Equations Of Nonequilibrium Thermodynamics Research Articles

Diffusion kinetics in a multicomponent thermodynamic system at small deviations from the equilibrium state

The theory of diffusion processes in solids has achieved significant results in recent decades, but the development of methods for calculating diffusion in a multicomponent thermodynamic system is still an urgent task. Problems of diffusion in solid and liquid solutions with small deviations from the equilibrium state, or fluctuations, are of significant interest. The work develops a general methodology for calculating diffusion flows in a multicomponent thermodynamic system for small deviations from the equilibrium state. A connection has been established between the mechanical approach to the analysis of generalized systems and the phenomenological equations of nonequilibrium thermodynamics. Examples are given of the use of the developed methodology for the analysis of carbide transformations in chromium steel.

Galilean Bulk-Surface Electrothermodynamics and Applications to Electrochemistry.

In this work, the balance equations of non-equilibrium thermodynamics are coupled to Galilean limit systems of the Maxwell equations, i.e., either to (i) the quasi-electrostatic limit or (ii) the quasi-magnetostatic limit. We explicitly consider a volume Ω, which is divided into Ω+ and Ω- by a possibly moving singular surface S, where a charged reacting mixture of a viscous medium can be present on each geometrical entity (Ω+,S,Ω-). By the restriction to the Galilean limits of the Maxwell equations, we achieve that only subsystems of equations for matter and electromagnetic fields are coupled that share identical transformation properties with respect to observer transformations. Moreover, the application of an entropy principle becomes more straightforward and finally helps estimate the limitations of the more general approach based the full set of Maxwell equations. Constitutive relations are provided based on an entropy principle, and particular care is taken in the analysis of the stress tensor and the momentum balance in the general case of non-constant scalar susceptibility. Finally, we summarise the application of the derived model framework to an electrochemical system with surface reactions.

Thermodynamic Analysis of Relaxation Model for Non-Equilibrium Phase Behavior of Hydrocarbon Mixtures

Mathematical models of phase behavior are widely used to describe multiphase oil and gas-condensate systems during hydrocarbon recovery from natural petroleum reservoirs. Previously a non-equilibrium phase behavior model was proposed as an extension over generally adopted equilibrium models. It is based on relaxation of component chemical potentials difference between phases and provides accurate calculations in some typical situations when non-instantaneous changing of phase fractions and compositions in response to variations of pressure or total composition is to be considered. In this paper we present a thermodynamic analysis of the relaxation model. General equations of non-equilibrium thermodynamics for multiphase flows in porous media are considered, and reduced entropy balance equation for the relaxation process is obtained. Isotropic relaxation process is simulated for a real multicomponent hydrocarbon system with different values of characteristic relaxation time using the non-equilibrium model implemented in the PVT Designer module of the RFD tNavigator simulation software. The results are processed with a special algorithm implemented in Matlab to calculate graphs of the total entropy time derivative and its constituents in the entropy balance equation. It is shown that the constituents have different signs, and the greatest influence on the entropy is associated with the interphase flow of the major component of the mixture and the change of the total system volume in the isotropic process. The characteristic relaxation time affects the rate at which the entropy is approaching its maximum value.

Local versus nonlocal constitutive theories of nonequilibrium thermodynamics: the Guyer\u2013Krumhansl equation as an example

On the example of the celebrated Grad’s 13-moment system of kinetic theory of rarefied gases and phonon hydrodynamics, it is proved that the constitutive equations of nonequilibrium thermodynamics must be nonlocal. A thermodynamic model of Guyer–Krumhansl heat-transport equation is derived within the frame of weakly nonlocal Rational Thermodynamics. The constitutive equation for the entropy flux is analyzed as well. Some nonlinear generalizations of Maxwell–Cattaneo equation are studied in view of the experiments on thermal wave propagation.

Landscape-Flux Framework for Nonequilibrium Dynamics and Thermodynamics of Open Hamiltonian Systems Coupled to Multiple Heat Baths.

