On statistical thermodynamics of multicomponent systems II

The equivalent neighbour (EN) model of interacting continuous spherically symmetric spins with the length equal or smaller than unity is solved exactly for all values of the spin dimensionality, D. A surprising result is that only for D 628 the system undergoes phase transitions at temperatures lower than those of the EN model of interacting spins with the hypercubic symmetry. In the D → ∞ limit the equation of state in zero field is derived analytically and gives the Landau type behaviour in the whole range of temperatures.     

Nonequilibrium thermodynamics of multicomponent systems

A general formalism of nonequilibrium thermodynamics of multicomponent fluids is given. Reversible parts of balance equations for the extensive variables are constructed in such a way that the growth rate of the extensive variables are related to their conjugate intensive parameters in an antisymmetric manner. This enables us to give the balance equation for the diffusional flow which plays an important role in the study of superfluidity and complex fluids.     

A characteristic class of quantities in nonequilibrium thermodynamics and a statistical justification of the local equilibrium approximation

A theory of bandwidth anisotropy in metallic ferromagnets developed previously is specialised to the case of 5d electrons in a hexagonal close-packed lattice. This theory is combined with a model for 5d electrons in the heavy rare earth metals to give a new theory for the low temperature values of the magnetic anisotropy coefficientsκ ₂ ⁰ andκ ₄ ⁰ in Tb, Dy, Ho, Er and Tm. In this theory the magnetic anisotropy is due to a combination of (i) crystal fields acting on 5d and 4f electrons and (ii) bandwidth anisotropy associated with a dependence of 5d bandwidths on magnetization direction. After use is made of empirical upper limits on the eighth order magnetic anisotropy in Gd, there remain four partially adjustable parameters of importance in the theory. These can be chosen to give a good fit to the six observed values forκ ₂ ⁰ andκ ₄ ⁰ in Tb, Dy and Ho. Crystal fields corresponding to negative point charges are seen by 5d electrons, but because of 4f – 5d interactions effective fields of larger magnitude and opposite sign act on 4f shells. Bandwidth anisotropy gives a significant contribution toκ ₄ ⁰ of opposite sign to that due to crystal fields, and dominates the latter in Tb and Er.     

On the connection of nonequilibrium information thermodynamics with non-hamiltonian quantum mechanics of open systems

A critical and improved version of the non-Hamiltonian quantum mechanics and nonequilibrium information thermodynamics is presented. It has been shown that the latter is connected with the former through the set of macroscopical observables defining the additional conditions by maximization of information. This set of macroscopical observables occurs in the asymptotic solution of the equation of motion of the non-Hamiltonian mechanics if the latter is ergodic. The identity of the two sets of observables is the consistency condition of information thermodynamics of open systems. The general theory has been illustrated and discussed on two rigorously solvable examples. Using a $\text{C}{}^1\text{-}\text{algebraic}$ approach a connection has been shown with the results of field-theoretical axiomatic methods concerning the thermodynamical limit.     

Relationship between the thermodynamics of nonequilibrium state and the thermodynamics of nonequilibrium processes

The concept and equations of the thermodynamics of nonequilibrium state are compared with the postulates of linear thermodynamics of nonequilibrium processes. It is shown that all the four basic postulates can be obtained as consequences of the equations of the thermodynamics of nonequilibrium state. In the limiting case of small deviations from equilibrium, the thermodynamics of nonequilibrium state are reduced to linear thermodynamics of nonequilibrium processes.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 10–14, December, 1990.     

Quantum kinetic equation for nonequilibrium dense systems

Using the density matrix method in the form developed by Zubarev, equations of motion for nonequilibrium quantum systems with continuous short range interactions are derived which describe kinetic and hydrodynamic processes in a consistent way. The T-matrix as well as the two-particle density matrix determining the nonequilibrium collision integral are obtained in the ladder approximation including the Hartree-Fock corrections and the Pauli blocking for intermediate states. It is shown that in this approximation the total energy is conserved. The developed approach to the kinetic theory of dense quantum systems is able to reproduce the virial corrections consistent with the generalized Beth-Uhlenbeck approximation in equilibrium. The contribution of many-particle correlations to the drift term in the quantum kinetic equation for dense systems is discussed.     

Renormalized kinetic theory of nonequilibrium many-particle classical systems

The far-from-equilibrium statistical dynamics of classical particle systems is formulated in terms of self-consistently determined phase-space density response, fluctuation, and vertex functions. Collective and single-particle effects are treated on an equal footing. Two approximations are discussed, one of which reduces to the Vlasov equation direct interaction approximation of Orszag and Kraichnan when terms that are explicitly due to particles are removed.Work performed under the auspices of the U.S. Department of Energy.     

