Geometrization of Statistical Mechanics

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Brody, D C and Hughston, L P. 1999. Geometrization of Statistical Mechanics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 455(1985), pp. 1683-1715. ISSN 1364-5021 [Article]

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Official URL: https://royalsocietypublishing.org/doi/10.1098/rsp...

Abstract or Description

Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert-space formulation of classical statistical thermodynamics, we show that the specification of a canonical polarity on the real projective space RP^n induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, equilibrium thermal states are determined by a Hamiltonian gradient flow with respect to this metric. This flow is characterized by the property that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is extended to establish a new framework for the quantum-mechanical dynamics of equilibrium thermal states. In this case, the relevant phase space is the complex projective space CP^n, here regarded as a real manifold Γ endowed with the Fubini–Study metric and a compatible symplectic structure. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the non-uniqueness of the infinite-temperature state. We are then led to formulate a geometric characterization of the standard KMS relation often considered in the context of C* algebras. Finally, we develop a theory of the quantum microcanonical and canonical ensembles, based on the geometry of the quantum phase space Γ . The example of a quantum spin-1/2 particle in a heat bath is studied in detail.

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Article

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https://doi.org/10.1098/rspa.1999.0376