Extended Thermodynamics Research Articles

The role of the dynamic pressure in the behavior of an oscillating gas bubble

The paper contains a preliminary study on the role that dynamic pressure might play in the dynamics of a gas bubble oscillating in a liquid. To this aim, we introduce a mathematical model, proposed under the homobaricity hypothesis and deduced from the 14-moment theory of rational extended thermodynamics through significant simplifications, that makes the equations easily integrable over long time intervals. In the presence of a gas with high bulk viscosity, relevant effects can be observed in different physical conditions: isothermal or adiabatic regimes, small amplitude oscillations, non-linear oscillations, resonances, and sonoluminescence. To make the study more realistic, we always refer to carbon dioxide gas, which on the one hand could present high values of bulk viscosity and on the other hand is known for its peculiar behaviors in the framework of cavitation and gas bubbles.

A Novel ES-BGK Model for Non-polytropic Gases with Internal State Density Independent of the Temperature

A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of rational extended thermodynamics. The structure of planar shock waves in carbon dioxide (CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonic behavior across the shock structure, rather than the non monotonic behavior previously found.

On two possible ways to recover ordinary thermodynamics from extended thermodynamics of polyatomic gases

We consider two possible ways, i.e. the Maxwellian iteration (MI) and the Chapman–Enskog method (CEM), to recover relativistic ordinary thermodynamics from relativistic extended thermodynamics of Polyatomic gases with N moments. Both of these methods give the Eckart equations which are the relativistic version of the Navier–Stokes and Fourier laws as a first iteration. However, these methods do not lead to the same expressions of the heat conductivity χ, the shear viscosity µ, and the bulk viscosity ν which appear as coefficients in the Eckart equations. In particular, we prove that the expressions of χ, µ, and ν obtained via the CEM do not depend on N, while those obtained through the MI depend on N. Moreover, we also prove that these two methods lead to the same results in the nonrelativistic limit.

Thermodynamic topology of black holes from bulk-boundary, extended, and restricted phase space perspectives

In this article, we investigate the thermodynamic topology of some black holes, namely AdS Reissner Nordstrom (R–N), AdS Einstein–Gauss–Bonnet (EGB), and AdS Einstein-power-Yang–Mills (EPYM), from different frameworks: bulk-boundary (BB) and restricted phase space (RPS). Using the generalized off-shell Helmholtz free energy method, we calculate the thermodynamic topology of the selected black holes in each space separately and determine their topological classifications. We show that the addition of GB terms, dimensions, and other factors do not affect the topological classes of black holes in both spaces. The calculations and plots indicate that the AdS R–N and AdS EGB black holes show similar behavior and their topological numbers sets in both spaces, i.e., BB and RPS, are similar and equal to (W=+1). However, AdS EPYM black holes show an interesting behavior. In addition to BBT and RPS, we also consider the extended phase space thermodynamics (EPST) and evaluate the thermodynamic topology for AdS EPYM black hole. The changing (r−τ) in both spaces shows similar behavior. Also, the topological number and the total topological numbers for this black hole in the BB, RPS and EPS thermodynamics are completely same, i,e., (ωBBT=ωRPS=ωEPST=+1,−1) or WBBT=WRPS=WEPST=0. An important point is that the Einstein–Yang–Mills black hole has thermodynamic topology equivalence in three spaces. The present result may be due to the non-linear YM charge parameter and the difference between the gauge and gravity corrections in the above black holes.

Eckart equations, Maxwellian iteration and relativistic causal theories of divergence type

We consider a general causal relativistic theory of divergence type in the framework of rational extended thermodynamics (RET) for a compressible, possibly dense, gas. We require that the system converges in the Maxwellian iteration’s first step to the parabolic Eckart equations. This requirement implies a constraint between the two coefficients present in the triple tensor evaluated at equilibrium. Moreover, the production tensor is determined for prescript thermal and caloric state equations and given heat conductivity, shear and bulk viscosities. In the second part, we prove that if the original hyperbolic system satisfies the universal principles of RET, as can be put in the symmetric form using the main field , it always satisfies the previous compatibility condition. Therefore, any causal system of divergence type that satisfies the entropy principle with a convex entropy converges to the Eckart system in the Maxwellian iteration also when we have no information at the mesoscopic scale from the kinetic theory. The obtained results are tested on the RET theories of rarefied monatomic and polyatomic gases. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

Higher-Dimensional Origin of Extended Black Hole Thermodynamics.

