Ginzburg criterion

The large size of now is responsible for mean-field theory being reliable for large N Invoking the Ginzburg criterion one says mean field theory is self-consistent if the order parameter fluctuation in a correlation volume is much smaller than the order parameter itself. 

Ginzburg criterion. Moreover, theories for the correlation length are basic ingredients for developing theories of inhomogeneous fluids, as needed in the treatment of interfacial phenomena. 

We recall that comparatively sharp and even nonmonotonous crossover from Ising to mean-field behavior has been deduced from experiments for a diversity of ionic systems. We note that this unusually sharp crossover is a striking feature of some other complex systems as well we quote, for example, solutions of polymers in low-molecular-weight solvents [307], polymer blends [308-311], and microemulsion systems [312], Apart from the fact that application of the Ginzburg criterion to ionic fluids yields no particularly... 

Thermal fluctuations can contribute dominantly to the scattering intensity right after the isothermal phase separation starts [70,76], Therefore, conditions 1) and 3) must be fulfilled to ensure that the effect of thermal noise is negligible. The dynamics of phase separation can be adequately described by the mean-field model if condition 2) is satisfied. Condition 2) is a direct consequence of the Landau Ginzburg criterion [75]. Thus, one may establish prerequisites for Eqs. (27) and (33) are the conditions 1) and 3), while Eq. (34) requires conditions 2) and 3). For example, Eq. (27) and as a consequence Eq. (33) cannot be confirmed experimentally not even for small values of q if the quench depth e is too small [70]. Moreover, owing to the effect of thermal fluctuations, Eq. (33) fails at q as qc even if the Landau Ginzburg criterion is fulfilled [70,77]. Thus, in the former case condition 2) is violated whereas in the latter example conditions 1) and 3) are not satisfied. 

The validity of the linear theory observed for the early stage of spinodal decomposition is chiefly related to the large size of the chain molecules. As shown above, characteristic quantities as the time t or the wavelength Am(0) of the fastest growing fluctuation are proportional to Ro and Rg, respectively. Furthermore, the Landau-Ginzburg criterion (cf. condition 2)) ensures that the mean-field regime is sufficiently extended. 

Again one concludes that the scaling relations eqs. (80), (90) and (91) are satisfied, while the hyperscaling relation [eq. (93)] would only be satisfied at d — 8. Indeed, using the Lifshitz exponents in the Ginzburg criterion [eqs. (52)—(55)] one does find that the Landau description of Lifshitz points becomes self-consistent only for d > 8. Thus it is no surprise that the behavior at physical dimensionalities (d = 2, 3) is very different from the above predictions. In fact, in d = 2 one does not have Lifshitz points at non-zero temperature (Selke, 1992). 

For a quantitative discussion of real polymer mixtures, it clearly is important to consider prefactors in evaluating the Ginzburg criterion, Eqs. (93)-(96) or (108), respectively. Also the asymmetry of the two types of chains needs to be taken into account. This problem was considered by Bates et al. [69] who... 

The singular variation of the critical amplitude fi(N) with chain length N is simple seen from the fact that the variable eN which controls the Ginzburg criterion, Eq. (96b), appears as a crossover scaling variable in the order parameter, assuming a smooth crossover between the order parameter in the mean... 

Heermann and coworkers [296, 297] were the first to carry out simulations of spinodal decomposition in two space dimensions. In this case chains cannot penetrate into each other, so each chain can interact only with a few neighboring chains around it, and our discussion of the Ginzburg criterion (Sect. 2.5) implies that nonlinear phenomena are very important even during the early stages of the quench, and a stage where the structure factor increases exponentially fast with... 

Note that all these formulas also contain the result for the limiting case of short chains dynamics described by the Rouse model [139,140] if we formally put NjSsN in these equations. As will be discussed later (Sect. 2.5), there occurs a crossover in the static critical behavior from mean-field-like behavior where ocR e" with e = 1 — x/Xcrit> Sc u(0)cx N e to the nonclassical critical behavior with Ising model [73, 74] critical exponents cc8 , S ii(0) oce , vwO.63, y 1.24. This crossover occurs, as predicted by the Ginzburg criterion [76-79,148] for Ec°c 1/N or [78,9], equivalently for ccRg, N. It thus is seen that for N, = N the crossover from non-mode-coupled dynamics (for /R, to mode-coupled dynamics (for /R, P, /N) and the crossover... 

Gibbs free energy 184 Ginzburg criterion 196, 205, 213-221, 257 GPC, trace analysis 52-93 Guggenheim s approximation 188, 242, 243... 

Ginzburg criterion, are in the range H (T -T)/T - 0.1 - 0.7 [49-51]. However, it has been pointed out that the breakdown of mean-field behavior is progressive [51] the ability of mean-field theory to predict non-universal quantities Xprefactors, GL-parameCers and T ) is lost within a region ( - (( ), Brout criterion [52], which may cover most of Che superconducting regime. 

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