Exact density of states and its critical behavior

Abstract

We propose a fast algorithm for the exact computation of the density of states for arbitrary discrete systems on intermediate-size lattices. Results for the Ising model on lattices up to 13×13 are presented. We also discuss how signals for phase transitions may be observed by inspecting the density of states directly, and verify this with the numerical data for the Ising model. This procedure is based on the classical view of the density of states as the exponential of the entropy. Phase transition are then characterized by “straight sections” in the entropy, considered as a function of the energy. It is found that second- as well asfirst-order phase transitions may be so described, the difference between the two being in the way the length of the straight section goes to infinity with the size of the system. This viewpoint leads then to a short and physically transparent derivation of the Josephson inequality for critical exponents.