An introduction to non-archimedean electrostatics in the context of $\mathbb Z_p$.

- Background
- The $p$-adic Canonical Ensemble
- Quadratic Recurrences for the Canonical Partition Function
- The $p$-adic Grand Canonical Ensemble
- The $Q$-Transform on Formal Power Series
- Probabilities of Cylinder Sets

### Background

See The $p$-adics and Related Concepts for notation and basics on the $p$-adic numbers, Introduction to Boltzmann Statistics for notation and background on electrostatics, and Basics of Probability for notation and background on probability and $\sigma$-algebras. Haar measure $\lambda$ on $\mathbb Q_p$, normalized so that $\lambda(\mathbb Z_p) = 1$ is a probability measure on $\mathbb… Read…

### The $p$-adic Canonical Ensemble

As in Two-Dimensional Electrostatics, the joint density of particles and partition function in the canonical ensemble are then given by $$f(N,\beta; \mathbf x) = \frac{1}{Z(N,\beta,\lambda)} \prod_{m<n} |x_n – x_m |^{\beta}; \qquad Z(N, \beta, \lambda) = \int_{\mathbb Z_p^N} \prod_{m<n} |x_n – x_m|^{\beta} \, d\lambda^N(\mathbf x).$$ It turns out that the path to evaluating probabilities of cylinder… Read…

### Quadratic Recurrences for the Canonical Partition Function

We have one recurrence for $\widetilde Z(N, \beta, \lambda)$ but it involves a complicated combinatorial sum and a product involving $p$ different terms $\widetilde Z(n_j, \beta, \lambda)$. Here we demonstrate a simpler quadratic recurrence. Theorem: $\widetilde Z(N, \beta, \lambda)$ satisfies the quadratic recurrence $$\sum_{n=1}^N \left(\frac{N}{p+1} – n \right) p^{-\beta {n \choose 2} – n} \widetilde… Read…

### The $p$-adic Grand Canonical Ensemble

See The Grand Canonical Ensemble for details about the general setup, but the basic idea distinguishing the canonical ensemble and the grand canonical ensemble is that in the latter we allow the number of particles to vary. The grand canonical partition function can also be viewed as the generating function for the partition functions of… Read…

### The $Q$-Transform on Formal Power Series

We take a brief detour to talk about some operations on power series that arise in calculations of probabilities in the grand canonical ensemble. We already saw an example of this when we computed the probability that all particles lie in a ball of radius $p^{-r}$ was $$\frac{1}{Z(t, \beta, \lambda)} \sum_N p^{-r\beta {N \choose 2}… Read…

### Probabilities of Cylinder Sets

See Point Processes for the general set up of the cylinder $\sigma$-algebra, but it suffices to say here that we are interested in computing probabilities of events of the form $\{N_{B_1} = n_1, \ldots, N_{B_M} = n_M\}$. These events are called cylinder sets, and we call $\mathbf n = (n_1, \ldots, n_M)$ an occupation vector,… Read…