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We analyze the statistics of electrostatic energies (and their differences)for a quantum dot system composed of a finite number $K$ of electron islands(metallic grains) with random capacitance-inductance matrix $C$, for which thetotal charge is discrete, $Q=Ne$ (where $e$ is the charge of an electron and$N$ is an integer). The analysis is based on a generalized charging model,where the electrons are distributed among the grains such that theelectrostatic energy E(N) is minimal. Its second difference (inversecompressibility) $\chi_{N}=E(N+1)-2 E(N)+E(N-1)$ represents the spacing betweenadjacent Coulomb blockade peaks appearing when the conductance of the quantumdot is plotted against gate voltage. The statistics of this quantity has beenthe focus of experimental and theoretical investigations during the last twodecades. We provide an algorithm for calculating the distribution functioncorresponding to $\chi_{N}$ and show that this function is piecewisepolynomial.Comment: 21 pages, no figures, mathematical nomenclature (except for Abstract  and Introduction

Discrete charging of metallic grains: statistics of addition spectra