Crossover scaling predictions

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

It is more difficult to predict the critical amplitudes that depend on microscopic details of the system. The essential issue is to predict how the amplitudes depend on the molecular weight of polymer chains. In the absence of detailed renormalization group calculations some crossover scaling arguments have been evoked to suggest that the critical amplitudes vary with molecular weight according to simple power laws e.g.. 

One sees that the data for xi span the full range from the mean-field result (xi = 1 /4) to the scaling prediction (cf. Eq. (33)), while the data for X3 are compatible with the mean-field result. However, the experimental results for X2 seem to indicate a clear discrepancy with the mean-field result X2 = 1/2 (cf. Eq. (35)). Enders et al. then also allowed for a non-mean-field crossover exponent, writing... 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

Since in Fig. 19 critical temperatures have been estimated by a rough extrapolation only, this study [39] could not make too definitive statements about the shift of the critical temperature due to confinement. A more extensive study has been possible for the bond fluctuation model [55], extracting both critical temperatures and the coexistence curves by use of the finite size scaling technique [236-239,280]. Figs.21, 22 present the results for chain length N=32. While mean field theory Eqs. (36) and (37) in Sect. 2.1 has predicted a crossover from... 

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