the Mach number of the largest-eddy velocity. We adopt $\mathcal{M}=3$ for shattering and $\mathcal{M}=1$ for coagulation to effectively achieve the grain velocity levels suggested by [@2004ApJ...616..895Y]. In considering the collision rate between two grains with $v_\mathrm{gr}=v_1$ and $v_2$, we estimate the relative velocity $v_{1,2}$ by $v_{1,2}=(v_1^2+v_2^2-2v_1v_2\mu_{1,2})^{1/2}$, where $\mu =\cos\theta$ ($\theta$ is an angle between the two grain velocities) is randomly chosen between $-1$ and 1 in every calculation of $\alpha$. The above expression for the velocity (with $\mathcal{M}=1$) was originally derived by who assumed the maximum eddy size to be the Jeans length and the maximum velocity dispersion to be the sound velocity. We simply use their functional form for the purpose of avoiding grain velocity calculations in every time-step, but modify it to scale the maximum velocity (with the Mach number) to match with the velocity scale adopted for our previously adopted models based on [@2004ApJ...616..895Y]. As shown by , detailed dependence of grain velocities on the grain radius is not essential, but the overall velocity level is more important. [ In particular, it is essential that large ($a\gtrsim 0.1~\micron$) grains attain velocities higher than a few km s$^{-1}$ in the diffuse ISM, since shattering, which plays an important 