It is relatively straightforward to apply the Mori-Zwanzig formalism to the calculation of fluctuation expressions for linear transport coefficients. Our first application of the method will be the calculation of shear viscosity. Before we do this we will say a little more about constitutive relations for shear viscosity. The Mori-Zwanzig formalism leads naturally to a non-Markovian expression for the viscosity. Equation (4.31) refers to a memory function rather than a simple Markovian transport coefficient such as the Newtonian shear viscosity. We will thus be lead to a discussion of viscoelasticity (see §2.4).

We choose our test variable A, to be the x-component of the wavevector dependent transverse momentum current J^⊥(k,t).

For simplicity, we define the coordinate system so that k is in the y direction and J^⊥ is in the x direction.

(4.33)

In §3.8 we saw that

(4.34)

where for simplicity we have dropped the Cartesian indices for J and k. We note that at zero wavevector the transverse momentum current is a constant of the motion, . The quantities we need in order to apply the Mori-Zwanzig formalism are easily computed.

The frequency matrix iΩ, defined in (4.26), is identically zero. This is always so in the single variable case as ⟨A*A⟩=0, for any phase variable A. The norm of the transverse current is calculated

(4.35)

At equilibrium p_x1 is independent of p_x2 and (y₁ - y₂) so the correlation function factors into the product of three equilibrium averages. The values of ⟨p_x1⟩ and ⟨p_x2⟩ are identically zero. The random force, F, can also easily be calculated since, if we use (4.34)

(4.36)

we can write,

(4.37)

The time dependent random force (see (4.29)), is

(4.38)

A Dyson decomposition of e^iQLt in terms of e^iLt shows that,

(4.39)

Now for any phase variable B,

(4.40)

Substituting this observation into (4.39) shows that the difference between the propagators e^iQLt and e^iLt is of order k, and can therefore be ignored in the zero wavevector limit.

From equation (4.30) the memory kernel K(t) is ⟨F(t)F*(0)⟩∕⟨AA*⟩. Using equation (4.38), the small wavevector form for K(t) becomes,

(4.41)

The generalised Langevin equation (the analogue of equation 4.31) is

(4.42)

where we have taken explicit note of the Cartesian components of the relevant functions. Now we know that the rate of change of the transverse current is ikP_yx(k,t). This means that the left hand side of (4.42) is related to equilibrium fluctuations in the shear stress. We also know that J(k)=∫ dk'ρ(k'-k)u(k'), so, close to equilibrium, the transverse momentum current (our Langevin variable A), is closely related to the wavevector dependent strain rate γ(k). In fact the wavevector dependent strain rate γ(k) is -ikJ(k)∕ρ(k=0). Putting these two observations together we see that the generalised Langevin equation for the transverse momentum current is essentially a relation between fluctuations in the shear stress and the strain rate - a constitutive relation. Ignoring the random force (constitutive relations are deterministic), we find that equation (4.42) can be written in the form of the constitutive relation (2.76),

(4.43)

If we use the fact that, P_yxV=lim_k→0P_yx(k), η(t) is easily seen to be

(4.44)

Equation (4.43) is identical to the viscoelastic generalisation of Newton's law of viscosity equation (2.76).

The Mori-Zwanzig procedure has derived a viscoelastic constitutive relation. No mention has been made of the shearing boundary conditions required for shear flow. Neither is there any mention of viscous heating or possible non linearities in the viscosity coefficient. Equation (4.42) is a description of equilibrium fluctuations. However unlike the case for the Brownian friction coefficient or the self diffusion coefficient, the viscosity coefficient refers to nonequilibrium rather than equilibrium systems.

The zero wavevector limit is subtle. We can imagine longer and longer wavelength fluctuations in the strain rate γ(k). For an equilibrium system however γ(k=0)≡0 and ⟨γ(k=0)γ*(k=0)⟩≡0. There are no equilibrium fluctuations in the strain rate at k=0. The zero wavevector strain rate is completely specified by the boundary conditions.

If we invoke Onsager's regression hypothesis we can obviously identify the memory kernel η(t) as the memory function for planar (ie. k=0) Couette flow. We might observe that there is no fundamental way of knowing whether we are watching small equilibrium fluctuations at small but non-zero wavevector, or the last stages of relaxation toward equilibrium of a finite k, nonequilibrium disturbance. Provided the nonequilibrium system is sufficiently close to equilibrium, the Langevin memory function will be the nonequilibrium memory kernel. However the Onsager regression hypothesis is additional to, and not part of, the Mori-Zwanzig theory. In §6.3 we prove that the nonequilibrium linear viscosity coefficient is given exactly by the infinite time integral of the stress fluctuations. In §6.3 we will not use the Onsager regression hypothesis.

At this stage one might legitimately ask the question: what happens to these equations if we do not take the zero wavevector limit? After all we have already defined a wavevector dependent shear viscosity in (2.77). It is not a simple matter to apply the Mori-Zwanzig formalism to the finite wavevector case. We will instead use a method which makes a direct appeal to the Onsager regression hypothesis.

Provided the time and spatially dependent strain rate is of sufficiently small amplitude, the generalised viscosity can be defined as (2.77),

(4.45)

Using the fact that γ(k,t)=-iku_x(k,t)=ikJ(k,t)∕ρ, and equation (4.34), we can rewrite (4.45) as,

(4.46)

If we Fourier-Laplace transform both sides of this equation in time, and using Onsager's hypothesis, multiply both sides by J(-k,0) and average with respect to the equilibrium canonical ensemble we obtain,

(4.47)

where C(k,t) is the equilibrium transverse current autocorrelation function ⟨J(k,t)J(-k,0)⟩ and the tilde notation denotes a Fourier-Laplace transform in time,

(4.48)

We call the autocorrelation function of the wavevector dependent shear stress,

(4.49)

We can use the equation (4.34), to transform from the transverse current autocorrelation function C(k,t) to the stress autocorrelation function N(k,t) since,

(4.50)

This derivation closely parallels that for equation (4.10) and (4.11) in §4.1. The reader should refer to that section for more details. Using the fact that, ρ=Nm∕V, we see that,

(4.51)

The equilibrium average C(k,0) is given by equation (4.35). Substituting this equation into equation (4.47) gives us an equation for the frequency and wavevector dependent shear viscosity in terms of the stress autocorrelation function,

(4.52)

This equation is not of the Green-Kubo form. Green-Kubo relations are exceptional being only valid for infinitely slow processes. Momentum relaxation is only infinitely slow at zero wavevector. At finite wavevectors momentum relaxation is a fast process. We can obtain the usual Green-Kubo form by taking the zero k limit of equation (4.52 ). In that case

(4.53)

Because there are no fluctuations in the zero wavevector strain rate the function is discontinuous at the origin. For all nonzero values of k, ! Over the years many errors have been made as a result of this fact. Figure 4.3 above illustrates these points schematically. The results for shear viscosity precisely parallel those for the friction constant of a Brownian particle. Only in the Brownian limit is the friction constant given by the autocorrelation function of the Brownian force.

An immediate conclusion from the theory we have outlined is that all fluids are viscoelastic. Viscoelasticity is a direct result of the Generalised Langevin equation which is in turn an exact consequence of the microscopic equations of motion.