeq |H|^{1/\delta }\mathrm {sign} (H)} , for H → 0 {\displaystyle H\rightarrow 0} χ T ( t , 0 ) ≃ { ( t ) − γ , for   t ↓ 0 ( − t ) − γ ′ , for   t ↑ 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}}} c H ( t , 0 ) ≃ { ( t ) − α for   t ↓ 0 ( − t ) − α ′ for   t ↑ 0 {\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

t ≡ T − T c T c {\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

H ( t ) ≃ M | M | δ − 1 f ( t / | M | 1 / β ) {\displaystyle H(t)\simeq M|M|^{\delta -1}f(t/|M|^{1/\beta })} .

where f {\displaystyle f} has an expansion

f ( t / | M | 1 / β ) ≈ 1 + c o n s t × ( t / | M | 1 / β ) ω + … {\displaystyle f(t/|M|^{1/\beta })\approx 1+{\rm {const}}\times (t/|M|^{1/\beta })^{\omega }+\dots } ,

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

Derivation

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

f s ( λ p t , λ q H ) = λ d f s ( t , H ) {\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

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

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

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

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

β = d − q p ≡ ν 2 ( d − 2 + η ) . {\displaystyle \beta ={\frac {d-q}{p}}\equiv {\frac {\nu }{2}}(d-2+\eta ).}

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

δ = q d − q ≡ d + 2 − η d − 2 + η . {\displaystyle \delta ={\frac {q}{d-q}}\equiv {\frac {d+2-\eta }{d-2+\eta }}.}

Hence

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

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

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

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

γ = γ ′ = 2 q − d p {\displaystyle \gamma =\gamma '={\frac {2q-d}{p}}\,}

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

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

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

α = α ′ = 2 − d p = 2 − ν d {\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 p , q ∈ R {\displaystyle p,q\in \mathbb {R} } with the relations expressed as

α = α ′ = 2 − ν d , {\displaystyle \alpha =\alpha '=2-\nu d,} γ = γ ′ = β ( δ − 1 ) = ν ( 2 − η ) . {\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)

References

1. Kerson Huang, Statistical Mechanics. John Wiley and Sons, 1987 ↩