Extended Irreversible Thermodynamics Research Articles

Numerical Analysis of Non-Fourier Model-Based Bio-Heat Transfer in the Laser-irradiated Axisymmetric Living Tissue.

The current work is related to the numerical investigation of non-Fourier heat transfer inside the short-pulsed laser-irradiated axisymmetric soft tissue phantom. It utilizes the modified discrete ordinate method to solve the transient radiative transfer equation (TRTE) for determining the intensity field. The laser energy absorbed by the soft tissue phantom behaves like a source in the Fourier/non-Fourier heat conduction model based-bio-heat transfer equation (BHTE), which is solved by employing the finite volume method (FVM) to determine the temperature distribution. Despite the prevalent use of non-Fourier BHTE for this purpose, a second law analysis is considered crucial to detect any potential anomalies. Equilibrium entropy production rates (EPR) are initially computed based on classical irreversible thermodynamics (CIT), which may yield negative values, possibly contravening the second law. Consequently, the EPR based on CIT is adjusted using the extended irreversible thermodynamics (EIT) hypothesis to ensure positivity. After that, the current research findings are compared with the results from the literature, and found good agreement between them. Then, the independent study is performed to select the optimum grid size, control angle size, and time step. A comparative analysis of results between the traditional Fourier and non-Fourier models has been performed. The impact of different parameters on the temperature fields and EPRs, is discussed. The effect of the optical properties of the inhomogeneity on the temperature distribution has been investigated. This study may help to enhance the effectiveness of the laser-based photo-thermal therapy.

Revisit nonequilibrium thermodynamics based on thermomass theory and its applications in nanosystems

Abstract The development of non-Fourier heat conduction models is encouraged by the invalidity of Fourier’s law to explain heat conduction in ultrafast or ultrasmall systems. The production of negative entropy will result from the combination of traditional nonequlibrium thermodynamics and non-Fourier heat conduction models. To resolve this paradox, extended irreversible thermodynamics (EIT) introduces a new state variable. However, real dynamics variables like force and momentum are still missing from nonequilibrium thermodynamics and EIT’s generalized force and generalized flux. Heat has both mass and energy, according to thermomass theory and Einstein’s mass-energy relation. The generalized heat conduction model containing non-Fourier effects was established by thermomass gas model. The thermomass theory reshapes the concept of the generalized force and flux, temperature, and entropy production in nonequilibrium thermodynamics and revisits the assumption for the linear regression of the fluctuations in Onsager reciprocal relation. The generalized heat conduction model based on thermomass theory has been used to study thermal conductivity, thermoelectric effect, and thermal rectification effect in nanosystems.

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.

Extending Israel and Stewart hydrodynamics to relativistic superfluids via Carter’s multifluid approach

We construct a relativistic model for bulk viscosity and heat conduction in a superfluid. Building on the principles of Unified Extended Irreversible Thermodynamics, the model is derived from Carter's multifluid approach for a theory with 3 four-currents: particles, entropy, and quasi-particles. Dissipation arises directly from the fact that the quasi-particle four-current is an independent degree of freedom that does not necessarily comove with the entropy. For small deviations from local thermodynamic equilibrium, the model provides an extension of the Israel-Stewart theory to superfluid systems. It can, therefore, be made hyperbolic, causal and stable if the microscopic input is accurate. The non-dissipative limit of the model is the relativistic two-fluid model of Carter, Khalatnikov and Gusakov. The Newtonian limit of the model is an Extended-Irreversible-Thermodynamic extension of Landau's two-fluid model. The model predicts the existence of four bulk viscosity coefficients and accounts for their microscopic origin, providing their exact formulas in terms of the quasi-particle creation rate. Furthermore, when fast oscillations of small amplitude around the equilibrium are considered, the relaxation-time term in the telegraph-type equations for the bulk viscosities accounts directly for their expected dependence on the frequency.

