A Lower Bound on the Partition Function for a Strictly Neutral Charge-Symmetric System

Abstract

A lower bound on the grand partition function of a classical charge-symmetric system is adapted to the neutral grand canonical ensemble, in which the system is constrained to have zero total charge. This constraint permits us to consider two-body potentials that are only conditionally positive definite.

Appendix

The Gaussian approximation $\varXi _2$ can be computed explicitly. As an illustration, we give in this Appendix an explicit formula for the case of a Coulomb system in a torus

$$\begin{aligned} \varLambda =X= {\mathbb {R}}^d/a{\mathbb {Z}}^d. \end{aligned}$$

The interaction potential is assumed to be given by (1.2); that is,

$$\begin{aligned} u(x,y) = \frac{1}{|\varLambda |}\sum _{p \in \varLambda ^* \setminus \{0\}} \hat{u}_p \, {\mathrm {e}}^{\mathrm{i} p \cdot (x-y)}, \end{aligned}$$

where $|\varLambda | = a^d$, $\varLambda ^*=2 \pi a^{-1} {\mathbb {Z}}^d$, and

$$\begin{aligned} \hat{u}_p = \frac{{\mathrm {e}}^{-t |p |^2}}{|p |^2}. \end{aligned}$$

The parameter $t > 0$ is an ultraviolet cutoff. (For $d=1$ we may let $t = 0$.) We assume the term $u_0=u_0(t)$ introduced in (2.2) is chosen so that the energy

$$\begin{aligned} E_0 = \tfrac{1}{2}\lim _{t\rightarrow 0} \Bigg [{ u_0(t) - \frac{1}{|\varLambda |}\sum _{p\in \varLambda ^*\setminus \{0\}} \frac{{\mathrm {e}}^{-t|p |^2}}{|p |^2} } \Bigg ] \end{aligned}$$

is finite.

For $\phi \in \varPhi /\varTheta$ and $p\in \varLambda ^*\setminus \{0\}$, let

$$\begin{aligned} \hat{\phi }_p = \int _\varLambda \langle \phi , \delta '_x \rangle \, {\mathrm {e}}^{-\mathrm{i} p \cdot x} \, v({\mathrm {d}}x). \end{aligned}$$

(Recall that $\pi ^*(\delta '_x) = \delta _x - \delta _{x_0}$, and note that $\hat{\phi }_p$ is independent of the choice of basepoint $x_0$.) The Fourier components $\hat{\varrho }_p$ of a charge distribution $\varrho \in (\varPhi /\varTheta )^*$ satisfy

$$\begin{aligned} \frac{1}{|\varLambda |} \sum _{p \in \varLambda ^*\setminus \{0\}} \hat{\phi }_p \,\overline{\hat{\varrho }_p} = \langle \phi , \varrho \rangle \end{aligned}$$

for any $\phi \in \varPhi /\varTheta$. Explicitly, if $\varrho =\sum _{j=1}^n c_j \delta '_{x_j}$ with $\sum _{j=1}^n c_j = 0$, then

$$\begin{aligned} \hat{\varrho }_p = \sum _{j=1}^n c_j {\mathrm {e}}^{-\mathrm{i}p \cdot x_j}. \end{aligned}$$

We have

$$\begin{aligned} V_2(\phi ) = \tfrac{1}{2} \xi _0^{-2} |\varLambda |^{-1} \sum _{p\in \varLambda ^*\setminus \{0\}} |\hat{\phi }_p |^2 - \tfrac{1}{2} \xi _0^{-2} u_0 |\varLambda | \end{aligned}$$

and

$$\begin{aligned} \int {\mathrm {e}}^{\mathrm{i}\langle \phi , \varrho \rangle } \, \gamma ({\mathrm {d}}\phi ) = \exp \Bigg [{- \tfrac{1}{2}|\varLambda |^{-1} \sum _{p \in \varLambda ^*\setminus \{0\}} \hat{u}_p |\hat{\varrho }_p |^2 }\Bigg ]. \end{aligned}$$

A standard calculation gives

$$\begin{aligned} \varXi _2&= \varXi _0 \int {\mathrm {e}}^{-V_2(\phi )} \, \gamma ({\mathrm {d}}\phi ) \\&= \varXi _0 \exp \Bigg [ \tfrac{1}{2} \xi _0^{-2} u_0 |\varLambda | - \tfrac{1}{2} \sum _{p \in \varLambda ^*\setminus \{0\}} \log \left( 1 + \xi _0^{-2}\hat{u}_p \right) \Bigg ]. \end{aligned}$$

In dimensions $d < 4$, we can remove the ultraviolet cutoff in $\varXi _2$ by letting t go to zero. The result is the Debye–Hückel approximation

$$\begin{aligned} \lim _{t \rightarrow 0} \varXi _2 = \varXi _0 \exp \Bigg \{{ \xi _0^{-2} E_0 |\varLambda |- \tfrac{1}{2} \sum _{p \in \varLambda ^*\setminus \{0\}} \left[ \log \left( 1 + \xi _0^{-2}|p |^{-2} \right) -\xi _0^{-2}|p |^{-2}\right] }\Bigg \}. \end{aligned}$$