Probability distributions in quantum statistical mechanics
The monograph studies primarily probability distributions of observables associated to quantum statistical systems of noninteracting bosons and fermions as functions of the spectral measure of the Hamiltonians. In particular, the asymptotic behaviour for small energies of observables of the form \(\sum_{j}n_ jg(E_ j)\), where \(n_ j\) are occupation numbers on the canonical ensemble (the symmetric case being the nontrivial one), and \(\{E_ j\}\) the one-particle energy eigenvalues, is studied for several classes of Stieltjes-spectral measures dF (e.g., of the form \(F(E)=bE\) or \(F(E)=bE^ 2\), with b constant). Both Minkowski and Einstein space \(S^ 3\times {\mathbb{R}}\) are considered, the latter being connected to an interesting application to cosmology which underlies the physical motivation; cf. \textit{H. P. Jakobsen}, \textit{M. Kon} and \textit{I. E. Segal}, Phys. Rev. Lett. 42, 1788-1791 (1979). The basic formalism is a proper definition of ''integrals'' of independent random variables w.r.t. the appropriate spectral measures; depending on the latter several more or less ''singular'' central limit theorems are proved.
scientific article; zbMATH DE number 3901788