High-order singularities

Experiments have been performed on different systems where attractive interactions could be induced, such as colloid-polymer mixtures [39,66] or micellar systems [67]. Figure 7.23 shows the correlation functions of these systems obtained from dynamic light scattering, confirming the existence of the high-order singularity in the region where MCT predicts it. 

Note that for high-order modes, cos (A() = cosh (I AC) and isin( A AC) =-sinh(pA AC) -coshfyA/ l AC), and as a result, the matrix A(AC) becomes singular. An alternative approach to solve this problem, the immittance matrix method, will be discussed in the next section, another one, the scattering matrix method, is described in ". ... 

Y. C. Zhou, S. Zhao, M. Feig, and G. W. Wei. High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources./ Comput. Phys., 213(l) l-30,2006. 

Fade approximants were used in Section 3.B. as a tool for singularity analysis, but their primary function is as a summation method. Unfortunately, their convergence with dimensional expansions tends to be rather slow and uneven. They can be useful if the expansion is known to high order, but they are not appropriate as a low-order summation technique. However, their rate of convergence can be improved considerably if they are used in conjunction with the hybrid expansion, of Eq.(38). The Fade approximants of the E converge quite well even at low order. We will refer to this technique as hybrid Fade summation. 

This implies that the exponents and y defined above are 0 = y = 2( = d) for a first-order transition. Since the symmetry around if = 0 is preserved for finite L, there is no shift of the transition. This feature is different, however, if we consider temperature-driven first-order transitions , since there is no symmetry between the disordered high-temperature phase and the ordered low-temperature phase. In order to understand the rounding of the delta-function singularity of the specific heat, which measures the latent heat for L- oo, it now is useful to consider the energy distribution, for which again a double Gaussian approximation applies ... 

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