Suppose $k$ is a local field with ring of integers $\mathfrak o$. Let $\mf m \subseteq \mf o$ denote the unique maximal ideal, and define $q = [\mf o: \mf m]$.  The maximal ideal is principal and we fix a generator, or uniformizer, $\varpi$ so that $\mf m = \varpi \mf o$. There is a unique absolute value $| \cdot |$ on $k$ with
$$
| \varpi | = \frac{1}{q}.
$$
In this context $\mf o = \{ \alpha : |\alpha| \leq 1\}$ and $\mf m = \{\alpha : |\alpha| < 1\}$. As a locally compact abelian group, $k$ has a Haar measure $\mu$ which can be made unique by specifying $\mu(\mf o) = 1$.

Imagine two charged particles, $p$-adic electrons if you will, identified with $\alpha, \alpha’ \in \mf o$ and whose interaction energy is given by
$$
E(\alpha, \alpha’) = -\log|\alpha’ – \alpha|.
$$
This vaguely corresponds with our intuition as to how electrons ($p$-adic or not) should behave in the sense that the energy is minimized when the distance between $\alpha$ and $\alpha’$ is maximal, and the energy is infinite if the electrons are on top of each other, that is when $\alpha = \alpha’$. Since we have restricted $\alpha$ and $\alpha’$ to $\mf o$, the farthest apart they can be is $| \alpha – \alpha’| = 1$, and in this case the interaction energy is 0. This observation is central in understanding how these particles behave when we introduce thermal fluctuations to the mix.

Moving from $2$ particles to multiple particles, there are three main systems or ensembles we will consider.

The Microcanonical Ensemble
The system contains $N$ particles at a specified energy $E$.
The Canonical Ensemble
The system contains $N$ particles at a specified temperature $T$. Energy can be exchanged with a heat bath, and is now variable.
The Grand Canonical Ensemble
The system is now in contact with a heat bath and a particle reservoir so that $E$ and $N$ are both variable. The temperature $T$ and a new quantity, the chemical potential, are fixed and control the average energy and particle number.

The Microcanonical Ensemble

In this setting, the energy of the $N$ particles located at the coordinates of $\boldsymbol \alpha \in \mf o^N$ is given by
$$
E(\bs \alpha) = -\sum_{m < n} \log|\alpha_n - \alpha_m|. $$ All states with the same energy are assumed to be equally probable, and the main problem for the micro canonical ensemble is the determination of the measure of the set with prescribed energy. For reasons that will become apparent, it is sometimes useful to deal with the exponentiated energy $$ e^{-E(\bs \alpha)} = \prod_{n

Leave a Reply

Your email address will not be published. Required fields are marked *