The Scaling Hypothesis

If a quantity is dimensionless (for example, fractional volume), then it remains unchanged when the length scale is changed. On the other hand, any dimensional quantity (for example, volume) must be expressed in terms of some unit of length and it will change when that unit is changed. In general a system will have an (or more than one) intrinsic length scale, such as the mean distance between nearest lattice points (atoms) in a crystal. The scaling hypothesis posits that near the critical point the correlation length, $\xi$, is the only characteristic length scale in terms of which all other quantities with dimensions of length are to be measured. Using this assumption one can derive a number of scaling laws which can be compared with experiment. (A more rigorous derivation of scale invariance and critical exponents requires renormalization group theory which is beyond the level of this course. Ken Wislon received the Nobel prize for that developement [12]). Let us first determine the length dimensions of various quantities. The Gibbs free energy has the same dimension as energy so $G/kT$ is dimensionless. The Gibbs free energy per unit volume $g=G/ kTV$ therefore has the dimension (length)$^{-d}$. One writes this statement as

$$[g]= L^{-d}$$ (4.37)

Note: Although the real world is three dimensional, systems can be studied under conditions which effectively reduces their diemsnionality to two or one. Thus $d$ is the effective dimensionality of the system. The correlation function has by definition the length dimension $2-d- \eta$.

$$[\Gamma]= L^{2-d- \eta} .$$ (4.38)

Therefore it follows from the definition of $\Gamma$ that

$$[{M \over V}]= L^{(2-d-\eta)/2}.$$ (4.39)

Hence from (4.30), the dimension of the magnetic field is

$$[{\cal{H}}/kT]=L^{(\eta-2-d)/2}$$ (4.40)

and then from Eq.(4.28) the dimension of $\chi$ is

$$[kT \chi]= L^{2-\eta} .$$ (4.41)

Now since the scaling hypothesis states that $\xi$ is the only characteristic scale near the transition temperature, one replaces $L$ in the formulae above by $\xi$ and uses also the definition that $\xi$ $\sim t^{-\nu}$ to obtain the critical exponents. Comparing those with the definitions given one obtains four relations:

$$2-\alpha = \nu d$$ (4.42)

$$\beta = -\nu (2-d-\eta)/2$$ (4.43)

$$\gamma = \nu (2-\eta)$$ (4.44)

$$\beta \delta = \nu (2+d-\eta)/2 .$$ (4.45)

Normally a different set of four relations is listed in books. By linear manipulation one obtains the following named relations, Josephson: $\nu d = 2-\alpha$ ,
Rushbrooke: $\alpha + 2\beta + \gamma =2$ ,
Widom : $\gamma =\beta (\delta - 1)$ ,
Fisher: $\gamma = \nu (2- \eta)$ .
In the table the experimental values of the scaling relations are compared with the theoretical predictions. The agreement is very good, supporting the scaling hypothesis. Thus near a second order phase transition, as the correlation length diverges, a system loses memory of the underlying microscopic structure so that different systems obey the same universal relations. These relations may be considered as further examples of emergent laws.