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In order to determine finite-temperature properties of the infinite one-dimensional Hubbard model for the half-filled band, finite systems such as chains with 2 to 5 atoms and rings with 4 and 6 atoms are studied with the application of the grand canonical ensemble. The obtained temperature dependence of the specific heat, the entropy, the magnetic susceptibility and some correlation functions clearly shows a rearrangement of the electronic state of the system with the increase of the strength of the Coulomb interaction relative to the transfer integral. In the light of our new results the functional integral approach recently proposed by Kimball and Schrieffer as well as Hubbard's approximate theory is examined.

STATISTICAL mechanics considers any particular body at given temperature T as a member of a canonical ensemble, that is, the probability of finding it with energy E is given by where Ω is the multiplicity of the level E of this body. We can take it that, for the body under test, P is a maximum with regard to any parameter n so that The use of this relation appears to simplify appreciably the treatment of some problems in statistical mechanics. I have not, however, been able to find it mentioned anywhere in the known treatises on the subject.

The diagram expansions which are applicable to the degenerate Bose fluids at finite temperatures are derived within the framework of the canonical ensemble. They are natural extensions of the diagram expansions due to Bloch and Dominicis, Lee and Yang, and Montroll and Ward for the non-degenerate case. The formulation is performed as carefully as Kromminga and Bolsteri treated the Bose fluids at zero temperature.