dorsal/arxiv
View SchemaGeometry of Thermodynamic States
| Authors | Dorje C. Brody, Lane P. Hughston |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9706030 |
| URL | https://arxiv.org/abs/quant-ph/9706030 |
| DOI | 10.1016/S0375-9601(98)00385-5 |
| Journal | Phys.Lett. A245 (1998) 73-78 |
Abstract
A novel geometric formalism for statistical estimation is applied here to the canonical distribution of classical statistical mechanics. In this scheme thermodynamic states, or equivalently, statistical mechanical states, can be characterised concisely in terms of the geometry of a submanifold ${\cal M}$ of the unit sphere ${\cal S}$ in a real Hilbert space ${\cal H}$. The measurement of a thermodynamic variable then corresponds to the reduction of a state vector in ${\cal H}$ to an eigenstate, where the transition probability is the Boltzmann weight. We derive a set of uncertainty relations for conjugate thermodynamic variables in the equilibrium thermodynamic states. These follow as a consequence of a striking thermodynamic analogue of the Anandan-Aharonov relations in quantum mechanics. As a result we are able to provide a resolution to the controversy surrounding the status of `temperature fluctuations' in the canonical ensemble. By consideration of the curvature of the thermodynamic trajectory in its state space we are then able to derive a series of higher order variance bounds, which we calculate explicitly to second order.
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"abstract": "A novel geometric formalism for statistical estimation is applied here to the\ncanonical distribution of classical statistical mechanics. In this scheme\nthermodynamic states, or equivalently, statistical mechanical states, can be\ncharacterised concisely in terms of the geometry of a submanifold ${\\cal M}$ of\nthe unit sphere ${\\cal S}$ in a real Hilbert space ${\\cal H}$. The measurement\nof a thermodynamic variable then corresponds to the reduction of a state vector\nin ${\\cal H}$ to an eigenstate, where the transition probability is the\nBoltzmann weight. We derive a set of uncertainty relations for conjugate\nthermodynamic variables in the equilibrium thermodynamic states. These follow\nas a consequence of a striking thermodynamic analogue of the Anandan-Aharonov\nrelations in quantum mechanics. As a result we are able to provide a resolution\nto the controversy surrounding the status of `temperature fluctuations\u0027 in the\ncanonical ensemble. By consideration of the curvature of the thermodynamic\ntrajectory in its state space we are then able to derive a series of higher\norder variance bounds, which we calculate explicitly to second order.",
"arxiv_id": "quant-ph/9706030",
"authors": [
"Dorje C. Brody",
"Lane P. Hughston"
],
"categories": [
"quant-ph"
],
"doi": "10.1016/S0375-9601(98)00385-5",
"journal_ref": "Phys.Lett. A245 (1998) 73-78",
"title": "Geometry of Thermodynamic States",
"url": "https://arxiv.org/abs/quant-ph/9706030"
},
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