- Widom scaling
Widom scaling is a hypothesis in
Statistical mechanics regarding the free energy of amagnetic system near itscritical point which leads to thecritical exponent s becoming no longer independent so that they can be parameterized in terms of two values.Widom scaling is an example of universality.
Definitions
The critical exponents alpha, alpha', eta, 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)^{eta}, for t uparrow 0 :M(0,H) simeq |H|^{1/ delta} mathrm{sign}(H), for H ightarrow 0 :chi_T(t,0) simeq egin{cases} (t)^{-gamma}, & extrm{for} t downarrow 0 \ (-t)^{-gamma'}, & extrm{for} t uparrow 0 end{cases}
:c_H(t,0) simeq egin{cases} (t)^{-alpha} & extrm{for} t downarrow 0 \ (-t)^{-alpha'} & extrm{for} t uparrow 0 end{cases}
where
:t equiv frac{T-T_c}{T_c} measures the temperature relative to the
critical point .Derivation
The scaling hypothesis is that near the critical point, the free energy f(t,H) 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, ie, a
homogeneous function , so that:f_s(lambda^p t, lambda^q H) = lambda 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 M(t, H)
Setting H=0 and lambda = (-t)^{-1/p} in the preceding equation yields
:M(t,0) = (-t)^{frac{1-q}{p M(-1,0), for t uparrow 0
Comparing this with the definition of eta yields its value,
:eta = frac{1-q}{p}
Similarly, putting t=0 and lambda = H^{-1/q} into the scaling relation for "M" yields
:delta = frac{q}{1-q}
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 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 -1}{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 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{1}{p}
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 - eta(delta +1) = 2 - frac{1}{p} :gamma = gamma' = eta(delta -1)
The relations are experimentally well verified for magnetic systems and fluids.
References
H.E. Stanley, "Introduction to Phase Transitions and Critical Phenomena"
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