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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 the plane. We usually refer to these points as particles and not wires, and we often identify the plane with the complex plane (or some subset such as $\mathbb R$).

In such a situation, Coulomb’s Law implies that the potential energy felt by a pair of wires (particles) identified points $z_1, z_2 \in \mathbb C$ with respective uniform charge densities $q_1$ and $q_2$ is proportional to $q_1 q_2 \log|z_1 – z_2|$. If there are $N$ particles located $z_1, \ldots, z_N \in \mathbb C$ each with unit charge density, then we define the energy functional $$E_N(\mathbf z) = -\sum_{m < n} \log|z_n – z_m|.$$ This is the sum of the pairwise interaction energies between all pairs of particles.

In this situation, the Boltzmann factor is $e^{-\beta E(\mathbf z)}$ and hence the joint density of states is given by $$f_N(\mathbf z) = \frac{1}{Z_N(\beta)} \prod_{m<n} |z_n – z_m|^{\beta}; \qquad Z_N(\beta) = \int_{\mathbb C^N} \prod_{m<n} |z_n – z_m|^{\beta} \, d\lambda_{\mathbb C}^N(\mathbf z).$$ Of course, if the particles are restricted to a subset of $\mathbb C$ then we integrate over the $N$-fold cartesian product of that subset. Actually, it is sensible to (perhaps) replace $\lambda_{\mathbb C}$ with some other measure $\mu$ (as it stands, the partition function does not converge for any $\beta > 0).$ We may envision $\mu$ as a measure which represents some background potential, and makes $Z_N(\beta)$ finite. If this measure is absolutely continuous with respect to $\lambda_{\mathbb C}$ (or $\lambda_{\mathbb R}$ is the particles are restricted to the real axis), then the Radon-Nikodym derivative is called the weight function of the ensemble. Specific weight functions make the analysis of the ensemble easier, but here we keep these generic and set $$Z_N(\beta) = \int \prod_{m<n} |z_n – z_m|^{\beta} \, d\mu^N(\mathbf z).$$ If necessary we may make the $\mu$-dependence explicit by writing $Z(N, \beta, \mu).$ Up to this renormalizing constant, the joint density of states $f(N, \beta, \mu; \mathbf z)$ is the integrand of $Z(N, \beta, \mu).$

The Solvability of the Canonical Ensemble when $\beta = 2$

The solvability of the canonical ensemble when $\beta = 2$ relies on the Vandermonde determinant identity. Given $\mathbf z \in \mathbb C^N$, the Vandermonde matrix is given by $$\mathbf V(\mathbf z) = \begin{bmatrix} 1 & 1 & & 1 \\ z_1 & z_2 & \cdots & z_N \\ & \vdots & \ddots & \vdots \\ z_1^{N-1} & z_2^{N-1} & \cdots & z_N^{N-1} \end{bmatrix}.$$

Theorem: $$\det \mathbf V(\mathbf z) = \prod_{m < n} (z_n – z_m).$$

Exercise: Prove this. Hint: show both sides are homogeneous polynomials of the same degree that vanish whenever $z_m = z_n$. Argue one must be a constant times the other.

Theorem: $$\frac{1}{N!} Z(N, 2, \mu) = \det\left[ \int z^{n-1} \overline z^{m-1} \, d\mu(z) \right]_{n,m=1}^N.$$


We can expand the determinant as a sum over the symmetric group $$\det \mathbf V(\mathbf z) = \sum_{\sigma \in S_N} \mathrm{sgn}(\sigma) \prod_{n=1}^N z_n^{\sigma(n) – 1}.$$

We can expand $| \det \mathbf V(\mathbf z)|^2$ as a double sum over the symmetric group and insert it into the integral in $Z(N, 2, \mu)$ to find $$Z(N, 2, \mu) = \sum_{\sigma \in S_N} \sum_{\tau \in S_N} \mathrm{sgn}(\sigma) \mathrm{sgn}(\tau) \int \prod_{n=1}^N z_n^{\sigma(n) – 1} \overline{z_m}^{\tau(n) – 1} \, d\mu^N(\mathbf z).$$ Fubini’s Theorem them implies $$Z(N, 2, \mu) = \sum_{\sigma \in S_N} \sum_{\tau \in S_N} \mathrm{sgn}(\sigma) \mathrm{sgn}(\tau) \prod_{n=1}^N \int z^{\sigma(n) – 1} \overline{z}^{\tau(n) – 1} \, d\mu(\mathbf z).$$

We may reindex the inner sum by $\tau \mapsto \sigma^{-1} \circ \tau$ rendering it superfluous. That is $$Z(N, 2, \mu) = N! \sum_{\sigma \in S_N} \mathrm{sgn}(\sigma) \prod_{n=1}^N \int z^{\sigma(n) – 1} \overline{z}^{n – 1} \, d\mu(\mathbf z)$$ and the theorem follows.

Exercise: Show that if $\mathbf p = (p_0, \ldots, p_{N-1})$ is a vector of monic (univariate) polynomials satisfying $\mathrm{deg}(p_n) = n$. Show

  • $\det \left[ p_m(x_n) \right]_{m,n=1}^N = \det \mathbf V(\mathbf z)$.
  • $Z(N, 2, \mu) = \det\left[ \langle p_m | p_n \rangle\right]_{m,n=1}^N$ where $\langle f | g \rangle = \int f \overline g \, d\mu.$ is the natural inner product on $L^2(\mu)$.

A monic family of polynomials $(\pi_n)$ is orthogonal in $L^2(\mu)$ if $\langle \pi_m | \pi_n \rangle = 0$ whenever $m \neq n$.

Theorem: Let $\pi_0, \ldots, \pi_{N-1}$ be monic orthogonal polynomials in $L^2(\mu)$ with $\mathrm{deg} \pi_n = n$. Then $$Z(N, 2, \mu) = \prod_{n=0}^{N-1} \langle \pi_n | \pi_n \rangle.$$

To generate the correlation functions in this situation, let $x_1, \ldots, x_n$ be points in $\mathbb C$ (or more generally in the support of $\mu$). Let $c_1, \ldots, c_N$ be positive parameters. Let $\delta_x$ represent a point mass probability measure at $x$, and define the measure $\nu = \mu + c_1 \delta_{x_1} + \cdots + c_N \delta_{x_N}.$ Then, $\frac{Z(N,2,\nu)}{Z(N,2,\mu)}$ is the generating function for the correlation functions.

Theorem: The coefficient of $c_1 \cdots c_n$ in $\frac{Z(N,2,\nu)}{Z(N,2,\mu)}$ is the $n$th correlation function $R_n(x_1, \ldots, x_N)$.

Exercise: Prove this.

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