Here we give a practical representation of $p$-adic numbers in terms of a power series in powers of $p.$ This perspective will connect our understanding of the $p$_adics as a Cauchy completion and as an… Read More »Series Expansions in $\mathbb Q_p$
Because $\mathbb Q_p$ is a locally compact abelian group it has a natural translation invariant measure, Haar measure, which allows us to formulate a natural theory of integration of (certain) real valued functions over subsets… Read More »Measures on $\mathbb Q_p$
Once we define distance appropriately, we’ll find a distance preserving embedding of $\mathbb Z_p$ into $\mathbb R^2$—that is we’ll be able to draw a picture of $\mathbb Z_p$ and determine the distance between points by… Read More »The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$
The microcanonical ensemble is a probability space which models physical systems at a fixed energy. We imagine a physical system whose states are described by some set $\Omega$ (which we will assume is a measurable… Read More »The Microcanonical Ensemble
The canonical ensemble is a probability space which models physical systems at a fixed temperature, but for which the energy can vary. In this situation we imagine that our system, described by states $(\Omega, \mathcal… Read More »The Canonical Ensemble
The grand canonical ensemble is a probability space which models physical systems with a variable number of particles and variable energy at a fixed temperature. Here we imagine our physical system is in contact with… Read More »The Grand Canonical Ensemble
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
An introduction to non-archimedean electrostatics in the context of $\mathbb Z_p$.
An introduction to Boltzmann statistics.
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