When theories and experiments meet: Rarefied gases as a benchmark of non-equilibrium thermodynamic models

The role of thermodynamics in deriving constitutive equations is unique, and various approaches have been developed in the last decades. In the present paper, the frameworks of Extended Irreversible Thermodynamics (EIT) and Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) are discussed and compared to each other on the basis of a particular problem of rarefied gases. In this comparison, both theoretical and experimental aspects are taken into account. Eventually, an experiment by Meyer and Sessler covering a wide range of pressures and frequencies is investigated. Here, concentrating on the scaling properties and the density dependence of parameters, the change of speed of sound in terms of frequency and pressure is recovered using NET-IV, and this fitting is compared to the results of Lebon and Cloot using EIT. NET-IV and EIT are seen to be more flexible than Rational Extended Thermodynamics (RET) concerning the functional form of the transport coefficients, which gives them some advantages over RET in some ranges of values of gas density.

Unified Extended Irreversible Thermodynamics and the Stability of Relativistic Theories for Dissipation

In a relativistic context, the main purpose of Extended Irreversible Thermodynamics (EIT) is to generalize the principles of non-equilibrium thermodynamics to the domain of fluid dynamics. In particular, the theory aims at modeling any diffusion-type process (like heat as diffusion of energy, viscosity as diffusion of momentum, charge-conductivity as diffusion of particles) directly from thermodynamic laws. Although in Newtonian physics this task can be achieved with a first-order approach to dissipation (i.e. Navier–Stokes–Fourier like equations), in a relativistic framework the relativity of simultaneity poses serious challenges to the first-order methodology, originating instabilities which are, instead, naturally eliminated within EIT. The first part of this work is dedicated to reviewing the most recent progress made in understanding the mathematical origin of this instability problem. In the second part, we present the formalism that arises by promoting non-equilibrium thermodynamics to a classical effective field theory. We call this approach Unified Extended Irreversible Thermodynamics (UEIT), because it contains, as particular cases, EIT itself, in particular the Israel-Stewart theory and the divergence-type theories, plus Carter’s approach and most branches of non-equilibrium thermodynamics, such as relativistic chemistry and radiation hydrodynamics. We use this formalism to explain why all these theories are stable by construction (provided that the microscopic input is correct), showing that their (Lyapunov) stability is a direct consequence of the second law of thermodynamics.

Review of Extended Irreversible Thermodynamics

In non-equilibrium thermodynamics, Extended irreversible thermodynamics (EIT), together with classical irreversible thermodynamics (CIT) and rational thermodynamics (RT) has been among the mainstream ofresearch. Critical analysis of EIT has been discussed in this paper.

On extended thermodynamics: From classical to the relativistic regime

The recent monumental detection of gravitational waves by LIGO, the subsequent detection by the LIGO/VIRGO observatories of a binary neutron star merger seen in the gravitational wave signal [Formula: see text], the first photo of the event horizon of the supermassive black hole at the center of Andromeda galaxy released by the EHT telescope and the ongoing experiments on Relativistic Heavy Ion Collisions at the BNL and at the CERN, demonstrate that we are witnessing the second golden era of observational relativistic gravity. These new observational breakthroughs, although in the long run would influence our views regarding this Kosmos, in the short run, they suggest that relativistic dissipative fluids (or magnetofluids) and relativistic continuous media play an important role in astrophysical-and also subnuclear-scales. This realization brings into the frontiers of current research theories of irreversible thermodynamics of relativistic continuous media. Motivated by these considerations, we summarize the progress that has been made in the last few decades in the field of nonequilibrium thermodynamics of relativistic continuous media. For coherence and completeness purposes, we begin with a brief description of the balance laws for classical (Newtonian) continuous media and introduce the classical irreversible thermodynamics (CIT) and point out the role of the local-equilibrium postulate within this theory. Tangentially, we touch the program of rational thermodynamics (RT), the Clausius–Duhem inequality, the theory of constitutive relations and the emergence of the entropy principle in the description of continuous media. We discuss at some length, theories of non equilibrium thermodynamics that sprang out of a fundamental paper written by Müller in 1967, with emphasis on the principles of extended irreversible thermodynamics (EIT) and the rational extended irreversible thermodynamics (REIT). Subsequently, after a brief introduction to the equilibrium thermodynamics of relativistic fluids, we discuss the Israel–Stewart transient (or causal) thermodynamics and its main features. Moreover, we introduce the Liu–Müller–Ruggeri theory describing relativistic fluids. We analyze the structure and compare this theory to the class of dissipative relativistic fluid theories of divergent type developed in the late 1990 by Pennisi, Geroch and Lindblom. As far as theories of nonequilibrium thermodynamics of classical media are concerned, it is fair to state that substantial progress has been made and many predictions of the extended theories have been placed under experimental scrutiny. However, at the relativistic level, the situation is different. Although the efforts aiming to the development of a sensible theory (or theories) of nonequilibrium thermodynamics of relativistic fluids (or continuous media) spans less than a half-century, and even though enormous steps in the right direction have been taken, nevertheless as we shall see in this review, still a successful theory of relativistic dissipation is lacking.

