  The total momentum of $N$ interacting bosons or fermions in a cube equipped
with periodic boundary conditions is a conserved quantity. Its eigenvalues
follow a probability distribution, determined by the thermal equilibrium state.
While in non-interacting systems the distribution is normal with variance $\sim
N$, interaction couples the single-particle momenta, so that the distribution
of their sum is unpredictable, except for some implications of Galilean
invariance. First, we present these implications which are strong in 1D,
moderately strong in 2D, and weak in 3D. Then, we speculate about the possible
form of the distribution in fluids, crystals, and superfluids. The existence of
phonons suggests that the total momentum can remain finite when $N\to\infty$.
We argue that in fluids the finite momenta distribute continuously, but their
integrated probability is smaller than 1, because the momentum can also tend to
infinity with $N$. In the fluid-crystal transition we expect that the total
momentum becomes finite with full probability and distributed over a lattice,
and that in the fluid-superfluid transition a delta peak appears only at zero
total momentum. Based on this picture, we discuss the superfluid flow in both
the frictionless and the dissipative cases, and derive a temperature-dependent
critical velocity. Finally, we show that Landau's criterion for excitations in
moving superfluids is an in some cases correct result of an erroneous
derivation.
