The $p$-adic Canonical Ensemble

$p$-adic Electrostatics

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 sets runs through the partition function. This is not terribly unsurprising from the viewpoint of statistical physics, where the identification (in some useful form) of the partition function allows for the calculation of macroscopic thermodynamic quantities. The establishment of a useful form of the partition function is sometimes called solvability of the ensemble, though that word means a lot of different things to a lot of different people.

The Canonical Partition Function

The secret to evaluating the canonical partition function is exploiting the self-similarity of in $\mathbb Z_p$.

Theorem: $(Z(N, \beta, \lambda))_N$ satisfies the recurrence $$\frac{1}{N!} Z(N, \beta, \lambda) = \sum_{n_0 + \cdots + n_{p-1} = N} \prod_{j=0}^{J-1} \frac{p^{-\beta{n_j \choose 2} – n_j}}{n_j!} Z(n_j, \beta, \lambda).$$

On first inspection this might look like a less-than-ideal recursion because $Z(N, \beta, \lambda)$ appears on both sides of the equation. However, the coefficient on the right hand side for terms involving $Z(N, \beta, \lambda)$ are not equal to one, and so one can solve for $Z(N, \beta, \lambda)$.

Corollary: $$\frac{1}{N!} Z(N, \beta, \lambda) = (p^N – p^{1 – \beta{N \choose 2}})^{-1} \sum_{\mathbf n} \prod_{j=0}^{J-1} \frac{p^{-\beta{n_j \choose 2} – n_j}}{n_j!} Z(n_j, \beta, \lambda),$$ where the sum is over all non-negative integer vectors $\mathbf n = (n_0, \ldots, n_{p-1})$ whose coordinates sum to $N$, but for which none of the $n_j = N$.

We will prove the Theorem through a sequence of (elementary) lemmas.

The Additivity of Energy over Cosets: Suppose $x$ and $y$ are in different cosets of $p \mathbb Z_p$. Then $E_N(x, y) = 0.$

There is nothing miraculous about this fact: if $x$ and $y$ are in different cosets then $|x – y|_p = 1$ and hence $E_N(x,y) = -\log|x – y|_p = 0$. In spite of this the fact is remarkable in that it has no analog in the real or complex case—this will lead to new and interesting tools unavailable in the archimedean situation. This being said, we also lose some other tools exploited in classical two-dimensional electrostatics. As usual, moving from the reals to the $p$-adics introduces new techniques but also new problems.

The additivity of energy over cosets can be better exploited if we reorder and relabel our particles according to which coset they are in. This does not change the probability of the underlying state—it is a choice of different representation (under the action of permutation of coordinates) of the underlying state. Regardless, when convenient, instead of writing $\mathbf x = (x_1, \ldots, x_N)$ or $\mathbf X = (X_1, \ldots, X_N)$ for the particle locations (and their corresponding random variables) we will write $\mathbf x^j = (x_1^j, \ldots, x_{n_j}^j)$ and $\mathbf X^j = (X_1^j, \ldots, X_{n_j}^j)$ for the particle location and associated random variables in the coset $j + p \mathbb Z_p$. This implies in particular that $n_0 + n_1 + \cdots + n_{p-1} = N$. For future consideration, when we move to the grand canonical ensemble we will lose this constraint.

Note that $\mathbf x$ and $(\mathbf x^0, \ldots, \mathbf x^j)$ are not identical in general—they differ by a permutation—but they do refer to the same state of the system. Thus we may use either representation as is convenient. Note that, in general, each $(\mathbf x^0, \ldots, \mathbf x^j)$ corresponds to ${N \choose n_0, \ldots, n_{p-1}}$ different vectors $\mathbf x$. This counting factor will arise when we move from integrating over $\mathbf x$ to integrating over $(\mathbf x^0, \ldots, \mathbf x^j)$.

The following version of the additivity of energy over cosets explains its name.

The Additivity of Energy over Cosets: $E_N(\mathbf x) = E_{n_0}(\mathbf x^0) + E_{n_1}(\mathbf x^1) + \cdots + E_{n_{p-1}}(\mathbf x^{p-1}).$ This in turn implies the multiplicativity of the Boltzmann Factor over cosets. $$\prod_{m<n}^N |x_n – x_m|_p^{\beta} = \prod_{j=0}^{p-1} \prod_{m<n}^{n_j} |x_n^j – x_m^j|_p^{\beta}.$$

The multiplicativity of the Boltzmann Factor is perfectly set up for Fubini’s Theorem.