The statistical foundation of entropy in extended irreversible thermodynamics

In the theory of extended irreversible thermodynamics (EIT), the flux-dependent entropy function plays a key role; it is a fundamental distinction between EIT and the usual flux-independent entropy function adopted by classical irreversible thermodynamics (CIT). However, its existence, as a prerequisite for EIT, and its statistical origin have never been justified. In this work, by studying the macroscopic limit of an ϵ-dependent Langevin dynamics, which admits a large deviations (LD) principle, we show that the stationary LD rate functions of probability density p ϵ (x, t) and joint probability density actually turn out to be the desired flux-independent entropy function in CIT and flux-dependent entropy function in EIT respectively. The difference of the two entropy functions is determined by the time resolution for Brownian motions times a Lagrangian, the latter arises from the LD Hamilton–Jacobi equation and can be used for constructing conserved Lagrangian/Hamiltonian dynamics.

Propagation of coupled porosity and fluid-concentration waves in isotropic porous media

This article presents an application of a theory, previously formulated in the framework of rational extended irreversible thermodynamics, to describe the thermal, mechanical and transport properties of a porous medium filled by a fluid. Starting from the anisotropic rate equations for the porosity field, its flux, and for the heat and fluid-concentration fluxes, the isotropic case is studied when the body has symmetry properties invariant for all rotations and inversions of the frame axes. Furthermore, the phenomenological tensors have special symmetry properties coming from the used theoretic model. Then, the propagation in one direction of coupled porosity and fluid-concentration waves is investigated. The dispersion relation is carried out and the wave propagation velocities as functions of the wavenumber are calculated and represented in a diagram for a given numerical set of the several coefficients characterizing the considered porous media. The results obtained in this article can be applied in several sciences such as seismology, medical sciences, geology and nanotechnology, where there is propagation of high-frequency waves.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/73/abstr.html

Two-dimensional fluids and their holographic duals

We describe the dynamics of two-dimensional relativistic and Carrollian fluids. These are mapped holographically to three-dimensional locally anti-de Sitter and locally Minkowski spacetimes, respectively. To this end, we use Eddington–Finkelstein coordinates, and grant general curved two-dimensional geometries as hosts for hydrodynamics. This requires to handle the conformal anomaly, and the expressions obtained for the reconstructed bulk metrics incorporate non-conformal-fluid data. We also analyze the freedom of choosing arbitrarily the hydrodynamic frame for the description of relativistic fluids, and propose an invariant entropy current compatible with classical and extended irreversible thermodynamics. This local freedom breaks down in the dual gravitational picture, and fluid/gravity correspondence turns out to be sensitive to dissipation processes: the fluid heat current is a necessary ingredient for reconstructing all Bañados asymptotically anti-de Sitter solutions. The same feature emerges for Carrollian fluids, which enjoy a residual frame invariance, and their Barnich–Troessaert locally Minkowski duals. These statements are proven by computing the algebra of surface conserved charges in the fluid-reconstructed bulk three-dimensional spacetimes.

