It is relatively straightforward to derive Green-Kubo relations for the other Navier-Stokes transport coefficients, namely bulk viscosity and thermal conductivity. In §6.3 when we describe the SLLOD equations of motion for viscous flow we will find a simpler way of deriving Green-Kubo relations for both viscosity coefficients. For now we simply state the Green-Kubo relation for bulk viscosity as (Zwanzig, 1965),
(4.54)
The Green-Kubo relation for thermal conductivity can be derived by similar arguments to those used in the viscosity derivation. Firstly we note from (2.26), that in the absence of a velocity gradient, the internal energy per unit volume ρU obeys a continuity equation, ρdU∕dt=-∇⋅JQ. Secondly, we note that Fourier's definition of the thermal conductivity coefficient λ, from equation (2.56a), is JQ=-λ∇T. Combining these two results we obtain
(4.55)
Unlike the previous examples, both U and T have nonzero equilibrium values; namely, ⟨U⟩ and ⟨T⟩. A small change in the left-hand side of equation (4.55) can be written as (ρ+∆ρ)d(⟨U⟩+∆U)∕dt. By definition d⟨U⟩∕dt=0, so to first order in ∆, we have ρd∆U∕dt. Similarly, the spatial gradient of ⟨T⟩ does not contribute, so we can write
(4.56)
The next step is to relate the variation in temperature ∆T to the variation in energy per unit volume ∆(ρU). To do this we use the thermodynamic definition,
(4.57)
where cV is the specific heat per unit mass. We see from the second equality, that a small variation in the temperature ∆T is equal to ∆(ρU)∕ρcV. Therefore,
(4.58)
If DT≡λ∕ρcV is the thermal diffusivity, then in terms of the wavevector dependent internal energy density equation (4.58) becomes,
(4.59)
If C(k,t) is the wavevector dependent internal energy density autocorrelation function,
(4.60)
then the frequency and wavevector dependent diffusivity is the memory function of energy density autocorrelation function,
(4.61)
Using exactly the same procedures as in §4.1 we can convert (4.61) to an expression
for the diffusivity in terms of a current correlation function. From (4.7 & 10)
if 
then,
(4.62)
Using equation (4.10), we obtain the analogue of (4.11),
(4.63)
If we define the analogue of equation (4.49), that is Φ(k,t)=k2NQ(k,t), then equation (4.63) for the thermal diffusivity can be written in the same form as the wavevector dependent shear viscosity equation (4.52). That is
(4.64)
Again we see that we must take the zero wavevector limit before we take the zero frequency limit, and using the canonical ensemble fluctuation formula for the specific heat,
(4.65)
we obtain the Green-Kubo expression for the thermal conductivity
(4.66)
This completes the derivation of Green-Kubo formula for thermal transport coefficients. These formulae relate thermal transport coefficients to equilibrium properties. In the next chapter we will develop nonequilibrium routes to the thermal transport coefficients.