  Shear-strain and shear-stress correlations in isotropic elastic bodies are
investigated both theoretically and numerically at either imposed mean
shear-stress $\tau$ ($\lambda=0$) or shear-strain $\gamma$ ($\lambda=1$) and
for more general values of a dimensionless parameter $\lambda$ characterizing
the generalized Gaussian ensemble. It allows to tune the strain fluctuations
$\mu_{\gamma\gamma} \equiv \beta V \la \delta \gamma^2 \ra =
(1-\lambda)/G_{eq}$ with $\beta$ being the inverse temperature, $V$ the volume,
$\gamma$ the instantaneous strain and $G_{eq}$ the equilibrium shear modulus.
Focusing on spring networks in two dimensions we show, e.g., for the stress
fluctuations $\mu_{\tau\tau} \equiv \beta V \la \delta\tau^2 \ra$ ($\tau$ being
the instantaneous stress) that $\mu_{\tau\tau} = \mu_{A} - \lambda G_{eq}$ with
$\mu_{A} = \mu_{\tau\tau}|_{\lambda=0}$ being the affine shear-elasticity. For
the stress autocorrelation function $c_{\tau\tau}(t) \equiv \beta V \la \delta
\tau(t) \delta \tau(0) \ra$ this result is then seen (assuming a sufficiently
slow shear-stress barostat) to generalize to $c_{\tau\tau}(t) = G(t) - \lambda
\Geq$ with $G(t)$ being the shear-stress relaxation modulus.