Effective thermal conductivity of nanostructures: A review

We present a synthesis of recent results on thermal heat conductivity in nano-composites and nano-structures. The model is a mixt of the Effective Medium Approximation (EMA) and Extended Irreversible Thermodynamics (EXIT). The latter is particularly well adapted to the description of small scaled systems and will be used to derive the expression of the thermal conductivity of nanoparticles. The model is applied to spherical, cylindrical (nanowires) and porous nanoparticles, respectively, being embedded in host media, like polymeric matrices and semi-conductors. Good agreement is observed with other models, experimental data and Monte-Carlo simulations.

Change of entropy for the one-dimensional ballistic heat equation: Sinusoidal initial perturbation.

This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). A formula for calculating the entropy within the framework of EIT for the ballistic heat equation is derived. The entropy is calculated for a sinusoidal initial temperature perturbation by using both approaches. The results obtained from CIT show that the entropy is a non-monotonic function and that the entropy production can be negative. The results obtained for EIT show that the entropy is a monotonic function and that the entropy production is nonnegative. A comparison between the entropy behaviors predicted for the ballistic, for the ordinary Fourier-based, and for the hyperbolic heat equation is made. A crucial difference of the asymptotic behavior of the entropy for the ballistic heat equation is shown. It is argued that mathematical time reversibility of the partial differential ballistic heat equation is not consistent with its physical irreversibility. The processes described by the ballistic heat equation are irreversible because of the entropy increase.

Inhomogeneous Flow of Wormlike Micelles: Predictions of the Generalized BMP Model with Normal Stresses

In this work, we examine the shear-banding flow in polymer-like micellar solutions with the generalized Bautista-Manero-Puig (BMP) model. The couplings between flow, structural parameters, and diffusion naturally arise in this model, derived from the extended irreversible thermodynamics (EIT) formalism. Full tensorial expressions derived from the constitutive equations of the model, in addition to the conservation equations, apply for the case of simple shear flow, in which gradients of the parameter representing the structure of the system and concentration vary in the velocity gradient direction. The model predicts shear-banding, concentration gradients, and jumps in the normal stresses across the interface in shear-banding flows.

On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations.

Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using the relaxation time approximation and power series expansion, macroscopic approximations are derived for these entropic concepts. For the standard BTE, our results can recover the entropic frameworks of classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT) as if there exists a well-defined effective thermal conductivity. For the fractional BTEs corresponding to the generalized Cattaneo equation (GCE) class, the entropy flux and entropy production rate will deviate from the forms in CIT and EIT. In these cases, the entropy flux and entropy production rate will contain fractional-order operators, which reflect memory effects.

Extended Irreversible Thermodynamics Versus Second Law Analysis of High-Order Dual-Phase-Lag Heat Transfer

This study introduces an analysis of high-order dual-phase-lag (DPL) heat transfer equation and its thermodynamic consistency. The frameworks of extended irreversible thermodynamics (EIT) and traditional second law are employed to investigate the compatibility of DPL model by evaluating the entropy production rates (EPR). Applying an analytical approach showed that both the first- and second-order approximations of the DPL model are compatible with the traditional second law of thermodynamics under certain circumstances. If the heat flux is the cause of temperature gradient in the medium (over diffused or flux precedence (FP) heat flow), the DPL model is compatible with the traditional second law without any constraints. Otherwise, when the temperature gradient is the cause of heat flux (gradient precedence (GP) heat flow), the conditions of stable solution of the DPL heat transfer equation should be considered to obtain compatible solution with the local equilibrium thermodynamics. Finally, an insight inspection has been presented to declare precisely the influence of high-order terms on the EPRs.

Evaluation of temperature history of a spherical nanosystem irradiated with various short-pulse laser sources.

