Shear-strain and shear-stress fluctuations in generalized Gaussian ensemble simulations of isotropic elastic networks
Shear-strain and shear-stress correlations in isotropic elastic bodies are investigated both theoretically and numerically at either imposed mean shear-stress $τ$ ($λ=0$) or shear-strain $γ$ ($λ=1$) and for more general values of a dimensionless parameter $λ$ characterizing the generalized Gaussian ensemble. It allows to tune the strain fluctuations $μ_{γγ} \equiv βV \la δγ^2 \ra = (1-λ)/G_{eq}$ with $β$ being the inverse temperature, $V$ the volume, $γ$ 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 $μ_{ττ} \equiv βV \la δτ^2 \ra$ ($τ$ being the instantaneous stress) that $μ_{ττ} = μ_{A} - λG_{eq}$ with $μ_{A} = μ_{ττ}|_{λ=0}$ being the affine shear-elasticity. For the stress autocorrelation function $c_{ττ}(t) \equiv βV \la δτ(t) δτ(0) \ra$ this result is then seen (assuming a sufficiently slow shear-stress barostat) to generalize to $c_{ττ}(t) = G(t) - λ\Geq$ with $G(t)$ being the shear-stress relaxation modulus.