In §2.3 we introduced the linear transport coefficients as the first term in an expansion, about equilibrium, of the thermodynamic flux in terms of the driving thermodynamic forces. The nonlinear Burnett coefficients are the coefficients of this Taylor expansion. Before we address the question of the nonlinear Burnett coefficients we will consider the differential susceptibility of a nonequilibrium steady state. Suppose we expand the irreversible fluxes in powers of the forces, about a nonequilibrium steady state. The leading term in such an expansion is called the differential susceptibility. As we will see, difficulties with commutation relations force us to work in the Norton rather than the Thévenin ensemble. This means that we will always be considering the variation of the thermodynamic forces which result from possible changes in the thermodynamic fluxes.
Consider an ensemble of N-particle systems satisfying the following equations of motion. For simplicity we assume that each member of the ensemble is electrostatically neutral and consists only of univalent ions of charge, ±e = ±1. This system is formally identical to the colour conductivity system which we considered in §6.2.
(9.23)
(9.24)
In these equations, λ and α are Gaussian multipliers chosen so that the x-component of the current per particle, J=Σ eivxi∕N and the temperature T = Σ m(v i - i eiJ)2∕3NkB are constants of the motion. This will be the case provided that,
(9.25)
and
(9.26)
In more physical terms λ can be thought of as an external electric/colour field which takes on precisely those values required to ensure that the current J is constant. Because it precisely fixes the current, it is a phase variable. It is clear from (9.4.3) that the form of the phase variable λ is independent of the value of the current. Of course the ensemble average of λ will depend on the average value of the current. It is also clear that the expression for α is similarly independent of the average value of the current for an ensemble of such systems.
These points can be clarified by considering an initial ensemble characterised by the canonical distribution function, f(0),
(9.27)
In this equation J 0 is a constant which is equal to the canonical average of the current,
(9.28)
If we now subject this ensemble of systems which we will refer to as the J-ensemble, to the equations of motion (9.23 and 9.24), the electrical current and the temperature will remain fixed at their initial values and the mean value of the field multiplier λ, will be determined by the electrical conductivity of the system.
It is relatively straightforward to apply the theory of nonequilibrium steady states to this system. It is easily seen from the equations of motion that the condition known as the Adiabatic Incompressibility of Phase Space (AIΓ) holds. Using equation (9.23) to (9.27), the adiabatic time derivative of the energy functional is easily seen to be,
(9.29)
This equation is unusual in that the adiabatic derivative does not factorise into the product of a dissipative flux and the magnitude of a perturbing external field. This is because in the J-ensemble the obvious external field, λ, is in fact a phase variable and the current, J, is a constant of the motion. As we shall see this causes us no particular problems. The last equation that we need for the application of nonlinear response theory is the derivative,
(9.30)
If we use the isothermal generalisation of the Kawasaki expression for the average of an arbitrary phase variable, B, we find,
(9.31)
In distinction to the usual case we considered in §7.2, the Kawasaki exponent involves a product of two phase variables J and λ, rather than the usual product of a dissipative flux (ie. a phase variable), and a time-dependent external field. The propagator used in (9.31) is the field-dependent thermostatted propagator implicit in the equations of motion (9.23) to (9.26). The only place that the ensemble averaged current appears in (9.31) is in the initial ensemble averages. We can therefore easily differentiate (9.31) with respect to J0 to find that (Evans and Lynden-Bell, 1988),
(9.32)
where Δ(B(t)) ≡ B(t) - <B(t)> and Δ(J(t)) ≡ J(t) - <J(t)> = J(0) - J0. This is an exact canonical ensemble expression for the J-derivative of the average of an arbitrary phase variable. If we let t tend toward infinity we obtain a steady state fluctuation formula which complements the ones we derived earlier for the temperature and pressure derivatives. Equation (9.32) gives a steady state fluctuation relation for the differential susceptibility of, B.
One can check that this expression is correct by rewriting the right hand side of (9.32) as an integral of responses over a set of Norton ensembles in which the current takes on specific values. Using equation (9.27) we can write the average of B(t) as,
(9.33)
We use the notation < B(t) ; J > to denote that subset of the canonical ensemble, (9.27), in which the current takes on the exact value of J. The probability of the J-ensemble taking on an initial x-current of J is easily calculated from (9.27) to be proportional to, exp[-βmN∆J2∕2]. Since the current is a constant of the motion we do not need to specify a time at which the current takes on the specified value.
Differentiating (9.33) we can write the derivative with respect to the average current as a superposition of ∆J-ensemble contributions,
(9.34)
This expression is of course identical to equation (9.32) which was derived using the Kawasaki distribution. (9.34) was derived however, without the use of perturbative mechanical considerations such as those implicit in the use of the Kawasaki distribution. This second derivation is based on two points: the initial distribution is a normal distribution of currents about J0, and; the dynamics preserves the value of the current for each member of the ensemble. Of course the result is still valid even when J is not exactly conserved provided that the time-scale over which it changes is much longer than the time-scale for the decay of steady state fluctuations. This derivation provides independent support for the validity of the renormalized Kawasaki distribution function.
We will now derive relations between the J-derivatives in the J-ensemble and in the constrained ensemble in which J takes on a precisely fixed value (the ΔJ-ensemble). In the thermodynamic limit, the spread of possible values of ΔJ will become infinitely narrow suggesting that we can write a Taylor expansion of <B(t);J> in powers of ΔJ about J0.
(9.35)
Substituting (9.35) into (9.34) and performing the Gaussian integrals over J, we find that,
(9.36)
This is a very interesting equation. It shows the relationship between the derivative computed in a canonical ensemble and a ΔJ-ensemble. It shows that differences between the two ensembles arise from non-linearities in the local variation of the phase variable with respect to the current. It is clear that these ensemble corrections are of order 1∕N compared to the leading terms.