## What is/are Irreversible Thermodynamics?

Irreversible Thermodynamics - This work proposes a fundamental thermodynamic description of structural relaxation in glasses by establishing a link between the Prony series solution to volume relaxation derived from the principles of irreversible thermodynamics and asymmetric Lévy stable distribution of relaxation rates.^{[1]}The time evolution equations are obtained through the usual methods of irreversible thermodynamics and from the MEPP.

^{[2]}The thermodynamic driving force for martensitic transformation and the heat equilibrium equation are deduced in the framework of irreversible thermodynamics.

^{[3]}We investigate the applicability of Onsager reciprocal relation, irreversible thermodynamics, and continuum fluid mechanics at the nanoscale.

^{[4]}We explore the physical and cosmological implications of the nonconservation of the energy–momentum tensor by using the formalism of irreversible thermodynamics of open systems in the presence of matter creation/annihilation.

^{[5]}A viscoelastic stress-strain relationship, which considered the proposed damage evolution model, was presented according to the principles of irreversible thermodynamics.

^{[6]}It has been regarded as an important application and direct experimental verification of the Onsager reciprocal relation (ORR) that is a cornerstone of irreversible thermodynamics.

^{[7]}The irreversible thermodynamics of a multicomponent fluid is reviewed.

^{[8]}In order to determine the variations in pressure and composition with depth and to be able to indicate if/where a gas-oil contact exists, we have developed a model based on the principles of irreversible thermodynamics, within the approach to thermodiffusion in porous media proposed by Montel et al.

^{[9]}In CCR theory, additional terms are invoked in the constitutive relations of the NSF equations originating from the arguments of irreversible thermodynamics as well as being consistent with the kinetic theory of gases.

^{[10]}Herein, we test an ostensibly complete model of LiPF6 in ethyl-methyl carbonate (EMC) based on the Onsager–Stefan–Maxwell theory from irreversible thermodynamics.

^{[11]}It is developed within the formalism of irreversible thermodynamics with internal state variables based on the Eulerian logarithmic strain and its corrotational objective rate.

^{[12]}The field theory of pH-sensitive hydrogel deformation is presented coupling electro-magnetic (EM) field (external or internal) and diffusion (Poisson–Nernst–Planck and Maxwell equations) of mobile ionic species through irreversible thermodynamics.

^{[13]}However, a desirable kinetic equation should satisfy the four balance equations of irreversible thermodynamics, i.

^{[14]}This model is a result of applying the irreversible thermodynamics to both the active layer and the support layer of the asymmetric FO membrane.

^{[15]}We propose a viscoelastic/plastic constitutive equation for viscoelastic media based on irreversible thermodynamics and the viscoelastic theory.

^{[16]}All model equations are derived rigorously from the balance laws of mass, momentum, energy, and entropy in the framework of irreversible thermodynamics, thus exposing all the coupling present in the field equations and constitutive relations.

^{[17]}In this paper, we take the view that both issues are closely tied to the principle of irreversible thermodynamics, and that, by considering the phase transition from the glassy to the rubbery state in the polymer, and the temperature-affected degradation of the interphase, two independent state variables can be chosen and implemented into a two-scale homogenization scheme.

^{[18]}In the background of a spatially flat, isotropic and homogeneous universe, we obtain set of equations describing the evolution of dynamical variables by combining the formalism of irreversible thermodynamics and gravitational field equations.

^{[19]}Using the Onsager formalism of irreversible thermodynamics we develop exact relations for the heats of transport of each component in terms of various measurable diffusion quantities under steady state conditions and with the vacancies being at local equilibrium.

^{[20]}The mathematical problem is governed by the coupling of the Kirchhoff-Love shell PDE with the Cahn-Hilliard PDE for phase transitions, which can be derived from surface mass balance in the framework of irreversible thermodynamics.

^{[21]}Based on the model of a photon-enhanced thermionic emission solar cell (PTESC), we investigate the optimum performance of the PTESC by using the theory of semiconductor physics and the irreversible thermodynamics.

^{[22]}The Maxwell–Stefan (M–S) formulation, that is grounded in the theory of irreversible thermodynamics, is widely used for describing mixture diffusion in microporous crystalline materials such as zeolites and metal–organic frameworks (MOFs).

^{[23]}In the present work, the governing equations based on theory of irreversible thermodynamics is deduced by introducing two internal variables to characterize the phase transformation and finite plastic deformation evolution for NiTi shape memory alloy.

^{[24]}An emphasis is given with regard to time-dependent problems in irreversible thermodynamics that arise from Biot-type variational formulations if conjugate variables are employed by means of a Legendre–Fenchel transformation.

^{[25]}Based on the entropy balance equation for a multidimensional linear Fokker–Planck equation we have developed connections between the several thermodynamically inspired quantities such as the total entropy production, the entropy production of the irreversible thermodynamics and the non-equilibrium temperature.

