9.5 The Inverse Burnett Coefficients

We will now use the TTCF formalism in the Norton ensemble, to derive expressions for the inverse Burnett coefficients. The Burnett coefficients, Li, give a Taylor series representation of a nonlinear transport coefficient L(X), defined by a constitutive relation between a thermodynamic force X, and a thermodynamic flux J(Γ),

(9.37)

It is clear from this equation the Burnett coefficients are given by the appropriate partial derivatives of < J >, evaluated at X=0. As mentioned in §9.4 we will actually be working in the Norton ensemble in which the thermodynamic force X, is the dependent rather than the independent variable. So we will in fact derive expressions for the inverse Burnett coefficients, L-1

(9.38)

The TTCF representation for a steady state phase average for our electrical/colour diffusion problem is easily seen to be.

(9.39)

We expect that the initial values of the current will be clustered about J0. If we write,

(9.40)

it is easy to see that if B is extensive then the two terms on the right hand side of (9.40) are O(1) and O(1∕N) respectively. For large systems we can therefore write,

(9.41)

It is now a simple matter to calculate the appropriate J-derivatives.

(9.42)

This equation relates the J-derivative of phase variables to TTCFs. If we apply these formulae to the calculation of the leading Burnett coefficient we of course evaluate the derivatives at J0=0. In this case the TTCFs become equilibrium time correlation functions. The results for the leading Burnett coefficients are (Evans and Lynden-Bell, 1988):

(9.43)

(9.44)

(9.45)

Surprisingly, the expressions for the Burnett coefficients only involve equilibrium, two-time correlation functions. At long times assuming that the system exhibits mixing they each factor into a triple product <B(s→∞)><λ(0)><cum(J(0))>. The terms involving λ(0) and the cumulants of J(0) factor because at time zero the distribution function (9.27), factors into kinetic and configurational parts. Of course these results for the Burnett coefficients could have been derived using the ∆J-ensemble methods discussed in §9.4.

It is apparent that our discussion of the differential susceptibility and the inverse Burnett coefficients has relied heavily on features unique to the colour conductivity problem. It is not obvious how one should carry out the analogous derivations for other transport coefficients. General fluctuation expressions for the inverse Burnett coefficients have recently been derived by Standish and Evans (1989). The general results are of the same form as the corresponding colour conductivity expressions. We refer the reader to the above reference for details.