is assumed to occur only in the diffuse medium ($n_\mathrm{H}<1$cm$^{-3}$) where the grain velocities are high enough. For coagulation, since we cannot spatially resolve the dense and cold medium where it occurs, we adopt a sub-grid model. We assume that coagulation occurs in gas particles which satisfy $n_\mathrm{H}>10$cm$^{-3}$ and $T_\mathrm{gas}<1000$K, and that a mass fraction of $f_\mathrm{dense}=0.5$ of such a dense particle is condensed into dense clouds, hosting coagulation, with $n_\mathrm{H}=10^3$cm$^{-3}$ and $T_\mathrm{gas}=50$K on subgrid scales.

The time evolution of grain size distribution by shattering or coagulation is expressed as $$\begin{aligned}
\left[\frac{\partial\rho_\mathrm{d}(m,\, t)}{\partial t}\right]_\mathrm{shat/coag}
= -m\rho_\mathrm{d}(m,\, t)\int_0^\infty\alpha (m_1,\, m)\rho_\mathrm{d}(m_1,\, t)
\mathrm{d}m_1\nonumber\\
+ \int_0^\infty\int_0^\infty\alpha (m_1,\, m_2)\rho_\mathrm{d}(m_1,\, t)\rho_\mathrm{d}(m_2,\, t)
\mu_\mathrm{shat/coag}(m;\, m_1,\, m_2)\mathrm{d}m_1
\mathrm{d}m_2,\label{eq:shat}\end{aligned}$$ where $\mu_\mathrm{shat/coag}$ describes the grain mass distribution function of newly formed shattered fragments or coagulated grains, and $\alpha$ is expressed in collisions between grains with masses $m_1$ and $m_2$ (radii $a_1$ and $a_2$, respectively) as $$\begin{aligned}
\alpha (m_1,\, m_2)\equiv\frac{\sigma_{1,2}v_{1,2}}{m_1m_2},\label{eq:alpha}\end{aligned}$$ where $\sigma_{1,2}=\upi (a_1+a_2)^2$ and $v_{1,2}$ are the collisional cross-section and the relative velocity between the two grains (we explain how to evaluate $v_{1,2}$ later).

We adopt the following formula for the grain velocity in a turbulent medium: $$\begin{aligned}
v_\mathrm{gr}(a) &= 1.1\mathcal{M}^{3/2}\left(
\frac{a}{0.1~\micron}\right)^{1/2}\left(\frac{T_\mathrm{gas}}{10^4~\mathrm{K}}\right)^{1/4}
\left(\frac{n_\mathrm{H}}{1~\mathrm{cm}^{-3}}
\right)^{-1/4}\nonumber\\
&\times \left(\frac{s}{3.5~\mathrm{g~cm}^{-3}}\right)^{1/2}~\mathrm{km~s}^{-1},
\label{eq:vel}\end{aligned}$$ where $\mathcal{M}$ is 