^{[26]}A passivity based control method is presented for the diffusion process by the use of the concept of available storage and irreversible thermodynamics.

^{[27]}Within the context of irreversible thermodynamics, we demonstrate that the expression of efficiency at maximum power satisfies a general form derived from nonlinear steady state heat engines.

^{[28]}In this work, based on irreversible thermodynamics, a three-dimensional (3D) single crystal constitutive model is constructed by considering the aforementioned four mechanisms simultaneously.

^{[29]}It is formulated within the framework of irreversible thermodynamics.

^{[30]}In this work, a new adsorption kinetic model with dispersion-type (D-type) has been proposed within the framework of irreversible thermodynamics for investigating the adsorption behavior of a single ionic species onto a solid adsorbent.

^{[31]}The expressions for dynamic (complex) dipolar and quadrupolar susceptibilities are obtained within the framework of the linear theory of irreversible thermodynamics in the Blume-Emery-Griffiths model.

^{[32]}Prigozhin's theory of irreversible thermodynamics.

^{[33]}In this study, the irreversible thermodynamics of the universe in the framework of f(R) gravity has been studied following standard Eckart theory of non-equilibrium thermodynamics.

^{[34]}The coupling between the electron flux and the heat flux in this type of semiconductor heterostructures, not only allows to obtain transport coefficients (electrical and thermal conductivities, and a Seebeck--like and Peltier--like coefficients), but also to study its operation as a thermionic generator or as a refrigerator within the context of irreversible thermodynamics.

^{[35]}Finally, based on irreversible thermodynamics and empirical support, a coupled elastoplastic damage constitutive model for low-rank bituminous coal was established by introducing the non-associated plastic flow laws and the damage evolution rules; the numerical simulation result of the model is in good agreement with the experimental measurements.

^{[36]}We consider a particular instance of the lift of controlled systems recently proposed in the theory of irreversible thermodynamics and show that it leads to a variational principle for an optimal control in the sense of Pontryagin.

^{[37]}As an example of irreversible thermodynamics, the thermomechanical model of DB will help understand heat engines manifested from microscopic to macroscopic systems.

^{[38]}In this study we analyze some aspects of the irreversible thermodynamics of this dissipative complex system.

^{[39]}The PDB has been a cornerstone for irreversible thermodynamics and chemical kinetics for a long time, and its wide application in geochemistry has mostly been implicit and without experimental testing of its applicability.

^{[40]}We will also explore how they are related, in a fundamental way, to the entropy production principles of irreversible thermodynamics.

^{[41]}Presented is a lithium-ion battery degradation model, based on irreversible thermodynamics, which was experimentally verified, using commonly measured operational parameters.

^{[42]}The plastic potential is defined in effective stress space and the damage evolution is based on the theory of irreversible thermodynamics.

^{[43]}Creating models in accordance with the principles of the irreversible thermodynamics is complete representation of the genesis of natural processes.

^{[44]}This description is compliant with the laws of irreversible thermodynamics.

^{[45]}Based on the experimental results, and considering the viscous effect in concrete, the viscoelastic stress–strain relationship was established in terms of the principles of irreversible thermodynamics and the generalized Onsager’s principle, providing the convergence of an iterative scheme in terms of the Lipschitz’s condition.

^{[46]}The constitutive relations for anisotropic damage of elastoplastic materials are developed based on irreversible thermodynamics.

^{[47]}We extend the Maxwell-Stefan (M-S) formulation of irreversible thermodynamics to multicomponent transport in mixed-matrix membranes (MMMs), using a simulation-based rigorous modeling approach (SMA) through finite-element method (FEM) solution of the three-dimensional (3-d) transport problem in full-scale MMMs.

^{[48]}For the phase transitions, the PDE is the Cahn–Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics.

^{[49]}

## Extended Irreversible Thermodynamics

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.^{[1]}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.

^{[2]}This is the second part of a two-part manuscript focused on the analysis of electronic circuits employing extended irreversible thermodynamics.

^{[3]}The main objective of this work is to demonstrate the agreement between the two-fluid linear Langevin formulation and that described by the extended irreversible thermodynamics (EIT).

^{[4]}The model is fully compatible with the model of heat transfer of extended irreversible thermodynamics, represents a generalization of the Guyer–Krumhansl proposal (Guyer & Krumhansl 1966 Phys.

^{[5]}In non-equilibrium thermodynamics, Extended irreversible thermodynamics (EIT), together with classical irreversible thermodynamics (CIT) and rational thermodynamics (RT) has been among the mainstream ofresearch.

^{[6]}The model is a mixt of the Effective Medium Approximation (EMA) and Extended Irreversible Thermodynamics (EXIT).