Holographic braneworlds are used to present a higher-dimensional origin of extended black hole thermodynamics. In this framework, classical, asymptotically anti-de Sitter black holes map to quantum black holes in one dimension less, with a conformal matter sector that backreacts on the brane geometry. Varying the brane tension alone leads to a dynamical cosmological constant on the brane, and, correspondingly, a variable pressure attributed to the brane black hole. Thus, standard thermodynamics in the bulk, including a work term coming from the brane, induces extended thermodynamics on the brane, exactly, to all orders in the backreaction. A microsopic interpretation of the extended thermodynamics of specific quantum black holes is given via double holography.

Extended phase space thermodynamics of black holes: A study in Einstein's gravity and beyond

In the extended phase space approach, one can define thermodynamic pressure and volume that gives rise to the van der Waals type phase transition for black holes. For Einstein's GR, the expressions of these quantities are unanimously accepted. Of late, the van der Waals phase transition in black holes has been found in modified theories of gravity as well, such as the $f(R)$ gravity and the scalar-tensor gravity. However, in the case of these modified theories of gravity, the expression of pressure (and, hence, volume) is not uniquely determined. In addition, for these modified theories, the extended phase space thermodynamics has not been studied extensively, especially in a covariant way. Since both the scalar-tensor and the $f(R)$ gravity can be discussed in the two conformally connected frames (the Jordan and the Einstein frame respectively), the arbitrariness in the expression of pressure, will act upon the equivalence of the thermodynamic parameters in the two frames. We highlight these issues in the paper. Before that, in Einstein's gravity (GR), we obtain a general expression of the equilibrium state version of first law and the Smarr-like formula from the Einstein's equation for a general static and spherically symmetric (SSS) metric. Here we directly obtain the first law as well as the Smarr-like formula in GR in terms of the parameters present in the metric (such as mass, charge \textit{etc.}). This study also shows how the extended phase space is formulated (by considering the cosmological constant as variable) and, also shows why the cosmological constant plays the role of thermodynamic pressure in GR in extended phase space. Moreover, obtaining the Smarr formula from the Einstein's equation for the SSS metric suggests that this dynamical equation encodes more information on BH thermodynamics than what has been anticipated before.

Acceleration Waves in Cylindrical Shrinking Gas Bubbles

The paper studies the case of shrinking cylindrical gas bubbles acting as a radial piston and generating acceleration waves. The behavior of such waves and their improbable transformation into shocks are illustrated theoretically, as well as through some examples inspired by experimental data. The use of rational extended thermodynamics enables us to highlight the relevance of the dissipation and the possible role played by dynamic pressure and stress tensor in bubble evolution or shock formation. These results constitute an extension and a completion of a previous work dedicated to the analysis of acceleration waves generated in oscillating spherical bubbles.

Thermodynamics of 5D charged rotating black holes: a counterterms treatment

We use the counterterms subtraction method to calculate various thermodynamical quantities for charged rotating black holes in five-dimensional minimal gauged supergravity (Chong et al. in Phys Rev Lett 95:161301, 2005, https://doi.org/10.1103/PhysRevLett.95.161301, arXiv:hep-th/0506029). Specifically, we analyze certain issues related to the first law and Smarr’s relation in the presence of a conformal anomaly. Among the bulk quantities calculated are the on-shell action, total mass, and angular momenta of the solution. All these quantities are consistent with previous calculations made using other methods. For the boundary theory, we calculate the renormalized stress tensor, conformal anomaly, and Casimir energy. Using the Papadimitriou–Skenderis analysis (Papadimitriou and Skenderis in J High Energy Phys 08:004, 2005), we show that the mass calculated via the counterterms method satisfies the first law of black hole thermodynamics. To discuss extended thermodynamics, we extend the definition of the thermodynamic volume to cases with conformal anomalies using a procedure similar to that of Papadimitriou–Skenderis. We show that this volume correctly accounts for extra terms due to boundary metric variation. This shows that the mass and volume calculated using counterterms satisfy Smarr’s relation as well as the first law.

Navier–Stokes–Fourier equations as a parabolic limit of a general hyperbolic system of rational extended thermodynamics

This paper aims to prove that a general system of 14 balance laws for a compressible, possibly dense, gas that satisfies the universal principles of Rational Extended Thermodynamics (RET) converges to the Navier–Stokes–Fourier equations in the first step of the Maxwellian iteration. Moreover, in a theory not far from equilibrium, we show that the production terms of the hyperbolic system are uniquely determined as soon as the heat conductivity, the shear viscosity, and the bulk viscosity are assigned. The obtained results are tested on the RET theories for rarefied monatomic and polyatomic gases.

