Diffusion and velocity autocorrelation at the jamming transition

We perform numerical simulations to examine particle diffusion at steady shear in a soft-disk model in two dimensions and zero temperature around the jamming density. We find that the diffusion constant depends on shear rate as $D\sim \dot{\gamma }$ below jamming and as $D\sim {\dot{\gamma }}^{q_D}$ with $q_D<1$ at the transition and set out to relate this to properties of the velocity autocorrelation function. It is found that this correlation function is governed by two processes with different time scales. The first time scale, the inverse of the externally applied shear rate, controls an exponential decay of the correlations whereas the second time scale, equal to the inverse shear stress, governs an algebraic decay with time. The obtained value of $q_D$ is related to these properties of the correlation function.

Figure 1

(Color online) Shear stress versus shear rate around ${\varphi }_J$. Well below ${\varphi }_J$, exemplified by $\varphi =0.82$, we find $\sigma \propto \dot{\gamma }$. On this log-log plot that is shown by a straight line with $\text{slope}=1$ (lower dashed line). At $\varphi =0.8433\approx {\varphi }_J$ the relation is algebraic, $\sigma \sim {\dot{\gamma }}^{q_{\sigma }}$, with $q_{\sigma }=0.33$ (upper dashed line). Data at slightly higher and lower densities (filled and open circles, respectively) show clear curvatures. In the $\dot{\gamma }\to 0$ limit they cross over to $\sigma =\text{const}$ and $\sigma \propto \dot{\gamma }$, respectively.

Figure 2

(Color online) Diffusion constant around ${\varphi }_J$. Panel (a) shows the determination of $D$ from the large $\gamma$ part of data from $\dot{\gamma }={10}^{-6}$ and $\varphi =0.8433$. The solid line is from fitting $⟨\Delta y^2⟩$ to Eq. (2). The dashed line corresponds to $D$. Panel (b) shows the expected behavior below jamming, $D\sim \dot{\gamma }$, together with the same data at jamming, $D\sim {\dot{\gamma }}^{q_D}$, with $q_D=0.78\left(2\right)$. This slope continues to decrease with increasing $\varphi$ and is $\approx 0.68$ at $\varphi =0.86$. Panel (c) is a scaling collapse of the diffusion constant with $q_D=0.78$.

Figure 3

(Color online) Scaling properties of the VCF below and at ${\varphi }_J$. The collapse of $G_v\left(\gamma \right)$ at $\varphi =0.82$ (upper points) nicely shows that the behavior is governed by $\gamma$, which implies that $1/\dot{\gamma }$ is the relevant time scale. The lower points which show $G_v\left(\gamma \right)$ at ${\varphi }_J$ clearly fail to scale. The solid line shows that $G_v$ decays exponentially, $e^{-\gamma /{\gamma }_1}$ with ${\gamma }_1=0.081$ down to a negative value. The meaning of the symbols for the data at $\varphi \approx {\varphi }_J$ is given in Fig. 4a.

Figure 4

(Color online) Attempted scaling collapse of the VCF at ${\varphi }_J$. Panel (a) shows a failure to collapse, except at very short times. It is then seen in panel (b) that the data for a small total shear, $\gamma <0.01$, actually collapse nicely and that the behavior at large times is algebraic. Finally, panel (c) shows that it is possible to make all data with $\dot{\gamma }\ge {10}^{-5}$ collapse by compensating for the exponential decay, $e^{-\gamma /{\gamma }_1}$, with ${\gamma }_1=0.081$. The simulation for the lowest shear rate was here done with twice as many particles $\left(N=131072\right)$ to avoid finite-size effects that become visible in $G_v\left(t\right)$ before they are seen in most other quantities.