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The total momentum of $N$ interacting bosons or fermions in a cube equippedwith periodic boundary conditions is a conserved quantity. Its eigenvaluesfollow a probability distribution, determined by the thermal equilibrium state.While in non-interacting systems the distribution is normal with variance $\simN$, interaction couples the single-particle momenta, so that the distributionof their sum is unpredictable, except for some implications of Galileaninvariance. First, we present these implications which are strong in 1D,moderately strong in 2D, and weak in 3D. Then, we speculate about the possibleform of the distribution in fluids, crystals, and superfluids. The existence ofphonons suggests that the total momentum can remain finite when $N\to\infty$.We argue that in fluids the finite momenta distribute continuously, but theirintegrated probability is smaller than 1, because the momentum can also tend toinfinity with $N$. In the fluid-crystal transition we expect that the totalmomentum becomes finite with full probability and distributed over a lattice,and that in the fluid-superfluid transition a delta peak appears only at zerototal momentum. Based on this picture, we discuss the superfluid flow in boththe frictionless and the dissipative cases, and derive a temperature-dependentcritical velocity. Finally, we show that Landau's criterion for excitations inmoving superfluids is an in some cases correct result of an erroneousderivation.

Total momentum and thermodynamic phases of quantum systems