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4 Critical Contribution, Dynamic Scaling and Crossover Theory

where

ΨD = arccos(1+q2 D ·ξ2)−1/2

1/2 tan(ΨD/2)

˜ω =  qcξ−1 qcξ+1

L( ˜ω) = ln 1+ ˜ω

 if qcξ > 1

1−˜ω

= 2·arctan| ˜ω| if qcξ ≤1

For large ε, the crossover function defined by Eq.(4.31) behaves as H(ξ(ε),qD,qc) →0, so that η →ηbg. In the asymptotic limit the Eq.(4.31) simplifies to the power law in Eq.(4.28). The influence of the crossover function is not only restricted to the shear vis- cosity. The cut-off wave numbers qc and qD play also an important role in the mutual diffusion coefficient, which is given by:

The value ∆D represents the singular contribution, which is shaped by the Kawasaki func- tion ΩK [66]:

with x = qξ. The mutual diffusion is then represented by:

with:

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