Fluctuations of addition spectra of independent quantum systems
arXiv:cond-mat/9811036 · doi:10.1088/0305-4470/31/40/003
Abstract
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) $Ï_{N} \equiv E(N+1)-2 E(N)+E(N-1)$ is a fluctuating quantity. Regarding the numbers $Ï_{N}$ as values assumed by a certain random variable $Ï$, we obtain a closed-form expression for its distribution $F(Ï)$. Its main feature is that the corresponding density $P(Ï)=\frac{dF(Ï)} {dÏ}$ has a maximum at the point $Ï=0$. As $K \to \infty$ the density is Poissonian, namely, $P(Ï) \to e^{-Ï}$.
13 pages, latex, no figures, to appear in J. Phys. A (Math. Gen.) 31, 8063 (1998)