Simplified SCGLE Theory

The main conclusion of the previous comparisons is that, except for very short times, in reality there is no practical reason for preferring the multiplicative version of the SCGLE theory over its additive counterpart, particularly if we are interested in intermediate and long times. Since the additive approximation is numerically simpler to implement, we shall no longer refer to the multiplicative approximation. Still, one of the remaining practical difficulties of the SCGLE theory is the involvement of the SEXP irreducible memory functions z) and z) the need to previously 

It is not difficult to see that the original self-consistent set of equations (involving Equations 1.33 and 1.34) shares the same long-time asymptotic solutions as its simplified version. It is then natural to ask what the consequences would be of replacing Equations 1.33 and 1.34 of the full SCGLE set of equations by the simpler approximations in Equations 1.37 and 1.38, that no longer contain the functions t) and t). The proposal of a simplified version of the SCGLE theory 

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To illustrate these ideas let us summarize the general system of equations that constitute the SCGLE theory. In principle, these are the exact results for A (f), F k, f), and t) in Equations 1.20,1.23, and 1.24, complemented with the simplified Vineyard approximation in Equation 1.37 and the simplified interpolating closure in Equation 1.38. This set of equations define the SCGLE theory of colloid dynamics. Its full solution also yields the value of the long-time self-diffusion coefficient which is the order parameter appropriate to detect the glass transition from the fluid side. This is, however, not the only method to detect dynamic arrest transition, as we now explain. 

The multicomponent extension of the SCGLE theory [21] may be summarized, in its simplified form [59], by Equations 1.30 through 1.32, 1.37, and 1.38. It was applied rather recently [72] to the description of dynamic arrest in two simple model colloidal mixtures, namely, the hard-sphere and the repulsive Yukawa binary mixtures. The main contribution of reference [72], however, is the extension to mixtures of Equation 1.39. Thus, the resulting equation for y , the localization length squared of particles of species a, was shown to be... 

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