Scaling hypothesis

The concept of homogeneity naturally extends to functions of more than one variable. For example, a generalized homogeneous function of two variables, f(x,y), can be written in the form 

We can now turn to a brief discussion of how these ideas relate to critical exponents. 

We will now show how some of the critical exponents defined in equation 7.13 can be deduced in terms of the generalized exponents a and b above. 

P - The critical exponent 0 is defined in equation 7.12 Mq H = 0,f) (— Recalling that M = —df/dH, we first differentiate equation 7.14 with respect to H  

Comparing this to the first expression in equation 7.13, we see that 

---

Figure 34. At the early stages of the phase separation, the scaling of the curvature distributions is in accordance with the dynamic scaling hypothesis [Eqs. (4)—(7)] (the order parameter is nonconserved).

For the symmetric system (< )0 = 0.5) the scaling exponent for the Euler characteristic has been found in accordance with the dynamic scaling hypothesis x L(t) 3 (see Section I.G). The homogeneity index, HI, of the interface defined as [222]... 

Known experimental values [126] are found to satisfy these relations within experimental error. The scaling hypothesis seems to be successful. 

This expression has the form of the scaling hypothesis (5) and the critical exponents may be written down in terms of / and g using (6). 

It is concluded that RG calculations provide a means of calculating Tc, a justification of the scaling hypothesis and a method of finding the critical exponents. 

All the above-listed experiments were interpreted as SIT in two-dimensional (2D) electron systems. This interpretation for those systems where the growth of the resistance was comparatively trifling was based on scaling hypothesis [8] which asserts that there is no non-superconducting metallic state at zero temperature in 2D. Hence any state of a film which does not display tendency to become superconducting, i.e. which has negative derivative dR/dT < 0, should be accepted as insulating. 

The next step in the argument is to consider how G L) changes with the size of the sample. Abrahams et al. (1979) make the scaling hypothesis that when small volumes are combined into a larger one of size bL, G L) is the only quantity needed to calculate the new G(bL), so that. 

As a generalization it is now proposed to introduce the following scaling hypothesis for the chemical potential of pure materials near their critical point ... 

Ml Ki is the thermal conductivity and x/ = 1 //r/. When the same scaling hypothesis is applied to the right-hand side of 9, /9p that has been adopted in setting up Eq. (7.2.2) it follows that... 

From this scaling hypothesis eq. (77) one readily derives a relation between the exponents y, rj and v. Using eq. (77) we find for the isothermal... 

In Sections 4,5 we used the scaling hypothesis to obtain the longitudinal conductivity. To verify this assumption in the framework of the method we use is very complicated since already... 

In the cases where the percolating networks of both A-rich and B-rich domains persist throughout the last stages, the data comply with the scaling hypothesis of Binder and Stauffer [178, 179]... 

Now the finite size scaling hypothesis [262-264] implies for the collective scattering functions... 

---