Spatiotemporal thermal response and characteristics of net entropy production rate of a gold nanosphere (radius: 50-200 nm), subjected to a short-pulse, femtosecond laser is reported. In order to correctly illustrate the temperature history of laser-metal interaction(s) at picoseconds transient with a comprehensive single temperature definition in macroscale and to further understand how the thermophysical response of the single-phase lag (SPL) and dual-phase lag (DPL) frameworks (with various lag-ratios') differs, governing energy equations derived from these benchmark non-Fourier frameworks are numerically solved and thermodynamic assessment under both the classical irreversible thermodynamics (CIT) as well as extended irreversible thermodynamics (EIT) frameworks is subsequently carried out. Under the frameworks of SPL and DPL with small lag ratio, thermophysical anomalies such as temperature overshooting characterized by adverse temperature gradient is observed to violate the local thermodynamic equilibrium (LTE) hypothesis. The EIT framework, however, justifies the compatibility of overshooting of temperature with the second law of thermodynamics under a nonequilibrium paradigm. The DPL framework with higher lag ratio was however observed to remain free from temperature overshooting and finds suitable consistency with LTE hypothesis. In order to solve the dimensional non-Fourier governing energy equation with volumetric laser-irradiation source term(s), the lattice Boltzmann method (LBM) is extended and a three-time level, fully implicit, second order accurate finite difference method (FDM) is illustrated. For all situations under observation, the LBM scheme is featured to be computationally superior to remaining FDM schemes. With detailed prediction of maximum temperature rise and the corresponding peaking time by all the numerical schemes, effects of the change of radius of the gold nanosphere, the magnitude of fluence of laser, and laser irradiation with multiple pulses on thermal energy transport and lagging behavior (if any) are further elucidated at different radial locations of the gold nanosphere. Last, efforts are further made to address the thermophysical characteristics when effective thermal conductivity (with temporal and size effects) is considered instead of the usual bulk thermal conductivity.

Entropic Constitutive Relation and Modeling for Fourier and Hyperbolic Heat Conductions

Most existing phenomenological heat conduction models are expressed by temperature and heat flux distributions, whose definitions might be debatable in heat conductions with strong non-equilibrium. The constitutive relations of Fourier and hyperbolic heat conductions are here rewritten by the entropy and entropy flux distributions in the frameworks of classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT). The entropic constitutive relations are then generalized by Boltzmann–Gibbs–Shannon (BGS) statistical mechanics, which can avoid the debatable definitions of thermodynamic quantities relying on local equilibrium. It shows a possibility of modeling heat conduction through entropic constitutive relations. The applicability of the generalizations by BGS statistical mechanics is also discussed based on the relaxation time approximation, and it is found that the generalizations require a sufficiently small entropy production rate.

Memory behaviors of entropy production rates in heat conduction

Based on the relaxation time approximation and first-order expansion, memory behaviors in heat conduction are found between the macroscopic and Boltzmann–Gibbs–Shannon (BGS) entropy production rates with exponentially decaying memory kernels. In the frameworks of classical irreversible thermodynamics (CIT) and BGS statistical mechanics, the memory dependency on the integrated history is unidirectional, while for the extended irreversible thermodynamics (EIT) and BGS entropy production rates, the memory dependences are bidirectional and coexist with the linear terms. When macroscopic and microscopic relaxation times satisfy a specific relationship, the entropic memory dependences will be eliminated. There also exist initial effects in entropic memory behaviors, which decay exponentially. The second-order term are also discussed, which can be understood as the global non-equilibrium degree. The effects of the second-order term are consisted of three parts: memory dependency, initial value and linear term. The corresponding memory kernels are still exponential and the initial effects of the global non-equilibrium degree also decay exponentially.

Mathematical and information-geometrical entropy for phenomenological Fourier and non-Fourier heat conduction.

The second law of thermodynamics governs the direction of heat transport, which provides the foundational definition of thermodynamic Clausius entropy. The definitions of entropy are further generalized for the phenomenological heat transport models in the frameworks of classical irreversible thermodynamics and extended irreversible thermodynamics (EIT). In this work, entropic functions from mathematics are combined with phenomenological heat conduction models and connected to several information-geometrical conceptions. The long-time behaviors of these mathematical entropies exhibit a wide diversity and physical pictures in phenomenological heat conductions, including the tendency to thermal equilibrium, and exponential decay of nonequilibrium and asymptotics, which build a bridge between the macroscopic and microscopic modelings. In contrast with the EIT entropies, the mathematical entropies expressed in terms of the internal energy function can avoid singularity paired with nonpositive local absolute temperature caused by non-Fourier heat conduction models.