Liu hard disk equation of state

The Liu equation of state for hard disks (2-dimensional hard spheres) is given by Eq. 1 of .

For the stable fluid:

$Z_{v}={\frac {1+\eta ^{2}/8+\eta ^{4}/18-4\eta ^{4}/21}{(1-\eta )^{2}}}$

where the packing fraction is given by $\eta =\pi \rho \sigma ^{2}/4$ where $\sigma$ is the diameter of the disks.

The EoS for the stable fluid, liquid-hexatic transition region and hexatic:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z_{lh}=Z_{v}+{\frac {b_{1}\eta ^{m_{1}}+b_{2}\eta ^{m_{2}}}{(1-c\eta )}}}

The global EoS for all phases:

$Z=Z_{lh}$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \eta <=0.72}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Z=Z_{solid}} , Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \eta >0.72}

where: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{solid} = \frac{2}{\alpha} + 1.9 + \alpha - 5.2 \alpha^2 + 114.48 \alpha^4}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{2}{3^{1/3} \rho}}