Para-Sasakian geometry in thermodynamic fluctuation theory

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.Comment: Significant improvements and corrections from the previous version. Additional material adde

Infinitesimal Legendre symmetry in the Geometrothermodynamics programme

The work within the Geometrothermodynamics programme rests upon the metric structure for the thermodynamic phase-space. Such structure exhibits discrete Legendre symmetry. In this work, we study the class of metrics which are invariant along the infinitesimal generators of Legendre transformations. We solve the Legendre-Killing equation for a $K$-contact general metric. We consider the case with two thermodynamic degrees of freedom, i.e. when the dimension of the thermodynamic phase-space is five. For the generic form of contact metrics, the solution of the Legendre-Killing system is unique, with the sole restriction that the only independent metric function -- $\Omega$ -- should be dragged along the orbits of the Legendre generator. We revisit the ideal gas in the light of this class of metrics. Imposing the vanishing of the scalar curvature for this system results in a further differential equation for the metric function $\Omega$ which is not compatible with the Legendre invariance constraint. This result does not allow us to use the regular interpretation of the curvature scalar as a measure of thermodynamic interaction for this particular class.Comment: Submitted to Advances in Mathematical Physics. 8 pages, 3 figure

Conformally invariant thermodynamics of a Maxwell-Dilaton black hole

The thermodynamics of Maxwell-Dilaton (dirty) black holes has been extensively studied. It has served as a fertile ground to test ideas about temperature through various definitions of surface gravity. In this paper, we make an independent analysis of this black hole solution in both, Einstein and Jordan, frames. We explore a set of definitions for the surface gravity and observe the different predictions they make for the near extremal configuration of this black hole. Finally, motivated by the singularity structure in the interior of the event horizon, we use a holographic argument to remove the micro-states from the disconnected region of this solution. In this manner, we construct a frame independent entropy from which we obtain a temperature which agrees with the standard results in the non-extremal regime, and has a desirable behaviour around the extremal configurations according to the third law of black hole mechanics.Comment: amendments from the previous versio

Maximally Symmetric Spacetimes emerging from thermodynamic fluctuations

In this work we prove that the maximally symmetric vacuum solutions of General Relativity emerge from the geometric structure of statistical mechanics and thermodynamic fluctuation theory. To present our argument, we begin by showing that the pseudo-Riemannian structure of the Thermodynamic Phase Space is a solution to the vacuum Einstein-Gauss-Bonnet theory of gravity with a cosmological constant. Then, we use the geometry of equilibrium thermodynamics to demonstrate that the maximally symmetric vacuum solutions of Einstein's Field Equations -- Minkowski, de-Sitter and Anti-de-Sitter spacetimes -- correspond to thermodynamic fluctuations. Moreover, we argue that these might be the only possible solutions that can be derived in this manner. Thus, the results presented here are the first concrete examples of spacetimes effectively emerging from the thermodynamic limit over an unspecified microscopic theory without any further assumptions.Comment: Preliminary version. Comments are welcome! Corrected equation referencin

Covariant Thermodynamics and Relativity

This thesis deals with the dynamics of irreversible processes within the context of the general theory of relativity. In particular, we address the problem of the 'infinite' speed of propagation of thermal disturbances in a dissipative fluid. The present work builds on the multi-fluid variational approach to relativistic dissipation, pioneered by Carter, and provides a dynamical theory of heat conduction. The novel property of such approach is the thermodynamic interpretation associated with a two-fluid system whose constituents are matter and entropy. The dynamics of this model leads to a relativistic generalisation of the Cattaneo equation; the constitutive relation for causal heat transport. A comparison with the Israel and Stewart model is presented and its equivalence is shown. This discussion provides new insights into the not-well understood definition of a non-equilibrium temperature. The variational approach to heat conduction presented in this thesis constitutes a mathematically promising formalism to explore the relativistic evolution towards equilibrium of dissipative fluids in a dynamical manner and to get a deeper conceptual understanding of non-equilibrium thermodynamic quantities. Moreover, it might also be useful to explore the more fundamental issues of the irreversible dynamics of relativity and its connections with the time asymmetry of nature.Comment: PhD Thesis, School of Mathematics, University of Southampton, UK, 201

