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Para-Sasakian geometry in thermodynamic fluctuation theory
Mathematical Physics · Submitted 22 Aug 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this way we show that the hyperbolic geometry and the Heisenberg commutation relations on the phase space naturally emerge from classical statistical mechanics.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
In this work we tie concepts derived from statistical mechanics, information theory and contact Riemannian geometry within a single consistent formalism for thermodynamic fluctuation theory. We derive the concrete relations characterizing the geometry of the Thermodynamic Phase Space stemming from the relative entropy and the Fisher-Rao information matrix. In particular, we show that the Thermodynamic Phase Space is endowed with a natural para-contact pseudo-Riemannian structure derived from a statistical moment expansion which is para-Sasaki and {\eta}-Einstein. Moreover, we prove that such manifold is locally isomorphic to the hyperbolic Heisenberg group. In this way we show that the hyperbolic geometry and the Heisenberg commutation relations on the phase space naturally emerge from classical statistical mechanics. Finally, we argue on the possible implications of our results.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this work we tie concepts derived from statistical mechanics, information theory and contact Riemannian geometry within a single consistent formalism for thermodynamic fluctuation theory. | contribution 0.98 | 0.92 | 0.10 | asserted 1.00 |
| s1 | We derive the concrete relations characterizing the geometry of the Thermodynamic Phase Space stemming from the relative entropy and the Fisher-Rao information matrix. | contribution 0.69 | 0.96 | 0.11 | asserted 1.00 |
| s2 | In particular, we show that the Thermodynamic Phase Space is endowed with a natural para-contact pseudo-Riemannian structure derived from a statistical moment expansion which is para-Sasaki and {\eta}-Einstein. | result 0.98 | 0.94 | 0.63 | asserted 1.00 |
| s3 | Moreover, we prove that such manifold is locally isomorphic to the hyperbolic Heisenberg group. | result 1.00 | 0.93 | 0.17 | asserted 1.00 |
| s4 | In this way we show that the hyperbolic geometry and the Heisenberg commutation relations on the phase space naturally emerge from classical statistical mechanics. | interpretation 0.94 | 0.87 | 0.46 | asserted 1.00 |
| s5 | Finally, we argue on the possible implications of our results. | interpretation 0.84 | 0.54 | 0.03 | hedged possibility 0.64 |
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The paper
Original abstract
In this work we tie concepts derived from statistical mechanics, information theory and contact Riemannian geometry within a single consistent formalism for thermodynamic fluctuation theory. We derive the concrete relations characterizing the geometry of the Thermodynamic Phase Space stemming from the relative entropy and the Fisher-Rao information matrix. In particular, we show that the Thermodynamic Phase Space is endowed with a natural para-contact pseudo-Riemannian structure derived from a statistical moment expansion which is para-Sasaki and {\eta}-Einstein. Moreover, we prove that such manifold is locally isomorphic to the hyperbolic Heisenberg group. In this way we show that the hyperbolic geometry and the Heisenberg commutation relations on the phase space naturally emerge from classical statistical mechanics. Finally, we argue on the possible implications of our results.
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How the literature uses it
2 arXiv papers cite this in 16 sentences
Papers citing it · 14 sentences without a confident label
Uses its method or tool 1 paper
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Following [ref] , from equations [ref] and [ref] one can construct an $(n+1)-$dimensional control manifold $𝒞$ embedded in $𝒯$ such that Φ: 𝒞 ⟶𝒯,
Contact Symmetries and Hamiltonian Thermodynamics · The Phase Space of Thermodynamics and the Hyperbolic Heisenberg Group · 0.81
Cites it as background 1 paper
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On the other hand, the theory of thermodynamical fluctuations can be recast using the geometric language of differential manifolds [ref] .
Generalised complex geometry in thermodynamical fluctuation theory · Introduction · 0.80
What it cites · 1 sentence without a confident label
Uses the method or tool of 1 paper
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To this end, we verify that the metric [ref] satisfies some formal definitions following the construction in [ref] and [ref] .
Geometry of $H$-paracontact metric manifolds · Geometric properties of the phase space of Equilibrium Thermodynamics · 0.87
Cites as background 1 paper
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In particular, it is a para-Sasakian and $η$-Einstein manifold whose Ricci scalar of the Levi-Civita connection is constant (see the standard references [ref] for Sasakian manifolds and [ref] for the para-Sasakian case).
Geometry of $H$-paracontact metric manifolds · Introduction · 0.93
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