Virial Equation of State for a Granular System

The equation of state for an ideal gas is simple, which is $P=nk_{\rm B}T$. In the case of imperfect gases where mutual interactions among the constituents are important, pressure $P$ can be expressed as the series expansion of density $n$ with appropriate coefficients, known as virial coefficients $B_m$. In this paper, we have obtained the first four virial coefficients for a model interaction potential $\Phi(r)$ using multidimensional Monte-Carlo integration and importance sampling methods. Next, we perform molecular dynamics simulations with the same $\Phi(r)$ for a many-particle system to obtain $P$ as a function of $T$ and $n$. We compare our numerical data with the virial equation of state.