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Nonlinear Burnett Coefficients

 

The nonlinear Burnett Coefficients have already been introduced in section      gif. Ever since the Green-Kubo formalism for calculating the linear transport coefficients was developed, there has been interest in a corresponding theory for the nonlinear Burnett coefficients. The discovery of long-time tails   in the velocity autocorrelation function by Alder and Wainwright [1970] indicated that the hydrodynamic transport coefficients do not exist in two dimensions, but do exist in three dimensions. By applying mode-coupling theories,     Ernst et al. [1978] showed that the relation between stress and strain rate should be for hard disks and for hard spheres, rather than the analytic form suggested by (gif). This result indicates that the nonlinear Burnett coefficients do not exist at all, so the interest has intensified for a numerical simulation to test the mode-coupling theories.

In a recent paper by Evans and Lynden-Bell [1988], equilibrium fluctuation expressions for inverse Burnett coefficients were derived for the colour conductivity problem. The coefficients, , give a Taylor series representation of a nonlinear transport coefficient L, in terms of the thermodynamic force F. Thus if a thermodynamic flux J is written in terms of the coefficient's defining constitutive relation as , then the Burnett coefficients are   related by . In order to derive closed form expressions for the Burnett coefficients, it was found necessary to work in the Norton ensemble, in which the   flux J, rather than the thermodynamic force F was the independent variable. The constitutive relation in this case is . In the thermodynamic limit, we may write , and so the non-linear Burnett coefficients can be computed by inverting the series.

Evans and Lynden-Bell [1988] applied constant current dynamics to a canonical ensemble with the currents distributed about an average current . This allowed the derivation of a transient time correlation function for the non-equilibrium    phase average <F>. It was then a simple matter to compute the derivatives of <F> with respect to the average current , as the constant current propagator commutes with the derivative   operator. However, this method appeared to be limited to colour currents, for which an appropriate canonical distribution could be found. In this chapter, we show that this method can be applied to the situation of an arbitrary thermodynamic flux. This result has been reported by Standish and Evans [1990].





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Next: Equations of Motion Up: On Various Questions in Previous: Colour Current



Russell Standish
Thu May 18 11:43:52 EST 1995