Math

Two-Dimensional Electrostatics

In two dimensional electrostatics we imagine infinite parallel wires in three dimensions carrying specified charge densities. By inserting a plane perpendicular to the charged wires we may identify each wire with its intersection point in… Read More »Two-Dimensional Electrostatics

$p$-adic Electrostatics

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

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… Read More »Background

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) =… Read More »The $p$-adic Canonical Ensemble

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:… Read More »Quadratic Recurrences for the Canonical Partition Function

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… Read More »The $p$-adic Grand Canonical Ensemble

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… Read More »The $Q$-Transform on Formal Power Series

Protected: A Tale of Two Hook Walks

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