We establish a nonequilibrium dynamic and thermodynamic formalism in the landscape-flux framework for open Hamiltonian systems in contact with multiple heat baths governed by stochastic dynamics. To systematically characterize nonequilibrium steady states, the nonequilibrium trinity construct is developed, which consists of detailed balance breaking, nonequilibrium potential landscape, and irreversible probability flux. We demonstrate that the temperature difference of the heat baths is the physical origin of detailed balance breaking, which generates the nonequilibrium potential landscape characterizing the nonequilibrium statistics and creates the irreversible probability flux signifying time irreversibility, with the latter two aspects closely connected. It is shown that the stochastic dynamics of the system can be formulated in the landscape-flux form, where the reversible force drives the conservative Hamiltonian dynamics, the irreversible force consisting of a landscape gradient force and an irreversible flux force drives the dissipative dynamics, and the stochastic force adds random fluctuations to the dynamics. The possible connection of the nonequilibrium trinity construct to nonequilibrium phase transitions is also suggested. A set of nonequilibrium thermodynamic equations, applicable to both nonequilibrium steady states and transient relaxation processes, is constructed. We find that an additional thermodynamic quantity, named the mixing entropy production rate, enters the nonequilibrium thermodynamic equations. It arises from the interplay between detailed balance breaking and transient relaxation, and it also relies on the conservative dynamics. At the nonequilibrium steady state, the heat flow, entropy flow, and entropy production are demonstrated to be thermodynamic manifestations of the nonequilibrium trinity construct. The general nonequilibrium formalism is applied to a class of solvable systems consisting of coupled harmonic oscillators. A more specific example of two harmonic oscillators coupled to two heat baths is worked out in detail. The example may facilitate connection with experiments.

Nonequilibrium thermodynamics of turbulence and stochastic fluid systems

Fluid systems are found in the Universe at various scales. Turbulence as a complex form of fluid motion far from thermodynamic equilibrium remains one of the most challenging problems in physics. In this work, we study the nonequilibrium thermodynamics of stochastic fluid systems in general and turbulence in particular. Our approach is based on a reinterpretation of the stochastic fluid system as an interacting many-body system in contact with multiple heat baths. A set of nonequilibrium thermodynamic equations for general stochastic fluid systems, applicable to turbulence in the far-from-equilibrium regime, is constructed using the potential landscape and flux field theory. In addition to the energy and entropy balance equations that represent the first and second laws of thermodynamics, a new thermodynamic equation is found to be crucial for relating the first law with the second law and connecting violation of detailed balance to entropy flow and entropy production at the steady state. It is demonstrated that steady-state entropy production and energy flow are manifestations of the nonequilibrium irreversible nature of fluid systems characterized by the nonequilibrium trinity construct that originates from temperature nonuniformity. We propose an intuitive thermodynamic picture of the turbulence energy cascade process as heat conduction in the scale domain, where energy flow across scales is conducted by nonlinear convection and driven by the temperature difference between the large and small scales. Nonequilibrium irreversibility of turbulence energy cascade is quantified by the steady-state entropy production rate. This work is rooted in both fluid dynamics and nonequilibrium statistical physics, fostering a deeper level of communication between these fields. Further extensions of this work have the potential to grow into a more complete nonequilibrium statistical theory, with a much wider range of applications encompassing general physical, chemical and biological nonequilibrium systems.

Non-equilibrium thermodynamics as gauge fixing

By considering a thermodynamic force as gauge field, we extend constitutive equations of Onsager's non-equilibrium thermodynamics to non-linear equations. In Onsager's non-equilibrium thermodynamics, the thermodynamic force corresponds to a pure gauge, for which the constitutive equations are obtained by gauge fixing. If we extend the thermodynamic force from pure gauge to physical one, we obtain the non-linear constitutive equations of non-equilibrium thermodynamics.

