Graph explorer

Non-Archimedean Electrostatics

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 nu