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Register a new userFluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited
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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 $G^{primeprime}(omega)$ 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 $C(t)$.
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We revisit the relation between the shear stress relaxation modulus $G(t)$, computed at finite shear strain $0 < gamma ll 1$, and the shear stress autocorrelation functions $C(t)|_{gamma}$ and $C(t)|_{tau}$ computed, respectively, at imposed strain $gamma$ and mean stress $tau$. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general $G(t) = C(t)|_{tau} = C(t)|_{gamma} + G_{eq}$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. $G(t)$ and $C(t)|_{gamma}$ thus must become different for solids and it is impossible to obtain $G_{eq}$ alone from $C(t)|_{gamma}$ as often assumed. We comment briefly on self-assembled transient networks where $G_{eq}(f)$ must vanish for a finite scission-recombination frequency $f$. We argue that $G(t) = C(t)|_{tau} = C(t)|_{gamma}$ should reveal an intermediate plateau set by the shear modulus $G_{eq}(f=0)$ of the quenched network.
Focusing on isotropic elastic networks we propose a novel simple-average expression $G(t) = mu_A - h(t)$ for the computational determination of the shear-stress relaxation modulus $G(t)$ of a classical elastic solid or fluid and its equilibrium modulus $G_{eq} = lim_{t to infty} G(t)$. Here, $mu_A = G(0)$ characterizes the shear transformation of the system at $t=0$ and $h(t)$ the (rescaled) mean-square displacement of the instantaneous shear stress $hat{tau}(t)$ as a function of time $t$. While investigating sampling time effects we also discuss the related expressions in terms of shear-stress autocorrelation functions. We argue finally that our key relation may be readily adapted for more general linear response functions.
Using molecular dynamics simulation of a standard coarse-grained polymer glass model we investigate by means of the stress-fluctuation formalism the shear modulus $mu$ as a function of temperature $T$ and sampling time $Delta t$. While the ensemble-averaged modulus $mu(T)$ is found to decrease continuously for all $Delta t$ sampled, its standard deviation $delta mu(T)$ is non-monotonous with a striking peak at the glass transition. Confirming the effective time-translational invariance of our systems, $mu(Delta t)$ can be understood using a weighted integral over the shear-stress relaxation modulus $G(t)$. While the crossover of $mu(T)$ gets sharper with increasing $Delta t$, the peak of $delta mu(T)$ becomes more singular. % It is thus elusive to predict the modulus of a single configuration at the glass transition.
We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasi-static and dynamical) shear-stress fluctuations as a function of temperature T and sampling time $Delta t$. The linear response is characterized using (ensemble-averaged) expectation values of the contributions (time-averaged for each shear plane) to the stress-fluctuation relation $mu_{sf}$ for the shear modulus and the shear-stress relaxation modulus $G(t)$. Using 100 independent configurations we pay attention to the respective standard deviations. While the ensemble-averaged modulus $mu_{sf}(T)$ decreases continuously with increasing T for all $Delta t$ sampled, its standard deviation $delta mu_{sf}(T)$ is non-monotonous with a striking peak at the glass transition. The question of whether the shear modulus is continuous or has a jump-singularity at the glass transition is thus ill-posed. Confirming the effective time-translational invariance of our systems, the $Delta t$-dependence of $mu_{sf}$ and related quantities can be understood using a weighted integral over $G(t)$. This implies that the shear viscosity $eta(T)$ may be readily obtained from the $1/Delta t$-decay of $mu_{sf}$ above the glass transition.
We show a direct connection between Kubos fluctuation-dissipation relation and Hawking effect that is valid in any dimensions for any stationary or static black hole. The relevant correlators corresponding to the fluctuating part of the force, computed from the known expressions for the anomalous stress tensor related to gravitational anomalies, are shown to satisfy the Kubo relation, from which the temperature of a black hole as seen by an observer at an arbitrary distance is abstracted. This reproduces the Tolman temperature and hence the Hawking temperature as that measured by an observer at infinity.
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