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The structure of the thermal equilibrium state of a weakly interacting Bosegas is of current interest. We calculate the density matrix of that state intwo ways. The most effective method, in terms of yielding a simple, explicitanswer, is to construct a generating function within the traditional frameworkof quantum statistical mechanics. The alternative method, arguably moreinteresting, is to construct the thermal state as a vector state in anartificial system with twice as many degrees of freedom. It is well known thatthis construction has an actual physical realization in the quantumthermodynamics of black holes, where the added degrees of freedom correspond tothe second sheet of the Kruskal manifold and the thermal vector state is astate of the Unruh or the Hartle-Hawking type. What is unusual about thepresent work is that the Bogolubov transformation used to construct the thermalstate combines in a rather symmetrical way with Bogolubov's originaltransformation of the same form, used to implement the interaction of thenonideal gas in linear approximation. In addition to providing a densitymatrix, the method makes it possible to calculate efficiently certainexpectation values directly in terms of the thermal vector state of the doubledsystem.

Interacting Bosons at Finite Temperature: How Bogolubov Visited a Black Hole and Came Home Again