Static self-consistent mean field models

When the dimensions of the scatterers are much smaller than the wavelength of sound simple expressions for f(9) and c are obtained in terms of the complex elastic moduli of the inclusion and host materials, and the volume fraction of the inclusions. Alternately, static self-consistent mean field models can be used to derive expressions for the complex effective moduli of the composite material in terms of the complex elastic moduli and volume fractions of the component materials [32,33,34,35]. The propagation wavenumber c can then be expressed in terms of the effective complex moduli of the composite using Eqs.9 and 10. Particularly interesting. 

We have briefly reviewed methods which extend the self-consistent mean-field theory in order to investigate the statics and dynamics of collective composition fluctuations in polymer blends. Within the standard model of the self-consistent field theory, the blend is described as an ensemble of Gaussian threads of extension Rg. There are two types of interactions zero-ranged repulsions between threads of different species with strength /AT and an incompressibility constraint for the local density. 

Both the Anderson and the Kondo (or Coqblin-Schrieffer) model have been solved exactly for thermodynamic properties such as the 4f-electron valence, specific heat, static magnetic and charge susceptibilities, and the magnetization as a function of temperature and magnetic field B by means of the Bethe ansatz (see Schlottmann 1989, and references therein). This method also allows one to calculate the zero-temperature resistivity as a function of B. Non-equilibrium properties, such as the finite temperature resistivity, thermopower, heat conductivity or dynamic susceptibility, could be calculated in a self-consistent approximation (the non-crossing approximation), which works well and is based on an /N expansion where N is the degeneracy of the 4f level. 

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