How Flexible Is the Concept of Local Thermodynamic Equilibrium?

It has been demonstrated by using generalized phenomenological irreversible thermodynamic theory (GPITT) that by replacing the conventional composition variables {xk} by the quantum level composition variables {x˜k,j} corresponding to the nonequilibrium population of the quantum states, the resultant description remains well within the local thermodynamic equilibrium (LTE) domain. The next attempt is to replace the quantum level composition variables by their respective macroscopic manifestations as variables. For example, these manifestations are, say, the observance of fluorescence and phosphorescence, existence of physical fluxes, and ability to register various spectra (microwave, IR, UV-VIS, ESR, NMR, etc.). This exercise results in a framework that resembles with the thermodynamics with internal variables (TIV), which too is obtained as a framework within the LTE domain. This TIV-type framework is easily transformed to an extended irreversible thermodynamics (EIT) type framework, which uses physical fluxes as additional variables. The GPITT in EIT version is also obtained well within the LTE domain. Thus, GPITT becomes a complete version of classical irreversible thermodynamics (CIT). It is demonstrated that LTE is much more flexible than what CIT impresses upon. This conclusion is based on the realization that the spatial uniformity for each tiny pocket (cell) of a spatially non-uniform system remains intact while developing GPITT and obviously in its other versions.

Six-Field Theory for a Polyatomic Gas Mixture: Extended Thermodynamics and Kinetic Models

Polyatomic gases may be characterized by internal molecular degrees of freedom. As a consequence, at a macroscopic level, dynamic pressure appears, which may be related to the bulk viscosity of the gas. Inspired by the models of a single polyatomic gas with six fields, developed within rational extended thermodynamics (RET) and the kinetic theory of gases, this paper presents a six-field theory for the mixture of polyatomic gases. First, the macroscopic mixture model is developed within the framework of RET. Second, the mixture of gases with six fields is analyzed in the context of the kinetic theory of gases, and corresponding moment equations are derived. Finally, complete closure of the RET model, i.e., computation of the phenomenological coefficients, is achieved by means of a combined macroscopic/kinetic closure procedure.

Acceleration waves and oscillating gas bubbles modelled by rational extended thermodynamics

The study of acceleration waves for a rarefied polyatomic gas is carried out in planar, cylindrical and spherical geometry referring to the rational extended thermodynamics theory with 14 moments. The case of a rarefied monatomic gas is determined as a limit case, and the role of geometry and molecular degrees of freedom is investigated. In addition, the behaviour of an acceleration wave travelling inside an oscillating gas bubble is modelled by the 14-moment PDE system under adiabatic condition. We show that dissipation combined with hyperbolicity tends to inhibit shock formation, and that the dynamic pressure cannot be zero inside the oscillating bubble. This fact can produce observable effects even in the Navier–Stokes approximation, if the gas exhibits high bulk viscosity.

A complete classification of sub-shocks in the shock structure of a binary mixture of Eulerian gases with different degrees of freedom

The shock structure in a binary mixture of polyatomic Eulerian gases with different degrees of freedom of a molecule is studied based on the multi-temperature model of rational extended thermodynamics. Since the system of field equations is hyperbolic, the shock-structure solution is not always regular, and discontinuous parts (sub-shocks) can be formed. For given values of the mass ratio and the specific heat of the constituents, we identify the possible sub-shocks as the Mach number M0 of the shock wave and the equilibrium concentration c0 of the constituents change. In the plane (c0, M0), we identify the possible regions for the sub-shock formation. The analysis is obtained to verify when the velocity of the shock wave meets a characteristic velocity in the unperturbed or perturbed equilibrium states, which gives a necessary condition for the sub-shock formation. The condition becomes necessary and sufficient when the velocity of the shock becomes greater than the maximum characteristic velocity in the unperturbed state, namely, the regions with no sub-shocks, a sub-shock for only one constituent, or sub-shocks for both constituents are comprehensively classified. The most interesting case is that the lighter molecule has more degrees of freedom than that of the heavy one. In this situation, the topology of the various regions becomes different. We also solve the system of the field equations numerically using the parameters in various regions and confirm whether the sub-shocks emerge or not. Finally, the relationship between an acceleration wave in one constituent and the sub-shock in the other constituent is explicitly derived.