^{[7]}In a previous paper, in the framework of extended irreversible thermodynamics with internal variables, a model for nanostructures with thin porous channels filled by a fluid flow was developed.

^{[8]}We consider a system of balance laws arising in extended irreversible thermodynamics of rigid heat conductors, together with its differential conse- quences, namely the higher-order system obtained by taking into account the time and space derivatives of the original system.

^{[9]}The couplings between flow, structural parameters, and diffusion naturally arise in this model, derived from the extended irreversible thermodynamics (EIT) formalism.

^{[10]}Diffusion and chemical reactions in nanosystems are described by the methods of extended irreversible thermodynamics based on a postulate according to which additional variables are time derivatives of usual thermodynamic variables.

^{[11]}For this paper, extended irreversible thermodynamics will be utilised to produce consistent thermal equations of motion that directly include the exergy destruction terms.

^{[12]}In the framework of extended irreversible thermodynamics, the compatibility of our theoretical models with second law is proved.

^{[13]}

## Linear Irreversible Thermodynamics

A brief overview of the development from classical linear irreversible thermodynamics to the modern rational thermodynamics with Coleman–Noll and Müller–Liu procedures is presented, emphasizing the basic assumptions and formulation details.^{[1]}This readily explains the second law of thermodynamics and also yields the entropy law for non-linear irreversible thermodynamics: maximum entropy production within the restraints of the system.

^{[2]}The analysis is made according to the Navier–Stokes-Fourier (NSF) hydrodynamics and a recent phenomenological Linear Irreversible Thermodynamics (LIT) model, both models consider the HC to study the role played by the longitudinal temperature.

^{[3]}In particular, through linear irreversible thermodynamics (small amplitude of the cycle), we give an explicit formula for the optimal cycle period and phase delay (between the two modulated parameters, stiffness and temperature) achieving maximum power with Curzon-Ahlborn efficiency.

^{[4]}This means that linear irreversible thermodynamics can be applied independently of the simulations to obtain the torques determining the orientations of the system and that the predictions of this theory can be cross-checked by the simulations.

^{[5]}We use the framework of linear irreversible thermodynamics to discuss the relevant force, time and length scales involved in these processes.

^{[6]}In this paper, the methodology of the so-called Linear Irreversible Thermodynamics (LIT) is applied to analyze the properties of an energetic-converting biological process using simple model for an enzymatic reaction that couples one exothermic and one endothermic reaction in the same fashion as Diaz-Hernandez et al.

^{[7]}We start from linear irreversible thermodynamics, we derive a basic equation set for ion transfer in terms of gradients of ion electrochemical potentials and transmembrane volume flux.

^{[8]}The model is based on the principles of linear irreversible thermodynamics, where we have been aware of the flow anisotropy caused by the shock-wave propagation.

^{[9]}Finally, we show how results from linear irreversible thermodynamics can be used to predict the dissipative terminal states of the active assembly process in terms of parameters of the toggling protocol.

^{[10]}Hence, the Rayleigh-Onsager dissipation is only applicable to linear irreversible thermodynamics.

^{[11]}

## Classical Irreversible Thermodynamics

Our result not only solves the long-lasting question on the origin of entropy function in classical irreversible thermodynamics, but also reveals an interesting emergent phenomenon that both the time-reversible dynamics equipped with a Hamiltonian function and the time-irreversible dynamics equipped with an entropy function could arise automatically during the study of deterministic limits of a stochastic dynamics and its large deviations rate function.^{[1]}To model the behavior of each component, we have used: classical thermodynamics to define the equilibrium states, the local equilibrium hypothesis of Classical Irreversible Thermodynamics to model the changes of state, and the port-Hamiltonian approach to obtain the equations of the system dynamics.

^{[2]}In the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, the Levy-Fokker-Planck equation is connected to near-equilibrium thermal transport.

^{[3]}This work presents a thermodynamic analysis of the ballistic heat equation from two viewpoints: classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT).

^{[4]}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.

^{[5]}After an introduction to the balance laws for continuous media, we discuss the field of the classical irreversible thermodynamics (CIT) and the role of the local-equilibrium postulate within this branch of non equilibrium thermodynamics.

^{[6]}

## irreversible thermodynamics theory

Making use of irreversible thermodynamics theory, we also model the last two stages showing that the crowding induces a sub-diffusion process similar to that caused by particles trapped in cages, and that the saturation of the MSD is due to the existence of an entropic potential that limits the number of accessible states to the particles.^{[1]}The general stress-strain relationship and anisotropic stiffness matrix are derived based on the irreversible thermodynamics theory.

^{[2]}This article extends the single-fluid relativistic irreversible thermodynamics theory of {\it M{u}ller}, {\it Israel} and {\it Stewart} (hereafter the {MIS} theory) to a multi-fluid system with inherent species interactions.

^{[3]}