Thermodynamic equations of state for the dust grain using Langevin Dynamics simulations

The complex plasma is modeled as an isotropic system of $N$ charged grains (radius $a_{p}$ and charge $z_{p})$ interacting with each other through a screened Coulomb potential with a fixed Debye length $\lambda_{D}$ in an isotropic periodic domain, without including a confining potential and the systematic drift of charged species. The equilibrium thermodynamic state is investigated using Langevin Dynamics simulations to capture the effect of grain-neutral gas interactions (parameterized by the grain Knudsen number $Kn\mathrm{\equiv }\frac{\lambda_{g}}{a_{p}})$ across the various $\Gamma \mathrm{\equiv }\frac{z_{p}^{\mathrm{2}}e^{\mathrm{2}}}{\mathrm{4}\pi \varepsilon _{o}k_{b}T_{d}n_{p}^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{3}}}}\mathrm{,\thinspace }\kappa \mathrm{=}\frac{n_{p}^{\mathrm{-}\frac{\mathrm{1}}{\mathrm{3}}}}{\lambda _{D}}$-based electrostatic coupling regimes, where $n_{p}$ is number concentration and $k_{B}T_{d}$ is the kinetic temperature of the grains. The Langevin-computed internal energy $u_{d}$, pressure $p_{d}$ and $k_{B}T_{d}$ of the grain phase are parameterized as equations of state $f\left( u_{d}\mathrm{,}p_{d}\mathrm{,}k_{B}T_{d} \right)\mathrm{=0}$ and compared with experimental reports of dust kinetic temperature to refine the modeling assumptions. The non-trivial influence of grain-neutral gas interactions is discussed by calculating the pair correlation functions in the gas, liquid, and solid-like regimes of grain correlated behavior. A unified thermodynamic model will further the understanding of phase transitions and the transport properties of the dust phase across the entire $\Gamma \mathrm{,\thinspace }\kappa \mathrm{,\thinspace }Kn$ regimes.