Viscoelastic Constitutive Relation In the Mori-Zwanzig formalism, the following identification for the generalised shear viscosity $\eta(t)$ is given:
$$
\eta(t) = \frac{V}{k_B T} \langle \sigma(t) \sigma(0) \rangle,
$$
identified as such because it appears in what amounts to a viscoelastic constitutive relation:
$$
\sigma(t) = -\int_0^t \mathrm{d} \tau \ \eta(t-\tau) \gamma(\tau).
$$
Here $\sigma$ is the shear stress, $\gamma$ the shear strain rate, $V$ volume, $T$ temperature and $\langle \ldots \rangle$ the equilibrium phase average. This approach was done in the books by both Evans & Morriss and Hansen & McDonald on liquid theory. I'm happy with the general usage of memory functions coming from Liouvillian evolution, however here something strange seems to have happened. Substituting the expression for $\eta$ into the constitutive relation gives:
$$
\sigma(t) = -\int_0^t \mathrm{d} \tau \ \frac{V}{k_B T} \langle \sigma(t-\tau) \sigma(0) \rangle \gamma(\tau),
$$
which seems to just be an expression for the shear stress in terms of itself. The viscosity, which describes the relationship between stress and strain rate, is itself a function of the history of shear stress. This makes sense, but when you put it together you seem to get a sort of recursive equation. It doesn't seem like much use as a constitutive equation given that it contains the quantity (the stress) which is being solved for. How do I interpret this?
 A: The equations are correct, but some confusion appears to have arisen from the notation and the nomenclature.
Firstly, in your last equation
$$
\sigma(t) = -\frac{V}{k_BT} \int_0^t d\tau \, \langle \sigma(t-\tau)\sigma(0)\rangle \, \gamma(\tau)
$$
the quantity inside $\langle\cdots\rangle$ is an equilibrium time correlation function. It does not reflect the history of the shear stress in a particular nonequilibrium experiment involving a time-dependent strain rate. It is just a function of time. Secondly, a better notation on the left of the equation would be $\langle \sigma(t)\rangle_{\text{ne}}$ to emphasize that it is not an instantaneous dynamical variable, but is also an average, taken in the nonequilibrium ensemble produced by the applied strain rate. 
So it should be clear that there is no recursive element to the equation: the same quantities do not appear on both sides. It simply links a nonequilibrium average response $\langle \sigma(t)\rangle_{\text{ne}}$ to an applied perturbation $\gamma(t)$. The response is governed by the equilibrium time correlation function.
Remember that dynamical linear response equations take the same form, in general. If the perturbation can be represented as a term $-\alpha(t) A$ in the hamiltonian, where $A$ is a dynamical variable and $\alpha(t)$ a time-dependent field, the response in any other dynamical variable $B$ can be written
$$
\langle B(t)\rangle_{\text{ne}}
=  \frac{1}{k_B T}  \int_{-\infty}^t \alpha(t') \langle \dot{A}(0) B(t-t')\rangle \, dt' ,
$$
where $\dot{A}$ is the time derivative of $A$.
The presence of $B$ on both sides of this equation is not a cause for concern (and indeed, we can choose $B$ to be $A$ if we wish). The memory function approach is useful, especially when discussing some of the subtleties of transport coefficients and time scale separation, but this issue is not specific to memory functions. Cases like the shear viscosity need more careful handling, because the perturbation is not a hamiltonian one, but again this aspect does not seem to be crucial to your question.
Secondly, although it is useful to have a symbol to denote that response function, it can be confusing to call it $\eta(t)$ and to refer to it as a generalized viscosity. (I'm not actually sure that Evans and Morriss, and Hansen and McDonald, precisely refer to $\eta(t)$ as a generalized viscosity, but maybe you have seen a place where they do). It has the wrong dimensions for a viscosity. Remember the Green-Kubo relation
$$
\eta = \frac{V}{k_BT} \int_0^\infty dt \, \langle \sigma(0)\sigma(t)\rangle
$$
so clearly the function
$$
\eta(t) = \frac{V}{k_BT} \langle \sigma(0)\sigma(t)\rangle
$$
is missing a factor with dimensions of time. E & M introduce this firstly through its Fourier transform, in eqn (2.73), which they term a "frequency dependent Maxwell viscosity"
$$
\tilde{\eta}_M(\omega) = \frac{\eta}{1+i\omega\tau_M}
$$
which makes clear that, when inverse transformed, it will not have the same dimensions as $\eta$. Then when they do the inverse transform to get the time-dependent function $\eta_M(t)$ which appears in eqn (2.74) they refer to it as the "Maxwell memory function" (again, it does not have viscosity dimensions) and then in eqn (2.76), which is your viscoelastic constitutive equation, they drop the restriction to the Maxwell model, and the subscript $M$, so it looks like a time dependent viscosity, but still has the wrong dimensions. Call it a minor hobby horse of mine, but I wish people would use a different symbol.
