Macroscopic coefficients interference

In view of the very slow time decay of the time correlation functions, it is not clear to what extent the Navier-Stokes transport coefficients can be used even in three dimensions to describe phenomena that vary on a time scale of 50tc, for on this time scale there is not yet a clear separation of microscopic and macroscopic effects. However, usually the Navier-Stokes equations are applied to phenomena that vary on a much longer time scale, and then the slow decay of the correlation functions does not interfere with the hydrodynamic processes. Nevertheless, the divergences of the Burnett and higher-order transport coefficients do appear to have experimental consequences even for three-dimensional systems. In particular, it appears that the dispersion relation for the sound wave frequency wave number k can no longer be expressed as a power series in k as was done in Eq. (133) but instead that fractional powers of the form for /i = l,2,... 

In this chapter we have provided evidence of nontrivial interference between macroscopic behavior and dynamics at the microscopic level. In the vicinity of the Hopf bifurcation this interference is essentially manifested in spatially extended systems where the presence of inhomogeneous modes can lead to the loss of coherence of the uniform limit cycle, even if the latter is macroscopically stable. On the other hand, in well-stirred (0-dimensional) systems (phase) coherence is recovered in the large size limit, since both the phase diffusion coefficient and the variance of fluctuations of the radial variable tend to zero. 

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