The zero temperature limit of statistical mechanical systems in the grand canonical ensemble is calculated using the methods of tropical algebraic geometry. Particularly, the grand potentials are interpreted as tropical Laurent polynomials in two variables which parametrises the energy and chemical potential of the system. Plotting the roots of these tropical polynomials gives a visual representation of the states of systems at zero temperature. These curves satisfies the balancing condition obeyed by all tropical polynomials. Furthermore, it is proven that the balancing condition continues to hold for tropical ‘polynomials’ with rational exponents. These methods are then applied to various simple physical systems and toy models.

使用热带代数几何方法计算了正则系综中统计力学系统的零温极限。特别地，大势能在两个变量中被解释为热带洛朗多项式，这些变量参数化了系统的能量和化学势。绘制这些热带多项式的根可以直观地表示零温度下的系统状态。这些曲线满足所有热带多项式所遵循的平衡条件。此外，已证明平衡条件继续适用于具有有理指数的热带“多项式”。然后将这些方法应用于各种简单的物理系统和玩具模型。