Correction to scaling

C being another universal constant. It should be emphasized, however, that all these relations neglect corrections to scaling and hence are only asymptotically valid in the limit where both D / and /. The scahng rela-... 

S. Livne, H. Meirovitch. Computer simulation of long polymers adsorbed on a surface I. Corrections to scaling in an ideal chain. J Chem Phys . 4498-4506, 1988. 

There are other scenarios for an apparent mean-field criticality [15, 17]. The most likely one is crossover from asymptotic Ising behavior to mean-field behavior far from the critical point, where the critical fluctuations must vanish. For the vicinity of the critical point, Wegner [43] worked out an expansion for nonasymptotic corrections to scaling of the general form... 

Figure 8.42. Schematic diagram of the lower rotational levels of NO, showing the A -doublet splittings exaggerated for the sake of clarity, but not drawn correctly to scale. The levels of 14N160 studied by Meerts and Dymanus [124] were 2n1/2, J = 1/2 to 7/2, and2n3/2,. 7 = 3/2 to 17/2.

Coexistence of binary systems. Coexisting phases are characterized by different figures of the order parameter M. In pure fluids, one identifies M with the density difference of the coexisting phases. In solutions, M is related to some concentration variable, where theory now advocates the number density or the closely related volume fraction [101]. At a quantitative level, these divergences are described by crossover theory [86,87] or by asymptotic scaling laws and corrections to scaling, which are expressed in the form of a so-called Wegner series [104], The two branches of the coexistence curve are described by... 

Subtle deviations from ideal chain statistics are caused by three-body monomer interactions (see Section 3.3.2.2) at the i9-temperature, leading to logarithmic corrections to scaling. 

While there is reasonable experimental evidence for the universality of scaling functions, the experimental evidence for the universality of amplitude relations such as eq. (94) is not very convincing. One reason for this problem is that the true critical behavior can be observed only asymptotically close to Tc, and if experiments are carried out not close enough to Tc the results for both critical amplitudes and critical exponents are affected by systematic errors due to corrections to scaling. For example, eq. (6) must be written more generally as... 

Very dose to Tc these corrections Cjt r]-t higher order terms are also negligible, but how close one has to get to Tc to see unambiguously the leading behavior depends both on the (universal) correction-to-scaling exponent X and the associated (non-universal) correction-to-scaling amplitude... 

This function is strongly peaked around two values, i.e. x = Nu and x = — Nu, since the width of the Gaussian functions increases only like Nl,i (see Fig. 3.16). Therefore, the law given by (3.3.12) is quite compatible with eqn (3.3.3) and the broadening associated with the Gaussian functions must simply be considered as a correction to scaling laws. 

Another example is from the numerical study of phase transitions. Renormalization theory has proved accurate for the basic scaling properties of simple transitions. The attention of the research community is now shifting to corrections to scaling, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect, and it has been discovered that the random number generator can influence the results [3-6]. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. 

Figure 12. Order-parameter scaling plot, assuming Ising-like universality. In the absence of corrections-to-scaling terms and in a thermodynamic system, these should be straight lines. The sharp curvatures at low values arise from finite-size effects due to limitations of the density fluctuations in the small finite systems.

Fig. 2.19 The indicatrix, or refractive-index ellipsoid, for a general anisotropic medium. OxqX2X3 are the axes of the ellipsoid and PO represents the direction of propagation (wave-normal) of light through the medium. OA and OB are the principal axes of the section of the ellipsoid normal to OP, shown shaded. The possible D vectors for the light are parallel to these axes and their lengths represent the corresponding values of the refractive indices if the ellipsoid is drawn correctly to scale. (Reproduced by permission of Oxford University Press.)...

We note that the leading correction to asymptotic value of the exponent is proportional to 2 — dt>. The next correction term is of order l/log6, and is proportional to 2—Db, where )(, = 2 — e is fractal dimension. These are like the e-expansions, except that there are several inequivalent definitions of dimension for fractals. Thus there are several e s, and the exponents on fractals may require a multi-variable e-expansion. Interestingly, at higher orders, corrections to scaling to the finite-size scaling functions f x), g x) etc. would give corrections to the exponents of the type 1/6. These are of the type exp(—A/e), and such correction terms are not calculable within the conventional e-expansions framework. 

In such a way, one can obtain the index which determines the degree of approach to llie asymptotic behaviour, i.e. determines the nature of the first corrections to scaling. Dps Cloizeaux defines... 

A more phenomenological approach to describe crossover critical phenomena in simple fluids has been developed by Kiselev and coworkers [76-79]. This approach starts from the asymptotic power-law expansion including the leading correction-to-scaling terms which is then multiplied by an empirical crossover functions so that the equation becomes analytic far away from the critical point. A comparison of this approach with the crossover theory based on a Landau expansion has been discussed in earlier publications [13, 78]. One principal difference is that in the application of the results of the RG theory to the Landau expansion the leading correction to asymptotic scaling law is incorporated in the crossover function and recovered upon expanding the crossover function [18]. 

The equations presented thus far describe the behaviour of the thermodynamic properties of fluids asymptotically close to the critical point. The actual temperature range of asymptotic scaling behaviour depends on the magnitude of correction-to-scaling terms. They are incorporated by extending eq 10.1 to... 

In some liquid mixtures one may encounter a closed solubility loop between an upper critical solution point with temperature Ju and concentration Xu and a lower critical solution point with temperature Jl and concentration Xl. One can obtain a quantitative representation of such closed solubility loops if the temperature variable lAri is replaced by " At ji = T j-T) T-T IT jTi. This procedure has been applied successfully in the revised-scaling approximation i.e., without a contribution proportional to IATulI ), but with the addition of a correction-to-scaling contribution proportional to 1A7ul as discussed in Section 10.3.5 ... 

Table 10.8 Leading and correction-to-scaling amplitudes for the crossover Landau model (CLM) .

In this expression T, are and are universal functions with Pq and Pi representing an asymmetric and a first correction-to-scaling function, respectively, while A, are universal exponents with A2 = 2Ai and A3 = A4 = y-fjS — 1. In eqs 10.97 and 10.99 R q) plays the role of a crossover function defined... 

---