On the geometrical structure of the generalized quantum Gibbs states

We consider generalized Gibbs statistical states generated by a set of quantum mechanical observables. By using the logarithmic scale we show that the set of logarithms of these states (microscopical entropies) has the structure of a concave hypersurface embedded in a linear operator space. We treat...

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Bibliographic Details
Published in:Reports on mathematical physics 1986, Vol.24 (1), p.11-19
Main Author: Janyszek, Henryk
Format: Article
Language:English
Subjects:
Exact sciences and technology
Physics
Quantum ensemble theory
Quantum statistical mechanics
Statistical physics, thermodynamics, and nonlinear dynamical systems
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Summary:We consider generalized Gibbs statistical states generated by a set of quantum mechanical observables. By using the logarithmic scale we show that the set of logarithms of these states (microscopical entropies) has the structure of a concave hypersurface embedded in a linear operator space. We treat this hypersurface as a differentiable manifold with map given by the set of statistical temperatures. Next we construct the tangent space and introduce a Riemannian metric as a first quadratic form of the hypersurface. Because this hypersurface is isomorphic with the hypersurface given by the formula of the logarithm of partition function (geometrically interpreted as a hypersurface embedded in the Euclidean space) we also adopt this metric on that hypersurface. As example we take the grand canonical distribution for quantum continuous ideal Boson and Fermion gases and show that the Gaussian curvature has a physical meaning.
ISSN:0034-4877
1879-0674
DOI:10.1016/0034-4877(86)90037-6