Critical amplitudes

The linear visco-elastic range ends when the elastic modulus G starts to fall off with the further increase of the strain amplitude. This value is called the critical amplitude yi This is the maximum amplitude that can be used for non-destructive dynamic oscillation measurements... 

In computer simulations we had chosen the following parameters of the potential h 1, a = 2. With such a choice the coordinates of minima equal Xmin = 1, the barrier height in the absence of driving is A = 1, the critical amplitude Ac is around 1.5, and we have chosen A = 2 to be far enough from Ac. In order to obtain the correlation function K t + x, t] we solved the FPE (2.6) numerically, using the Crank-Nicholson scheme. 

The model of interacting oscillators was developed to describe the concerted decomposition of a molecule [10]. The decomposing molecule is treated as a collection of oscillators. The reaction of concerted decomposition is described as the transition of the system of oscillators from the thermal vibration with amplitude d to the vibration with critical amplitude d. If n bonds participate in the concerted decomposition and the activated energy is equal to En the rate constant of concerted decomposition kn depends on n and En according to the equation ... 

Fig. 20. Critical amplitude of acoustic treatment versus time...

Fig. 20 The forcing amplitude as above which the periodic solution A i is linearly stable. The solid line is the approximate analytical result according to (59) and the solid circles are obtained numerically. The critical amplitude ac above which the evolution of the initially homogeneous system after the quench is locked to the 2%/q-periodicity of the forcing (solid squares, obtained numerically). The results are given for e = 1...

We have found that in the 2D case, similar to ID, there exists a critical driving amplitude ac above which the spinodal decomposition ends up in the stationary periodic solution a striped structure with the period of the forcing. The critical amplitude turned out to be about three to five times larger than in the ID case. In particular, for q = 6n/Lx with Lx = 256 one has in 2D ac = 0.014 whereas for ID ac = 0.0045. Thus, for 2D the upper curve in Fig. 20 moves slightly upward (the linear stability curve as remains unchanged). 

D. Schwahn, G. Meier, K. Mortensen, and S. Janssen (1994) On the N-scaling of the ginzburg number and the critical amplitudes in various compatible polymer blends. J. Phys. II (France) 4, pp. 837-848 H. Frielinghaus, D. Schwahn, L. Willner, and T. Springer (1997) Thermal composition fluctuations in binary homopolymer mixtures as a function of pressure and temperature. Physica B 241, pp. 1022-1024... 

Another critical exponent (5) is defined considering the variation of the order parameter at Tc as a function of the conjugate field H, D being the associated critical amplitude,... 

Thus there exist relations between suitable ratios of critical amplitudes and derivatives of the scaling functions. We now consider the response function... 

At this point, we mention a further consequence of the universality principle alluded to above. For each universality class (such as that of the Ising model or that of the XY model, etc.) not just the critical exponents are universal, but also the scaling function F(H), apart from non-universal scale factors for the occurring variables (a factor for H we have expressed via the ratio C/B in eq. (84), for instance). A necessary implication then is the universality of certain critical amplitude ratios, where all scale factors for the variables of interest cancel out. In particular, ratios of critical amplitudes of corresponding quantities above and below Tc, A+j A [eq. (7)], C+jC [eq. (6)] and f+/ [eq. (38)] are universal (Privman et al., 1991). A further relation exists between the amplitude D and B and C 1 Writing M H -> oo) = XHl/ cf. eqs. (87) and (91), the universality of M(H) states that X is universal. But since 0 = B tfM H) = B t PXH = B] SC S H] X, a comparison with eq. (45) yields... 

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... 

For ordinary critical phenomena, such a spatial anisotropy is not very important — it gives rise to an anisotropy of the critical amplitude, of the correlation length in different lattice directions ( i = r[-1, fx = xffl 1 ), while the critical exponent dearly is the same for all spatial directions. Of course, this is no longer necessarily true at Lifshitz points There is no reason to assume that both functions / (p), K (p) in eq. (122) vanish for p = pt Let us rather assume that only ih(pl) — 0 while K (pO > 0 this yields the uniaxial Lifshitz point (Homreich et al., 1975). We then have to add a term f Ki (p)[32 (x)/9x ]2 to eq. (122) to find... 

The parameter Bm is the critical amplitude associated with the singularity (subscript a for atomic). 

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