Cesar S. Lopez-Monsalvo

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 Fishers 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.

Thermal dynamics in general relativity

We discuss a relativistic model for heat conduction, building on a convective variational approach to multi-fluid systems where the entropy is treated as a distinct dynamical entity. We demonstrate how this approach leads to a relativistic version of the Cattaneo equation, encoding the finite thermal relaxation time that is required to satisfy causality. We also show that the model naturally includes the non-equilibrium Gibbs relation that is a key ingredient in most approaches to extended thermodynamics. Focusing on the pure heat conduction problem, we compare the variational results with the second-order model developed by Israel and Stewart. The comparison shows that, despite the very different philosophies behind the two approaches, the two models are equivalent at first-order deviations from thermal equilibrium. Finally, we complete the picture by working out the non-relativistic limit of our results, making contact with recent work in that regime.

Relativistic two-stream instability

We study the (local) propagation of plane waves in a relativistic, non- dissipative, two-fluid system, allowing for a relative velocity in the “background” configuration. The main aim is to analyze relativistic two-stream instability. This instability requires a relative flow—either across an interface or when two or more fluids interpenetrate—and can be triggered, for example, when one-dimensional plane-waves appear to be left-moving with respect to one fluid, but right-moving with respect to another. The dispersion relation of the two-fluid system is studied for different two-fluid equations of state: (i) the “free” (where there is no direct coupling between the fluid densities), (ii) coupled, and (iii) entrained (where the fluid momenta are linear combinations of the velocities) cases are considered in a frame-independent fashion (e.g.no restriction to the rest-frame of either fluid). As a by-product of our analysis we determine the necessary conditions for a two-fluid system to be causal and absolutely stable and establish a new constraint on the entrainment.

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 η-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.

A consistent first-order model for relativistic heat flow

This paper revisits the problem of heat conduction in relativistic fluids, associated with issues concerning both stability and causality. It has long been known that the problem requires information involving second-order deviations from thermal equilibrium. Basically, any consistent first-order theory needs to remain cognizant of its higher order origins. We demonstrate this by carrying out the required first-order reduction of a recent variational model. We provide an analysis of the dynamics of the system, obtaining the conditions that must be satisfied in order to avoid instabilities and acausal signal propagation. The results demonstrate, beyond any reasonable doubt, that the model has all the features one would expect of a real physical system. In particular, we highlight the presence of a second sound for heat in the appropriate limit. We also make contact with previous work on the problem by showing how the various constraints on our system agree with previously established results.

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

Abstract We present an exact, axially symmetric, static, vacuum solution for f ( R ) -gravity in Weylʼs canonical coordinates. We obtain a general explicit expression for the dependence of d f ( R ) / d R 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 ) ≠ R , this solution corresponds to a naked singularity.

Conformal Gauge Transformations in Thermodynamics

In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the connection one-form, whilst, as a consequence of the non-integrability of the connection, its curvature is not and, therefore, neither is the associated pseudo-Riemannian geometry. We claim that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all of the elements of the geometric structure of the thermodynamic phase space change under a re-scaling of the connection one-form. We call this transformation of the geometric structure a conformal gauge transformation. As an example, we revisit the change of the thermodynamic representation and consider the resulting change between the two metrics on the thermodynamic phase space, which induce Weinhold’s energy metric and Ruppeiner’s entropy metric. As a by-product, we obtain a proof of the well-known conformal relation between Weinhold’s and Ruppeiner’s metrics along the equilibrium directions. Finally, we find interesting properties of the almost para-contact structure and of its eigenvectors, which may be of physical interest.

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

The zeroth law in quasi-homogeneous thermodynamics and black holes

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Thermodynamic optimization of a Penrose process: An engineers’ approach to black hole thermodynamics

-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 --