论文标题
非架构的静电
Non-Archimedean Electrostatics
论文作者
论文摘要
我们介绍了排斥带电粒子的合奏，这些粒子仅限于非架构的磁场（例如$ p $ - 亚种数），这些粒子在它们之间的（$ p $ - 亚种）距离的对数成比例的粒子之间相互作用。在{\ em canonical集合}中，将$ n $颗粒的系统与固定的反温度$β$接触，并允许能量在系统和热浴之间流动。使用统计物理的标准公理，状态的相对密度由vandermonde决定因素在粒子位置的（$ p $ -ADIC）绝对值（$ p $ - adic）的$β$给出。分区函数是该集合的归一化常数（作为$β$的函数），我们确定了递归，该递归允许在有限的时间内明确计算出来。感兴趣的概率，包括指定子集的概率将具有规定的职业粒子数量，并根据分区功能明确给出了子集中粒子在子集中的条件分布。然后，我们转向{\ em Grand Canonical Ensemble}，其中粒子的能量和数量都是可变的。我们计算与规范合奏中的概率相似的概率，并显示如何使用规范和宏伟的规范分区功能给出这些概率。最后，我们简要考虑了允许粒子采取不同整数电荷的多组分集合，我们将该集合的基本特性连接到规范和宏伟的规范合奏。
We introduce ensembles of repelling charged particles restricted to a ball in a non-archimedean field (such as the $p$-adic numbers) with interaction energy between pairs of particles proportional to the logarithm of the ($p$-adic) distance between them. In the {\em canonical ensemble}, a system of $N$ particles is put in contact with a heat bath at fixed inverse temperature $β$ and energy is allowed to flow between the system and the heat bath. Using standard axioms of statistical physics, the relative density of states is given by the $β$ power of the ($p$-adic) absolute value of the Vandermonde determinant in the locations of the particles. The partition function is the normalizing constant (as a function of $β$) of this ensemble, and we identify a recursion that allows this to be computed explicitly in finite time. Probabilities of interest, including the probabilities that specified subsets will have a prescribed occupation number of particles, and the conditional distribution of particles within a subset given a prescribed occupation number, are given explicitly in terms of the partition function. We then turn to the {\em grand canonical ensemble} where both the energy and number of particles are variable. We compute similar probabilities to those in the canonical ensemble and show how these probabilities can be given in terms the canonical and grand canonical partition functions. Finally, we briefly consider the multi-component ensemble where particles are allowed to take different integer charges, and we connect basic properties of this ensemble to the canonical and grand canonical ensembles.
