Generalization of Tropical Geometry and Amebas to the Region of Negative Pressures: Comparison with van der Waals Gas

In the region of negative pressures $P$, the addend $\gamma$ corresponding to the number of degrees of freedom $D=2\gamma+2$ is less than zero. The energy equal to $E=VP$ is also less than zero. Therefore, it is natural to introduce a negative relative temperature as the mean energy. The negative pressure on the negative spinodal tends to minus infinity as $T_r\to 0$. Accordingly, the negative relative temperature will also approach minus infinity. As the chemical potential $\mu\to-\infty$, $\mu/T\to 0$, the asymptotic behavior of the thermodynamic quantities gives rise to amebas of special form in tropical (idempotent) analysis. It is shown that the theoretical isotherms obtained in this paper coincide with the isotherms of van der Waals gas up to the third decimal place. For theoretical distributions, transitions to amorphic solids (glass) occurs.