On the Distribution of Eigenvalues of Grand Canonical Density Matrices

Using physical arguments and partition theoretic methods, we demonstrate under general conditions, that the eigenvalues w(m) of the grand canonical density matrix decay rapidly with their index m, like w(m)∼exp[−βB−1(ln m)1+1/α], where B and α are positive constants, O(1), which may be computed from the spectrum of the Hamiltonian. We compute values of B and α for several physical models, and confirm our theoretical predictions with numerical …