Lemma: $$Z(N, \beta, \lambda) = {N \choose n_0, \ldots, n_{p-1}} \prod_{j=0}^{p-1} \int_{(j + p \mathbb Z_p)^{n_j}} \prod_{m < n}^{n_j} |x_n – x_m|_p^{\beta} \, d\lambda^{n_j}(\mathbf x).$$

At this point (and maybe we should have done this earlier) is to renormalize $Z(N, \beta, \lambda)$ by $N!$. That is, we set $\widetilde Z(N, \beta, \lambda) = Z(N, \beta, \lambda)/N!$. Reasons for this are discussed in The Canonical Ensemble, but in short, if we alter the Boltzmann factor by the same $N!$ the probabilities are unchanged. This allows us to compensate for the counting term in the lemma by integrating over a very slightly different function.

Let $\lambda_1 = \boldsymbol 1_{p \mathbb Z_p} \lambda$ be Haar measure restricted to $p \mathbb Z_p$. We observe that the integral $\widetilde Z(n, \beta, \lambda_1)$ can be expressed in terms of $\widetilde Z(n, \beta, \lambda)$. To see this, note that if $x \in p \mathbb Z_p$ then there is a unique $y \in \mathbb Z_p$ such that $x = p y$. Note also (by properties of Haar & Lebesgue Measure) $d\lambda_1(p x) = |p|_p d\lambda(x).$ It follows from the homogeneity of the integrand, $$\widetilde Z(n, \beta, \lambda_1) = \frac{p^{-\beta{n \choose 2}-n}}{n!} \int_{\mathbb Z_p^n} \prod_{j<k}^n| y_k – y_j |^{\beta} \, d\lambda^N(\mathbf y)$$

Lemma: $\widetilde Z(n, \beta, \lambda_1) = p^{-\beta{n \choose 2}-n} \widetilde Z(n, \beta, \lambda).$

Next we look at the integrals over the cosets of $p \mathbb Z_p$, $$\int_{(j + p \mathbb Z_p)^{n_j}} \prod_{m < n}^{n_j} |x_n – x_m|_p^{\beta} \, d\lambda^{n_j}(\mathbf x).$$ Haar measure is translation invariant and thus if we write $j + \mathbf x^j = (j + \mathbf x^j_1, j + \mathbf x_{n_j}^j),$ then $d\lambda^{n_j}(j + \mathbf x^j) = d\lambda^{n_j}(\mathbf x^j).$ Moreover, if we write $$\Delta(\mathbf x) = \prod^{n_j}_{m < n} (x_n – x_m)$$ for the Vandermonde determinant, then it is clear that $\Delta(j + \mathbf x) = \Delta(\mathbf x).$ We have proved the following lemma.

Lemma: $$\frac{1}{n_j!} \int_{(j + p \mathbb Z_p)^{n_j}} \prod_{m < n}^{n_j} |x_n – x_m|_p^{\beta} \, d\lambda^{n_j}(\mathbf x) = \widetilde Z(n_j, \beta, \lambda_1).$$

We can put all this together to prove the recursion for the $\widetilde Z_N$ (and hence the $Z_N$) given in the Theorem. The final step is to observe when we integrate over $\widetilde Z_N$ over all $\mathbf x \in \mathbb Z_p^N$ we may first partition the domain of integration into disjoint subdomains where we have a prescribed number of particles in each coset of $p \mathbb Z_p$. This introduces a sum over all vectors of non-negative integers $\mathbf n = (n_0, \ldots, n_{p-1})$ which sum to $N$, and then using the previous lemmas to simplify each of the summands. That is,

$$\widetilde Z(N, \beta, \lambda) = \sum_{\mathbf n} \prod_{j=0}^{p-1} p^{-\beta{n \choose 2}-n} \widetilde Z(n, \beta, \lambda).$$.

Simple Probabilities in the Canonical Ensemble

We can (and have!) already computed some basic probabilities in the canonical ensemble. In particular, $(p \mathbb Z_p)^N$ is the event “all particles are in $p \mathbb Z_p$.” Hence, $$\mathbb P((p \mathbb Z_p)^N) = \frac{\widetilde Z(N, \beta, \lambda_1)}{\widetilde Z(N, \beta, \lambda)} = p^{-\beta}{N \choose 2} – N.$$ This result can be extended to give the probability that all particles are in any prescribed ball.

Theorem: Let $B$ be a ball of radius $p^{-r}$. Then the probability that all particles are in $B$ is given by $$\mathbb P\{N_B = N\} = p^{-r \beta {N \choose 2} – Nr}.$$

Conditioning on $N_B$

The additivity of energy over cosets has another major implication: Once we specify how many particles are in a ball, the distribution of these particles is independent of all the location of particles not in the ball.

Theorem: Let $B$ be a Borel set in $\mathbb Z_p$. On the event $\{N_B = n\}$, let $X_1, \ldots, X_n$ denote the particles in $B$ and $X_{n+1}, \ldots, X_N$ denote the particles in $B^c$. Then $X_1, \ldots, X_n$ and $X_{n+1}, \ldots, X_N$ are conditionally independent given $N_B = n$. Moreover, the conditional distribution of $X_1, \ldots, X_n$ is equal to that of the distribution of particles in the canonical ensemble on $B$ with $n$ particles.