Widom scaling

The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}, for t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \mathrm{sign}(H), for H \rightarrow 0 : \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 : 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) \simeq M|M|^{\delta-1} f(t/|M|^{1/\beta}). where f has an expansion : f(t/|M|^{1/\beta})\approx 1+{\rm const}\times( t/|M|^{1/\beta})^\omega +\dots , with \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), in d dimensions, can be written as the sum of a slowly varying regular part f_r and a singular part f_s, with the singular part being a scaling function, i.e., a homogeneous function, so that : 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 : \lambda^q M(\lambda^p t, \lambda^q H) = \lambda^d M(t, H) \, Setting H=0 and \lambda = (-t)^{-1/p} in the preceding equation yields : M(t,0) = (-t)^{\frac{d-q}{p}} M(-1,0), for t \uparrow 0 Comparing this with the definition of \beta yields its value, : \beta = \frac{d-q}{p}\equiv \frac{\nu}2(d-2+\eta). Similarly, putting t=0 and \lambda = H^{-1/q} into the scaling relation for M yields : \delta = \frac{q}{d-q} \equiv \frac{d+2-\eta}{d-2+\eta}. Hence : \frac{q}{p} = \frac{\nu}{2} (d+2-\eta),~\frac 1 p=\nu. Applying the expression for the isothermal susceptibility \chi_T in terms of M to the scaling relation yields : \lambda^{2q} \chi_T (\lambda^p t, \lambda^q H) = \lambda^d \chi_T (t, H) \, Setting H=0 and \lambda = (t)^{-1/p} for t \downarrow 0 (resp. \lambda = (-t)^{-1/p} for t \uparrow 0 ) yields : \gamma = \gamma' = \frac{2q -d}{p} \, Similarly for the expression for specific heat c_H in terms of M to the scaling relation yields : \lambda^{2p} c_H ( \lambda^p t, \lambda^q H) = \lambda^d c_H(t, H) \, Taking H=0 and \lambda = (t)^{-1/p} for t \downarrow 0 (or \lambda = (-t)^{-1/p} for t \uparrow 0) yields : \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 \in \mathbb{R} with the relations expressed as : \alpha = \alpha' = 2-\nu d, : \gamma = \gamma' = \beta(\delta -1)=\nu(2-\eta) . The relations are experimentally well verified for magnetic systems and fluids. ==References==