A thermodynamic approach to grain growth and coarsening

The evolution equations for grain growth and coarsening have been derived in the open literature mainly based on phenomenological considerations. Applying a thermodynamic extremal principle, the evolution equations are derived in a rigorous way. All kinetic parameters are provided directly. Existing relations are proved and generalized.     

The thermodynamic characteristics of multicomponent reaction mixtures in the sub- and supercritical states

The paper presents mathematical models and calculation methods for solving particular research problems related to the thermodynamic characteristics of multicomponent and multiphase mixtures. The special features of chemical and phase equilibria in such mixtures are considered in the ideal gas approximation and taking nonideality into account. The conditions of equilibrium phase stability are studied for multiphase systems. The results of calculations of characteristic phase diagrams and binodal and spinodal are given for model systems with a fixed chemical composition, and a new interpretation of the mathematical model for localizing the critical point of a multicomponent mixture with a given composition is presented. A new interpretation of the well-known classic homotopy method is suggested for solving complex nonlinear systems of equations. Some anomalies of phase portraits and critical curves that are necessary to take into account in selecting (planning) experimental conditions and calculating chemical processes and reaction parameters are considered separately. The possibility of calculating thermodynamic and thermophysical properties (entropy, enthalpy, heat capacity, heat effects of reactions, and adiabatic heating) is demonstrated for the example of particular multicomponent nonideal mixtures. The conclusion is drawn that cubic equations of state can be used for predicting the deviations of these properties from the ideal gas state and their anomalies in the vicinity of the critical points of mixtures.     

A field analysis of nonlinear irreversible thermodynamic processes

The generalized thermodynamic potential analysis of nonlinear irreversible processes precludes the analysis of rotational processes. The nonexistence of scalar potential functions necessitates a thermodynamic analysis of the system forces. A field analysis in the phase space of the generalized displacements and velocities treats the force components as tensors of second order that tend to deform and rotate the irreversible process, which is viewed as an elastic material. The analysis of chemical oscillatory processes involves the introduction of the thermodynamic vector potential, which is subsequently used in the formulation of a variational principle and to define an energy flux vector. The direction of energy flow elucidates the mechanism by which steady motion is maintained and it is a characteristic property of open systems. Field analyses of systems that are described by half and single degrees of freedom are contrasted.     

On the Onsager Variation Principle and its Generalization for Nonlinear Irreversible Thermodynamics

The Onsager variation principle is examined from the viewpoint of the thermodynamic analogue of the D'Alembert principle in mechanics when the irreversible processes are linear and thus the system is near equilibrium. The thermodynamic D'Alembert principle is shown to be a precursor to the Onsager variation principle. The thermodynamic D'Alembert principle is then generalised to the cases of nonlinear irreversible processes occurring removed from equilibrium and a generalised form of the Onsager variation principle is obtained under some restricting conditions. The restricted variation principle so deduced has an accompanying exact differential form generalising the Clausius entropy differential (equilibrium Gibbs relations) and contains in it the essence of the thermodynamics of irreversible processes in systems where non-linear transport processes occur. An example is given for the nonlinear dissipation function in the variation functional. The evolution equations for fluxes are shown to yield those known in the literature.     

Asymptotic thermodynamic criteria for the persistency of metastable states

The mean first exit time is logarithmically equivalent to the exponential of the smallest total free energy barrier separating metastable from stable states for a general class of discontinuous Markov processes in the thermodynamic limit. This asymptotic principle of minimum relative free energy difference constitutes a generalization of the corresponding entropy principle to systems in thermal contact with their environment and formalizes the results of a number of previous authors. The overwhelmingly most probable path of exit, in the thermodynamic limit, is the mirror image in time of the causal path. Along the anticausal path the free energy is an increasing function of time and hence does not provide any criterion of evolution. The kinetic mean-field model of a ferromagnetic serves for illustration.     

An Optimization Principle for Deriving Nonequilibrium Statistical Models of Hamiltonian Dynamics

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. Given a vector of resolved variables, selected to describe the macroscopic state of the system, a family of quasi-equilibrium probability densities on phase space corresponding to the resolved variables is employed as a statistical model, and the evolution of the mean resolved vector is estimated by optimizing over paths of these densities. Specifically, a cost function is constructed to quantify the lack-of-fit to the microscopic dynamics of any feasible path of densities from the statistical model; it is an ensemble-averaged, weighted, squared-norm of the residual that results from submitting the path of densities to the Liouville equation. The path that minimizes the time integral of the cost function determines the best-fit evolution of the mean resolved vector. The closed reduced equations satisfied by the optimal path are derived by Hamilton-Jacobi theory. When expressed in terms of the macroscopic variables, these equations have the generic structure of governing equations for nonequilibrium thermodynamics. In particular, the value function for the optimization principle coincides with the dissipation potential that defines the relation between thermodynamic forces and fluxes. The adjustable closure parameters in the best-fit reduced equations depend explicitly on the arbitrary weights that enter into the lack-of-fit cost function. Two particular model reductions are outlined to illustrate the general method. In each example the set of weights in the optimization principle contracts into a single effective closure parameter.     