Nonequilibrium Effects in Atmospheric Perturbations Caused by Solar Radiation Flux

The effects of atmospheric nonequilibrium in the generation of wave perturbations due to the solar radiation flux are studied. Equations of nonequilibrium thermodynamics are used to perform an assessment of the channels of solar energy transformation into the atmosphere for different altitudes. As a result of calculations of the dispersion relation for a nonequilibrium atmosphere, we consider how the flux of solar radiation changes the spectrum of natural atmospheric oscillations at different altitudes and for different solar activities. A qualitative relation between the results of wave spectra calculations and the data of ionosphere dynamics observations for different intensities of the solar radiation flux has been established.

Issues of analysis of the characteristic times of vacation relaxation in small solids in the case of adsorption and absorption

The process of leveling of vacancy density in solids determines the times of reaching the equilibrium state and conditions for using the equations of nonequilibrium thermodynamics, particularly using diffusion equations of atom transport. Issues of characteristic times of vacancy relaxation are analyzed in small single- and two-component solids (granules of solid-phase systems or aerosol particles). The differences in the values of relaxation times are discussed in the case of thermal transport of labels in the processes of self-diffusion and mass transport. It is shown that taking into account the thermodynamic cofactor in the diffusion mass-transport coefficient decreases the relaxation times of processes of the mixing of atoms in two-component mixtures by three or more orders of magnitude, as compared to taking into account the contribution of only the self-diffusion coefficient, while the transport coefficient of vacancies can exceed the self-diffusion coefficients of atoms by three to seven orders of magnitude at the initial stage of the vacancy transport. Size dependences of these relaxation times obtained in the diffusion approximation are considered.

Nonequilibrium thermodynamics of transport processes on membrane surfaces

Nonequilibrium processes have been considered on interfaces between media, with one medium being a permeable membrane. The nonequilibrium processes have been described in terms of nonequilibrium thermodynamics combined with the Fokker−Planck kinetic equation. Phenomenological equations derived within the framework of the combined approach have been discussed. It has been shown that the formal use of linear equations of nonequilibrium thermodynamics may be, in some cases, reduced to nonlinear equations for heat and mass transfer through an interface. It has been noted that the asymmetry of transport characteristics of asymmetric membranes may be explained in terms of the derived relations.

FEATURES INTERFACIAL INTERACTION IN Si-SiO2 STRUCTURES

The Si/SiO2 boundary interphase is a source of internal mechanical stresses, localized charges and points defects. It cause the unstability of electrical parameters of Si-SiO2 instrument structure. However, in sprite of the great importance of the problem, today we do not have appropriate model of interphase interaction and numerical data of it energy parameters. The aim is to develop the model of interphase interaction for numerical calculation of energy parameter within Si/SiO2 boundary: interfacial energy . ѓБm, interfacial tension . ѓРm. To conduct the investigation of such parameters under influence of external factors to predict the changes in these parameters for formation of thermo stable structures. Base on equations of nonequilibrium thermodynamics and surface physics of Si-SiO2 solid-state structures the energy parameters of interphase interaction on Si-SiO2 boundary: interfacial energy . ѓБm, interfacial tension . ѓРm were calculated. For the firs time those parameters were evaluated: ѓБm = 4,56 J/m2 „„„p ѓРm = 5,0 N/m. It was shown that energy parameters is significantly higher in investigated structure (Si-SiO2) in compare with ЃgSi-„}„u„„„p„|„y (Cu,Al,Au,Cr)Ѓh (ѓБm: 0,27; 0,8; 0,12; 0,31 J/m2, ѓРm: 0,61; 0,10; 0,37; 0,75 N/m) „„„p ЃgCu-SiO2Ѓh (ѓБm: 0,1 J/m2, ѓРm: 0,11 N/m).The influence of temperature and X-rays on such parameter was investigated. In the range of X-rays absorbed dose (8.101-102 Gray) the transition in equilibrium was found. It characterized with minimum interphase energy. The correlation with mechanical tension in structure and interphase tension in Si-SiO2 were established. The analyses of investigated energy parameters can be used for prediction of interphase interaction in choice of thermodynamically stable contacting boundary structures with pre-defined parameters of interphase interaction and mechanical tension.

