Geometrisation of Statistical Mechanics
Abstract
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 introduce projective geometry as a basis for analysing probabilistic aspects of statistical physics. In particular, the specification of a canonical polarity on $RP^{n}$ induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, we show that equilibrium thermal states are determined by the Hamiltonian gradient flow with respect to this metric. This flow is concisely characterised by the fact 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 then extended to include the quantum mechanical dynamics of equilibrium thermal states. In this case the relevant state space is complex projective space, here regarded as a real manifold endowed with the natural FubiniStudy metric. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the nonuniqueness of the infinite temperature state. We are then led to formulate a geometric characterisation of the standard KMSrelation often considered in the context of $C^{*}$algebras. The example of a quantum spin onehalf particle in heat bath is studied in detail.
 Publication:

arXiv eprints
 Pub Date:
 August 1997
 DOI:
 10.48550/arXiv.grqc/9708032
 arXiv:
 arXiv:grqc/9708032
 Bibcode:
 1997gr.qc.....8032B
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 31 pages, RevTeX