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.     

The construction of kinetic equations for nonequilibrium quantum systems

A method of construction of a Markoffian solution of the Liouville equation for nonequilibrium systems, leading to closed kinetic equations, is described. The method is illustrated on the basis of an electron-phonon system; higher orders in the interaction are taken into account.     

颗粒介质的结构及热力学

颗粒介质具有远程无序和近程有序的结构, 是产生动力学不均匀性(dynamical heterogeneity) 和复杂不可逆过程的根源. 本文分析了颗粒介质的结构特征、变形和能量耗散之间的内在关联, 讨论了颗粒介质的弹性, 提出了流变应变增量、耦合应变增量和弹性应变增量的应变增量分解方式. 沿用非平衡热力学框架, 引入表征运动无序的动理学颗粒温度T^k和表征弹性应力涨落的构型温度T^c, 作为非平衡态变量, 建立了双颗粒温度热力学(two-granular-temperature thermodynamics, TGT理论), 注重分析了不可逆过程中的热力学力和流, 并与著名的砂土内变量热力学进行了对比.     

Statistical thermodynamics of multicomponent systems (mixtures) in the quasilattice coordination approximation

Based on the well-known concepts of statistics of nearest neighbors in a lattice and on the BethePeierls-Guggenheim quasichemical approximation, combination rules are obtained allowing us to express the thermodynamic functions of multicomponent systems (mixtures) with given chemical and coordination composition in terms of thermodynamic functions of the pure subsystems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 85–89, January, 1976.     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

Statistical thermodynamics of multicomponent systems of nonspherical molecules taking account of the short-range order in a quasichemical approximation

It is shown that short-range order can be taken into account effectively for a liquid multicomponent system (mixture) of molecules with internal (particularly orientation) degrees of freedom within the framework of a quasichemical approximation (in the Guggenheim formulation) by using a vector in the space of internal states of the molecule (correlation vector) which is introduced together with the ordinary set (vector) of occupation numbers describing the longrange order. A closed system of equations is obtained in the correlation vector representation for the thermodynamic functions and occupation numbers of chemically distinct components in the space of internal states of the molecule, and its variational formulation is given. The possibilities for using the results obtained in the theory of the liquid-crystal state are discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 17–20, September, 1977.The author is grateful to Prof. V. K. Semenchenko for his interest in this work.     

On the functional integral for generalized Wiener processes and nonequilibrium phenomena

The attention will be focussed on a generalized Wiener diffusion process for which the macroscopic evolution $\text{y}\text{?}\text{= c}{}_1\text{(}\text{y}\text{)}$ equals zero, of course, and where the variance of the process obeys $\text{g}\text{?}\text{s}{}^2\text{= c}{}_2\text{(}\text{y}\text{)}$. The diffusion function c₂(y) may be state dependent in an arbitrary way. We invoke our treatment of the general time-local Gaussian process as presented in a previous paper. This process will be seen to define a generalized functional Wiener measure. This measure has already been used implicitly in earlier work being concerned with nonlinear, nonequilibrium Markov processes. The sum of the generalized measure over the entire function space will be shown to be exactly related to the general Fokker-Planck equation for the driftless diffusion process. The relation between the well-defined functional sum and its corresponding functional integral will be studied in detail. The analysis demonstrates in clear fashion the origin of the deviations from other approaches, and provides an extension of our previous results on nonequilibrium, nonlinear phenomena to include generalized diffusion processes.     

On the positivity of correlations in nonequilibrium spin systems

We consider Ising spin systems, equivalently lattice gases evolving under discrete- or continuous-time Markov processes, i.e., stochastic cellular automata or interacting particle systems. We show that for certain spin-flip probabilities or rates and suitable initial states the expectation values of products of spin variables taken at equal or different times are nonnegative; they satisfy the same inequalities as the equal-time correlations of ferromagnetic systems in equilibrium (first Griffiths inequality). Extensions of FKG inequalities to time-displaced correlations are also discussed.     

Differential relations of the nonlocal nonequilibrium thermodynamics of martensitic transformations

Differential relations describing the change in the degree of martensitic transformation with change in the external temperature and pressure are derived. As an illustration, the relations are applied to the analysis of the experimental situation of apparent violation of the Clausing-Clapeyron equation.