We study the equilibrium simplex of Nica-Pimsner algebras arising from product systems of finite rank on the free abelian semigroup. First we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube. Secondly we associate every gauge-invariant part to a sub-simplex of tracial states of the diagonal algebra. We show how this parametrization lifts to the full equilibrium simplices of non-infinite type. The finite rank entails an entropy theory for identifying the two critical inverse temperatures: (a) the least upper bound for existence of non finite-type equilibrium states, and (b) the least positive inverse temperature below which there are no equilibrium states at all. We show that the first one can be at most the strong entropy of the product system whereas the second is the infimum of the tracial entropies (modulo negative values). Thus phase transitions can happen only in-between these two critical points and possibly at zero temperature.

中文翻译：

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Nica-Pimsner 代数的平衡态和熵理论

我们研究由自由阿贝尔半群上的有限秩乘积系统产生的 Nica-Pimsner 代数的平衡单纯形。首先，我们证明每个平衡状态都有一个凸分解，分解为由单位超立方体上的理想参数化的部分。其次，我们将每个规范不变部分与对角代数的迹态的亚单纯形联系起来。我们展示了这种参数化如何提升到非无限类型的完全平衡单纯形。有限秩需要一个熵理论来确定两个临界逆温度：(a) 存在非有限类型平衡状态的最小上限，以及 (b) 最低正逆温度，低于该温度时根本没有平衡状态. 我们表明第一个最多可以是乘积系统的强熵，而第二个是迹熵的下界（模负值）。因此相变只能发生在这两个临界点之间，并且可能在零温度下发生。