Fluctuations of addition spectra of independent quantum systems

Motivated by recent experiments on large quantum dots, we consider the energy spectrum in a system consisting of $N$ particles distributed among $K<N$ independent sub-systems, such that the energy of each sub-system is a quadratic function of the number of particles residing on it. On a large scale, the ground state energy E(N) of such a system grows quadratically with $N$, but in general there is no simple relation such as $E(N)=a N +b N^{2}$. The deviation of E(N) from exact quadratic behavior implies that its second difference (the inverse compressibility) $\chi_{N} \equiv E(N+1)-2 E(N)+E(N-1)$ is a fluctuating quantity. Regarding the numbers $\chi_{N}$ as values assumed by a certain random variable $\chi$, we obtain a closed-form expression for its distribution $F(\chi)$. Its main feature is that the corresponding density $P(\chi)=\frac{dF(\chi)} {d\chi}$ has a maximum at the point $\chi=0$. As $K \to \infty$ the density is Poissonian, namely, $P(\chi) \to e^{-\chi}$.