Thermodynamic stability and P\u2013V criticality of nonsingular-AdS black holes endowed with clouds of strings

We investigate the extended phase space thermodynamics of nonsingular-AdS black holes minimally coupled to clouds of strings in which we consider the cosmological constant (Lambda ) as the pressure (P) of the black holes and its conjugate variable thermodynamical volume (V) of the black holes. Owing to the background clouds of strings parameter (a), we analyse the Hawking temperature, entropy and specific heat on horizon radius for fixed-parameter k. We find that the strings clouds background does not alter small/large black hole (SBH/LBH) phase transition but occurs at a larger horizon radius, and two second-order phase transitions occur at a smaller horizon radius. Indeed, the G–T plots exhibit a swallowtail below the critical pressure, implying that the first-order phase transition is analogous to the liquid–gas phase transition at a lower temperature and lower critical pressure. To further examine the analogy between nonsingular-AdS black holes and a liquid–gas system, we derive the exact critical points and probe the effects of a cloud of strings on P-V criticality to find that the isotherms undergo liquid–gas like phase transition for {tilde{T}},<,{tilde{T}}_c at lower {tilde{T}}_c. We have also calculated the critical exponents identical with Van der Walls fluid, i.e., same as those obtained before for arbitrary other AdS black holes, which implies that the background clouds of strings do not change the critical exponents.

Central charge criticality of charged AdS black hole surrounded by different fluids

We analyze the extended phase space thermodynamics of Kiselev black hole introducing a central charge and allowing the gravitational constant to vary. We also discuss the relation between the chemical potential and the size of the black hole, besides the new description of phase transitions. We obtain as a conclusion that the universality of the central charge does not remain valid in general.

Relativistic Rational Extended Thermodynamics of Polyatomic Gases with a New Hierarchy of Moments.

A relativistic version of the rational extended thermodynamics of polyatomic gases based on a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal mode is proposed. The moment equations associated with the Boltzmann–Chernikov equation are derived, and the system for the first 15 equations is closed by the procedure of the maximum entropy principle and by using an appropriate BGK model for the collisional term. The entropy principle with a convex entropy density is proved in a neighborhood of equilibrium state, and, as a consequence, the system is symmetric hyperbolic and the Cauchy problem is well-posed. The ultra-relativistic and classical limits are also studied. The theories with 14 and 6 moments are deduced as principal subsystems. Particularly interesting is the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This simplified model can be very useful when bulk viscosity is dominant and might be important in cosmological problems. Using the Maxwellian iteration, we obtain the parabolic limit, and the heat conductivity, shear viscosity, and bulk viscosity are deduced and plotted.

Phase transitions in four-dimensional AdS black holes with a nonlinear electrodynamics source

In this work we consider black hole solutions to Einstein’s theory coupled to a nonlinear power-law electromagnetic field with a fixed exponent value. We study the extended phase space thermodynamics in canonical and grand canonical ensembles, where the varying cosmological constant plays the role of an effective thermodynamic pressure. We examine thermodynamical phase transitions in such black holes and find that both first- and second-order phase transitions can occur in the canonical ensemble while, for the grand canonical ensemble, Hawking–Page and second-order phase transitions are allowed.

Joule-Thomson expansion of AdS black holes in Einstein Power-Yang-mills gravity

In this paper we study Joule-Thomson expansion of non-linearly charged AdS black holes in Einstein-power-Yang-Mills (EPYM) gravity in D dimensions. Within the framework of extended phase space thermodynamics we identify the cosmological constant as thermodynamic pressure and the black hole mass with the enthalpy to derive the Joule- Thomson coefficient μ. Furthermore we have presented equations for inversion curves and the exact expression for the minimum inversion temperature. We also have calculated the ratio between the minimum of inversion temperature and the critical temperature T c and obtained the analytic expression for the ratio that depends explicitly on the non- linearity parameter q and dimension D. We consider the isenthalpic curves in the T − P plane for different values of the fixed black hole mass and obtain heating and cooling region. Finally we have dealt with two limiting masses which characterizes the process of Joule-Thomson expansion in the EPYM black holes.

Molecular Extended Thermodynamics of Rarefied Polyatomic Gases with a New Hierarchy of Moments

The aim of this paper is to construct the molecular extended thermodynamics for classical rarefied polyatomic gases with a new hierarchy, which is absent in the previous procedures of moment equations. The new hierarchy is deduced recently from the classical limit of the relativistic theory of moments associated with the Boltzmann–Chernikov equation. The field equations for 15 moments of the distribution function, in which the internal degrees of freedom of a molecule are taken into account, are closed with the maximum entropy principle. It is shown that the theory contains, as a principal subsystem, the previously polyatomic 14 fields theory, and in the monatomic limit, in which the dynamical pressure vanishes, the differential system converges, instead of to the Grad 13-moment system, to the Kremer 14-moment system.