Virial equation of state for a granular system

The equation of state for an ideal gas is simple, which is \(P=nk_\textrm{B}T\). 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.

The plot of Mayer function f(r) as a function of radial distance r for \(\Theta (r)\) for different inverse temperature \(\beta \).