Quantum-Statistical Theory of Liquid Helium

The lambda transition in liquid helium is investigated on the basis of quantum statistical mechanics. In the first place, the partition function for an assembly of Bose particles is formulated in such a way that the effect of Bose statistics appears as a strong attractive potential between particles and gives rise to molecular clustering of various sizes, so that the assembly can be considered as a system of molecules and atoms obeying Boltzmann statistics. This formulation is then applied to the ideal Bose gas and to the actual helium liquid, and the following results are obtained: (a) An ideal Bose gas undergoes a phase transition similar in its formal property to the condensation of a classical imperfect gas. (b) By taking into account the interatomic potential, but neglecting all the quantum effects but the statistics, we obtain a jump in the specific heat curve at the transition temperature and a value of the transition temperature coming nearer to the actuality. (c) Beside the molecular clustering, or the “Bose-excitation,” the quantum effects bring in the “Debye phonons” and the “Landau rotons”, the former corresponding to the translational motions of the molecules and the atoms, and the latter to the rotational motions of the molecules. Although the quantum effects are not dealt with in detail, the agreement between theory and experiments is shown to be improved by taking them into account. It is also shown that these effects probably are not of very great importance even near the lambda point.