Statistical origin of Legendre invariant metrics

Pineda-Reyes, V.; Escamilla-Herrera, L. F.; Gruber, Christine; Nettel, Francisco; Quevedo, Hernando

doi:10.1016/j.physa.2019.04.003

Cited by 6 publications

(11 citation statements)

References 19 publications

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“…it is invariant under Legendre transformations. However, the metric is not a Hessian although there have been recent attempts to derive it from statistical mechanics [103].…”

Section: Discussionmentioning

confidence: 99%

Contact and metric structures in black hole chemistry

Ghosh

Bhamidipati

2023

Front. Phys.

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We review recent studies of contact and thermodynamic geometry for black holes in AdS spacetimes in the extended thermodynamics framework. The cosmological constant gives rise to the notion of pressure P = −Λ/8π and, subsequently a conjugate volume V, thereby leading to a close analogy with hydrostatic thermodynamic systems. To begin with, we review the contact geometry approach to thermodynamics in general and then consider thermodynamic metrics constructed as the Hessians of various thermodynamic potentials. We then study their correspondence to statistical ensembles for systems with two-dimensional spaces of equilibrium states. From the zeroes and divergences of the curvature scalar obtained from the metric, we carefully analyze the issue of ensemble non-equivalence and show certain complimentary behaviors in the description of a thermodynamic system. Following a thorough analysis of the familiar van der Waals system, we turn our attention to black holes in extended phase space. Considering the example of charged AdS black holes, we discuss the generic features of their thermodynamic geometry in detail. The relationship of the thermodynamic curvature(s) with critical points as well as microscopic interactions in black holes is also briefly explored. We finally set up the thermodynamic geometry for finite temperature gauge theories dual to black holes in AdS via holographic correspondence and comment on recent progress.

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“…it is invariant under Legendre transformations. However, the metric is not a Hessian although there have been recent attempts to derive it from statistical mechanics [103].…”

Section: Discussionmentioning

confidence: 99%

Contact and metric structures in black hole chemistry

Ghosh

Bhamidipati

2023

Front. Phys.

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“…The Legendre invariant metrics presented here were constructed with the sole objective of minimally modifying the structures defining a contact metric manifold. Some other works had reached equivalent results using as a motivation some physical criteria [19,20,32] and from a different perspective. For instance, in [31] it was studied the group of transformations that leave the Hessian metrics invariant.…”

Section: Closing Remarksmentioning

confidence: 98%

“…Other attempts to find metrics for the space of thermodynamic equilibrium states which are invarianat under Legendre transformations were conducted in a strongly dependent thermodynamic coordinate framework [19] yielding a set of metrics whose components are proportional to the components of the Hessian metrics. Recently, it was shown in [20,32] that Legendre invariant metrics can be related physically to reparametrizations of the thermodynamic state variables. The set of metrics found in [20] have as a particular case those of [19].…”

Section: Associated Metric To An Almost Para-contact Structurementioning

confidence: 99%

“…Recently, it was shown in [20,32] that Legendre invariant metrics can be related physically to reparametrizations of the thermodynamic state variables. The set of metrics found in [20] have as a particular case those of [19]. Hence, it is physically relevant and mathematically consistent to generate Legendre invariant metrics from purely geometric structures.…”

Section: Associated Metric To An Almost Para-contact Structurementioning

confidence: 99%

“…It is worth mentioning that there is another simple choice for Ω (a) that makes g Ω Legendre invariant, although not invariant under polarization scaling. It was explored in [20] from a different approach and considers the functions Ω (a) as the product of two functions. That is, Ω (a) = f (a) (q a )g (a) (p a ) where f and g have the same functional form and are odd functions.…”

Section: Polarization Independent Metricsmentioning

confidence: 99%

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Contact polarizations and associated metrics in geometric thermodynamics

Lopez-Monsalvo,

Nettel,

Pineda-Reyes

et al. 2020

Preprint

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In this work we show that a Legendre transformation is nothing but a mere change of symplectic polarization from the point of view of contact geometry. Then, we construct a set of Riemannian and pseudo-Riemannian metrics on a contact manifold by introducing almost contact and para-contact structures and we analyze their isometries. We show that it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent i.e. the Legendre transformations are the corresponding isometries and, on the other, that it induces a Hessian metric into the corresponding Legendre submanifolds. This second property is motivated by the well known Riemannian structures of the geometric description of thermodynamics which are based on Hessian metrics on the space of equilibrium states and whose properties are related to the fluctuations of the system. We find that to define a Riemannian structure with such properties it is necessary to abandon the idea of an associated metric to an almost contact or para-contact structure. We find that even extending the contact metric structure of the thermodynamic phase space the thermodynamic desiderata cannot be fulfilled.

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