Interdiffusion in Ionic Compounds

In this chapter, we review the Nernst-Planck equation describing the cation interdiffusion coefficient, the two tracer cation diffusion coefficients and the thermodynamic factor in ionic compounds and how it is related to the analogous Darken-Manning relationship in binary alloy systems. We make use of the Onsager flux equations of non-equilibrium thermodynamics to rigorously address the problem. The recently found correction factor to the Nernst-Planck equation is analyzed visually by means of computer simulation

Non-equilibrium thermodynamics of gravitational screens

We study the Einstein gravity equations projected on a timelike surface, which represents the time evolution of what we call a gravitational screen. We show that such a screen behaves like a viscous bubble with a surface tension and an internal energy, and that the Einstein equations take the same forms as non-equilibrium thermodynamic equations for a viscous bubble. We provide a consistent dictionary between gravitational and thermodynamic variables. In the non-viscous cases there are three thermodynamic equations that characterize a bubble dynamics: these are the first law, the Marangoni flow equation and the Young–Laplace equation. In all three equations the surface tension plays a central role: in the first law it appears as a work term per unit area, in the Marangoni flow its gradient drives a force, and in the Young–Laplace equation it contributes to a pressure proportional to the surface curvature. The gravity equations appear as a natural generalization of these bubble equations when the bubble itself is viscous and dynamical. In particular, this approach shows that the mechanism of entropy production for the viscous bubble is mapped onto the production of gravitational waves. We also review the relationship between surface tension and temperature, and discuss black-hole thermodynamics.

Potential and flux field landscape theory. II. Non-equilibrium thermodynamics of spatially inhomogeneous stochastic dynamical systems.

We have established a general non-equilibrium thermodynamic formalism consistently applicable to both spatially homogeneous and, more importantly, spatially inhomogeneous systems, governed by the Langevin and Fokker-Planck stochastic dynamics with multiple state transition mechanisms, using the potential-flux landscape framework as a bridge connecting stochastic dynamics with non-equilibrium thermodynamics. A set of non-equilibrium thermodynamic equations, quantifying the relations of the non-equilibrium entropy, entropy flow, entropy production, and other thermodynamic quantities, together with their specific expressions, is constructed from a set of dynamical decomposition equations associated with the potential-flux landscape framework. The flux velocity plays a pivotal role on both the dynamic and thermodynamic levels. On the dynamic level, it represents a dynamic force breaking detailed balance, entailing the dynamical decomposition equations. On the thermodynamic level, it represents a thermodynamic force generating entropy production, manifested in the non-equilibrium thermodynamic equations. The Ornstein-Uhlenbeck process and more specific examples, the spatial stochastic neuronal model, in particular, are studied to test and illustrate the general theory. This theoretical framework is particularly suitable to study the non-equilibrium (thermo)dynamics of spatially inhomogeneous systems abundant in nature. This paper is the second of a series.

Investigation into the diffusion of boron, phosphorus, and arsenic in silicon during annealing in a nonisothermal reactor

The influence of the temperature gradient on the diffusion of boron, phosphorus, and arsenic during annealing of silicon in a nonisothermal lamp reactor in the second and minute ranges is investigated experimentally and theoretically. Parameters of the thermodiffusion process are determined: for the boron diffusion in the second range, the effective diffusivity Deff ∼ 10−12 cm2/s and the effective measured heat of transport Qmeff* ∼ 103–104 eV. The results are interpreted based on the equations of nonequilibrium thermodynamics.

Solar radiation and atmospheric dynamics

This paper studies the problems of mathematical description of physical processes in open non-equilibrium atmosphere. It is proposed that the mathematical description should be based on the idea that properties of an open system are determined by the properties of its elements and external constraints. It is also explained why atmosphere should be represented by an open non-equilibrium system of gas and radiation. Difference between physical processes in the above mentioned system and equilibrium atmosphere is given. The modification of equations of non-equilibrium thermodynamics for a system of gas and radiation is proposed. Possible ways for further development of tools of non-equilibrium thermodynamics are considered.