Natural selection and thermodynamic optimality

Summary It is assumed that Darwin's principle translates into optimal regimes of operation along metabolical pathways in a biological system. Fitness is then defined in terms of the distance of a given individual's thermodynamic parameters from their optimal values. The method is illustrated testing maximum power as a criterion of merit satisfied in ATP synthesis. The author of this paper has agreed to not receive the proofs for correction. Support from Secretaría de Educación Pública and CONACYT-México is gratefully acknowledged. Permanent address.     

The effect of quaternary element on the thermodynamic parameters and structure of CuAlMn shape memory alloys

In this study, the Cu-based shape memory alloys were produced by arc melting. We have investigated the effects of the alloying elements on the characteristic transformation temperatures, enthalpy, entropy values, and the structure of Cu–Al–Mn ternary system. The evolution of the transformation temperatures was studied by the differential scanning calorimetry. The characteristic transformation temperatures can be controlled by the variations in the aluminum and manganese content. Additionally, the effect of magnesium and iron on the transformation temperatures and thermodynamic parameters was investigated in the Cu–Al–Mn ternary system. The addition of the magnesium decreases the characteristic transformation temperatures of the Cu–Al–Mn system, but that of the iron increases. The structural changes of the samples were studied by X-ray diffraction measurements and optical microscope observations. Due to the low solubility of the magnesium, the magnesium addition into the Cu–Al–Mn system forms precipitates in the matrix. It is evaluated that the transformation parameters of the CuAlMn shape memory alloys can be controlled by the change of the alloying elements and the weight percentages of alloying elements.     

Thermodynamics of one-dimensional nonlinear lattices

Analytic equations were obtained for the thermodynamic parameters of one-dimensional lattices of particles with the Toda and Morse interaction potentials in a canonical Gibbs ensemble. For the same systems, equations were derived for molecular dynamics simulations of thermodynamic processes. Stochastic differential equations were solved with simulating the thermostat by Langevin sources with random forced. Analytic equations for thermodynamic parameters (energy, temperature, and pressure) excellently coincided with molecular dynamics simulation results. The kinetics of system relaxation to the thermodynamic equilibrium state was analyzed. The advantages of simulating the physical properties of systems in a canonical compared with microcanonical ensemble were demonstrated.     

Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

Jerky flow model as a basis for research in deformation instabilities

A phenomenological jerky flow model was developed in which macroscale plastic strain rates are defined by dislocation kinetics. The model takes into account destructive processes governed by shear and bulk defect accumulation. At the heart of the model lie equations of solid mechanics and relaxation-type constitutive equations. A loaded elastoplastic solid is treated as a nonlinear dynamic system whose evolution, according to synergetic laws, is much contributed by negative and positive feedbacks expressed, respectively, through constitutive equations of the first group (relaxation equations) and constitutive equations of the second group (kinetic equations for deformation defect and damage accumulation rates). The negative feedback stabilizes deformation by relaxation, bringing the process to some local dynamic equilibrium. The positive feedback destabilizes deformation, driving the system to a critical state. Numerical experiment was performed in 2D and 3D statements. Statistical analysis of stress fluctuations about the average trend shows that the jerky flow model of an elastoplastic medium demonstrates evolution characteristic of nonlinear dynamic systems: through states of dynamic chaos and self-organized criticality to a global catastrophe.     

Equilibrium thermodynamics and thermodynamic processes in nonlinear systems

A general method for the calculation of thermodynamic quasi-equilibrium processes by molecular dynamics (MD) simulation in canonical ensemble is developed. The method is suitable for classical systems with arbitrary interaction potentials. Though this MD method does not allow to calculate directly partition function and entropy, it is possible to calculate necessary partial derivatives which enter into the expressions for the full derivatives dT/dV and d β/dV for adiabitic and isobaric processes. Namely the solutions of these ordinary differential equations define the corresponding thermodynamic processes. The adiabatic process for the 1D Toda lattice is analyzed in details. The usage of the Toda potential allows to perform all analytical calculus up to accurate answers and to compare numerical and analytical results. Exact analytical expressions for the thermodynamics of 1D lattices with few types of nearest neighbor interactions are obtained as a necessary interim solutions. MD-simulation of quasi equilibrium processes in canonical ensemble demands the achievement of thermodynamic equilibrium, thus the thermalization kinetics is briefly discussed.     

Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.     

Critical thermodynamic evaluation and optimization of the Fe-Mg-O system

A complete literature review, critical evaluation and thermodynamic modeling of the phase diagrams and thermodynamic properties at 1 bar total pressure of all oxide phases in the Fe-Mg-O system are presented. Optimized model equations for the thermodynamic properties of all phases are obtained which reproduce all available thermodynamic and phase equilibrium data within experimental error limits from 25 °C to above the liquidus temperatures at all compositions and oxygen partial pressures. The complex phase relationships in the system have been elucidated and discrepancies among the data have been resolved. The database of the model parameters can be used along with software for Gibbs energy minimization in order to calculate any type of phase diagram section. Sublattice models, based upon the compound energy formalism, were used for the spinel, pyroxene, olivine and monoxide phases. The use of physically reasonable models means that the models can be used to predict properties, phase equilibria, and cation site distributions in composition and temperature regions where data are not available.     

Critical evaluation and thermodynamic modeling of the Mn–Cr–O system for the oxidation of SOFC interconnect

A complete critical evaluation and thermodynamic modeling of the phase diagrams and thermodynamic properties of the Mn–Cr–O system at 1 bar total pressure are presented. Optimized equations for the thermodynamic properties of all phases are obtained, which reproduce all available and reliable thermodynamic and phase equilibrium data within experimental error limits from 25 °C to above the liquidus temperatures at all compositions and oxygen partial pressures. As results of optimization, the Gibbs energy function of MnCr₂O₄ is for the first time properly estimated and the discrepancies of the phase diagram experiments of the Mn–Cr–O system are resolved. In particular, unexplored phase diagrams and thermodynamic properties of the Mn–Cr–O system of importance for the oxidation of SOFC interconnect are predicted on the basis of the optimized model parameters. The database of the model parameters can be used along with software for Gibbs energy minimization in order to calculate any type of phase diagram sections and thermodynamic properties.     

Phase-field model for multiphase systems with different thermodynamic factors

A modified phase-field model for quantitative simulations of low-speed phase transitions in multiphase systems is proposed, which takes into account the difference between thermodynamic factors in all the phases. The presented model is based on the quantitative phase-field concept developed by Steinbach et al. [I. Steinbach, F. Pezolla, B. Nestler, M. Seeelberg, R. Prieler, G.J. Schmitz, J.L.L. Rezende, A phase field concept for multiphase systems, Physica D 94 (1996) 135] for multiphase systems allowing to consider the multiphase transition as a superposition of pairwise interactions between two phases. We complete this approach and develop a model, which uses parameters derived from chemical free energy functions of individual phases evaluated from experimental data by the CALPHAD method Lukas et al. (2007) [17]. Because the thermodynamic factors are different in various phases we need to evaluate a special form of total chemical free energy function of a multiphase mixture and use it in the phase-field model. It is shown, that for the developed model the thin-interface asymptotic and the anti-trapping term developed previously for the solidification of pure substances can be applied. The model is verified by an example of the Al-Ni system whose peritectic structural morphology during the directional solidification is investigated. The suggested model can be also extended to multicomponent systems.     

统计物理的微观动力学基础

统计的基本出发点是研究系统具有的随机性,不同系统在不同情形下的宏观热力学性质起源于系统内部随机性的差异,通过对宏观热力学系统的微观非线性动力学进行研究探索，我们可以进一步更为深入地理解物态方程、相变等诸多的宏观热力学现象。本文通过哈密顿系统的非线性动力学研究，以及遍历性理论的动力学随机性研究对此问题进行了分析，研究表明，动力学系统的全局性混沌是系统统计成立的根本要素，系统的无限大自由度（热力学极限）已不是决定性的因素，人们可以在此基础上建立少自由度系统的统计力学及热力学。     

流体混合物热力学性质计算的混合法则新进展

回顾近期关于流体混合物热物性混合法则的研究新进展,特别是余吉布斯自由能(GE)和余亥姆霍兹自由能(AE)型混合法则的拓展及应用,重点介绍一种基于AE,仅通过偏微分方程便可计算混合物所有热力学性质的混合法则。可为发展未知流体混合物热物性计算方法以及各种常见混合物热物性的工程应用提供参考和依据。