The total momentum of quantum fluids

Abstract

The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension the asymptotic distribution of P/\sqrt{N} is normal at all temperatures and densities, and in two dimensions the tail distribution of P/\sqrt{N} is normal. We introduce the notion of the density matrix reduced to the center of mass, and show that its eigenvalues are N times the probabilities of the different eigenvalues of ¶. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P=0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum systems above one dimension, in infinite space, the total momentum is finite with a nonzero probability at all temperatures and densities. In solids this probability is 1, and in a crystal it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase. For a superfluid we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and prove that Landau's criterion cannot explain superfluidity and its breakdown in a dissipative flow. We also comment on the relation between superfluidity and Bose-Einstein condensation.