Vibrational temperature of the adlayer in the ‘hot-atom’ reaction mechanism

The hot-atom reaction mechanism brings about reaction rates several orders of magnitude higher than those expected in the case of adatoms which have thermalized with the surface. This paper addresses the issue of a possible thermodynamic characterization of the adlayer under reactive conditions and at the steady state. In turn, this implies having to determine the temperature of the adatoms. This is done by means of a nonequilibrium statistical thermodynamic approach, by exploiting a suitable definition of the entropy. The interplay between reaction rate, vibrational temperature of the adatoms and adsorbed quantities is highlighted. This paper shows that the vibrational temperature depends on reaction rate, logarithmically and exhibits a non-linear scaling on physical quantities linked to the energetics of the reaction, namely the adsorption energy and the binding energy of the molecule. The present modeling is also discussed in connection with response equations of nonequilibrium thermodynamics.

Energetics of Glucose Metabolism: A Phenomenological Approach to Metabolic Network Modeling

A new formalism to describe metabolic fluxes as well as membrane transport processes was developed. The new flux equations are comparable to other phenomenological laws. Michaelis-Menten like expressions, as well as flux equations of nonequilibrium thermodynamics, can be regarded as special cases of these new equations. For metabolic network modeling, variable conductances and driving forces are required to enable pathway control and to allow a rapid response to perturbations. When applied to oxidative phosphorylation, results of simulations show that whole oxidative phosphorylation cannot be described as a two-flux-system according to nonequilibrium thermodynamics, although all coupled reactions per se fulfill the equations of this theory. Simulations show that activation of ATP-coupled load reactions plus glucose oxidation is brought about by an increase of only two different conductances: a [Ca2+] dependent increase of cytosolic load conductances, and an increase of phosphofructokinase conductance by [AMP], which in turn becomes increased through [ADP] generation by those load reactions. In ventricular myocytes, this feedback mechanism is sufficient to increase cellular power output and O2 consumption several fold, without any appreciable impairment of energetic parameters. Glucose oxidation proceeds near maximal power output, since transformed input and output conductances are nearly equal, yielding an efficiency of about 0.5. This conductance matching is fulfilled also by glucose oxidation of β-cells. But, as a price for the metabolic mechanism of glucose recognition, β-cells have only a limited capability to increase their power output.

A model of amorphous and nano-crystalline ribbon processing by planar-flow casting

This paper provides a mathematical model describing the processing of amorphous and nano-crystalline ribbon. The developed model is based on averaging of the equations of non-equilibrium thermodynamics over a forming zone. The model describes an effect of the various technological parameters on the geometry and temperature of the zone. These parameters are: the overpressure, the speed of rotation of a cooling disc, the value of a gap, the thickness of a nozzle, the initial overheating, the atmosphere pressure, and the evenness of the disc surface. Model allows to take into account various random factors such as vibrations, etc. Considered time-dependent quantities are: the temperature of the formation zone, the length of the zone, the thickness of the ribbon, the curvature radius of the rear part of the zone, the nozzle outflow speed. Model takes into account the following physical properties as determining the process: viscosity, density, surface tension, heat capacity, temperature of solidification, heat transfer. Relief of the free and contact surfaces can also be regarded as parameters of the model. Correlation between the quantities mentioned above is considered without introduction of any phenomenological constants.

Non-equilibrium thermodynamics of photophoretic motion of colloidal particles

The behavior of electromagnetic radiation in colloidal systems is studied by non-equilibrium thermodynamics methods. Phenomenological equations of non-equilibrium thermodynamics are derived preliminarily for discontinuous systems where photon exchange occurs. Phenomenological equations describing the radiative transfer in absorbing medium are also considered. The equations responsible for the motion of particles in a nonuniform absorbing medium with the directed flux of electromagnetic radiation are derived. These equations describe thermo-and photophoretic motion of colloidal particles. The effect related to the radiation of electromagnetic waves by a colloidal particle moving in a uniform medium under the action of an external force is a cross effect with respect to photophoresis. The Onsager’s relations for kinetic coefficients are shown to be valid for all considered systems.