Fluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited

The shear stress relaxation modulus $G(t)$ may be determined from the shear stress $\tau(t)$ after switching on a tiny step strain $\gamma$ or by inverse Fourier transformation of the storage modulus $G^{\prime}(\omega)$ or the loss modulus $G^{\prime\prime}(\omega)$ obtained in a standard oscillatory shear experiment at angular frequency $\omega$. It is widely assumed that $G(t)$ is equivalent in general to the equilibrium stress autocorrelation function $C(t) = \beta V \langle \delta \tau(t) \delta \tau(0)\rangle$ which may be readily computed in computer simulations ($\beta$ being the inverse temperature and $V$ the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general $G(t) = G_{eq} + C(t)$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. A similar relation holds for $G^{\prime}(\omega)$. $G(t)$ and $C(t)$ must thus become different for a solid body and it is impossible to obtain $G_{eq}$ directly from $C(t)$.