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 secondorder 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.

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. I. CONTEXTRelativistic thermodynamics continues to provide interesting challenges, in particular in the context of dissipative and nonlinear phenomena. The issues involved range from direct applications in various areas of physics to fundamental problems like the nature of time (visavi the second law of thermodynamics) and the formation of structures at nonlinear deviations from thermal equilibrium. Much recent work has been motivated by the modelling of complex astrophysical systems, like neutron stars [1], and cosmology [2]. There has also been a resurgence of interest in dissipative systems in the context of colliders like RHIC at Brookhaven and the LHC at CERN [3-5]. These latter developments, which have to a large extent been driven by the need to understand the dynamics of a hot quark-gluon plasma, are often linked with underlying principles like the AdS/CFT conjecture and holography [6]. Even though the problem dates back to the origins of relativity theory, it remains (in a slightly different guise) at the forefront of modern thinking.According to the established consensus view, one must account for second-order deviations from thermal equilibrium in order to achieve causality and stability. This is certainly the lesson from the celebrated work of Israel and Stewart [7,8], see [9][10][11] for recent work on the problem. We have recently revisited the key points in the context of heat conduction [12], taking a multi-fluid prescription based on Carter's convective variational formulation for relativistic fluids [13] as our starting point. This is a mathematically elegant approach that has the flexibility required to account for the physics that we need to consider. A particularly appealing feature of the variational approach is that, once an "equation of state" for matter is provided, the theory provides the relation between the various currents and their conjugate momenta [1]. The variational analysis leads to a second-order model which has the key elements required for causality and stability, in particular, it clarif...

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.

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 onedimensional 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 (eg. 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.

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.