the dense ISM, and dust disruption by shattering in the diffuse ISM. We assume that the dust grains are dynamically coupled with the gas, which is usually valid on the spatial scales of interest in this paper [@2018MNRAS.478.2851M]. In what follows, we only describe the outline of the model to be implemented for individual SPH gas particles (hereafter, referred to simply as gas particles). We refer the interested reader to HA19 for the complete set of equations and the details.

The grain size distribution at time $t$ is expressed by the grain mass distribution $\rho_\mathrm{d}(m,\, t)$, which is defined such that $\rho_\mathrm{d}(m,\, t)\,\mathrm{d}m$ ($m$ is the grain mass and $t$ is the time) is the mass density of dust grains whose mass is between $m$ and $m+\mathrm{d}m$. We assume grains to be spherical and compact, so that $m=(4\upi /3)a^3s$, where $a$ is the grain radius and $s$ is the material density of dust. We adopt $s=3.5$ g cm$^{-3}$ based on silicate in [@2001ApJ...548..296W]. The grain mass distribution is related to the grain size distribution, $n(a,\, t)$, as $$\begin{aligned}
\rho_\mathrm{d}(m,\, t)\,\mathrm{d}m=\frac{4}{3}\upi a^3sn(a)\,\mathrm{d}a.
\label{eq:rho_n}\end{aligned}$$ The total dust mass density $\rho_\mathrm{d,tot}(t)$ is $$\begin{aligned}
\rho_\mathrm{d,tot}(t)=\int_0^\infty\rho_\mathrm{d}(m,\, t)\,\mathrm{d}m.\end{aligned}$$ The gas density, $\rho_{\rm gas}$, is given by 