Contact Symmetries and Hamiltonian Thermodynamics

It has been shown that contact geometry is the proper framework underlying classical thermodynamics and that thermodynamic fluctuations are captured by an additional metric structure related to Fisher's Information Matrix. In this work we analyze several unaddressed aspects about the application of contact and metric geometry to thermodynamics. We consider here the Thermodynamic Phase Space and start by investigating the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics. Then we present a novel mathematical characterization of first order phase transitions as equilibrium processes on the Thermodynamic Phase Space for which the Legendre symmetry is broken. Moreover, we use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics and we show that the relevant Hamiltonian coincides with the irreversible entropy production along thermodynamic processes. Therefore, we use such property to give a geometric definition of thermodynamically admissible fluctuations according to the Second Law of thermodynamics. Finally, we show that the length of a curve describing a thermodynamic process measures its entropy production.Comment: 33 pages, 2 figures, substantial improvement of http://arxiv.org/abs/1308.674

A static axisymmetric exact solution of f(R)f(R)-gravity

We present an exact, axially symmetric, static, vacuum solution for $f(R)$ gravity in Weyl's canonical coordinates. We obtain a general explicit expression for the dependence of $df(R)/dR$ upon the $r$ and $z$ coordinates and then the corresponding explicit form of $f(R)$, which must be consistent with the field equations. We analyze in detail the modified Schwarzschild solution in prolate spheroidal coordinates. Finally, we study the curvature invariants and show that, in the case of $f(R)\neq R$, this solution corresponds to a naked singularity.Comment: To be published in Physics Letters B. Replaced with minor changes to match published versio

Legendre symmetry and first order phase transitions of homogeneous systems

In this work we give a characterisation of first order phase transitions as equilibrium processes on the thermodynamic phase space for which the Legendre symmetry is broken. Furthermore, we consider generalised theories of thermodynamics, where the potential is a homogeneous function of any order $\beta$ and we propose a (contact) Hamiltonian formulation of equilibrium processes. Indeed we prove that equilibrium corresponds to the zeroth levels of such function. Using these results we infer that the description in equilibrium of first order phase transitions is possible only when the potential is a homogeneous function of order one, unless a generalised Zeroth Law is postulated in order to allow for equilibrium between sub-parts of the system at different values of the intensive quantities. Finally, we show the example of the Tolman-Ehrenfest effect.Comment: Substantial improvements of the previous versio

Variational thermodynamics of relativistic thin disks

We present a relativistic model describing a thin disk system composed of two fluids. The system is surrounded by a halo in the presence of a non-trivial electromagnetic field. We show that the model is compatible with the variational multi-fluid thermodynamics formalism, allowing us to determine all the thermodynamic variables associated with the matter content of the disk. The asymptotic behaviour of these quantities indicates that the single fluid interpretation should be abandoned in favour of a two-fluid model.Comment: Einstein-Maxwell system and equations for the sources added. Clarifications regarding other disk solutions included. Revised solutions for matter models. Submitted to Phys Lett

Some Statistical Mechanical Properties of Photon Black Holes

We show that if the total internal energy of a black hole is constructed as the sum of $N$ photons all having a fixed wavelength chosen to scale with the Schwarzschild radius as $\lambda=\alpha R_{s}$, then $N$ will scale with $R_{s}^{2}$. A statistical mechanical calculation of the configuration proposed yields (\alpha = 4 \pi^2 / \ln(2)) and a total entropy of the system $S=k_{B}N \ln(2)$, in agreement with the Bekenstein entropy of a black hole . It is shown that the critical temperature for Bose-Einstein condensation for relativistic particles of $\lambda=\alpha R_{s}$ is always well below the Hawking temperature of a black hole, in support of the proposed internal configuration. We then examine our results from the point of view of recent loop quantum gravity ideas and find that a natural consistency of both approaches appears. We show that the Jeans criterion for gravitational instability can be generalised to the special and general relativistic regimes and holds for any type of mass--energy distribution.Comment: 7 pages, 1 figure. Added discussion on relativistic Jeans criterion and a more formal statistical mechanical treatment. Accepted for publication in the Revista Mexicana de Fisic