scholarly journals GENERALIZED KINETIC AND EVOLUTION EQUATIONS IN THE APPROACH OF THE NONEQUILIBRIUM STATISTICAL OPERATOR

2005 ◽  
Vol 19 (06) ◽  
pp. 1029-1059 ◽  
Author(s):  
A. L. KUZEMSKY
Keyword(s):  
Kinetic Equation ◽  
Evolution Equations ◽  
Kinetic Equations ◽  
Type Equation ◽  
Nonequilibrium State ◽  
Nonequilibrium Statistical Operator ◽  
Thermal Bath ◽  
Relaxation Processes ◽  
Collision Term ◽  
Statistical Operator

The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.

scholarly journals THEORY OF TRANSPORT PROCESSES AND THE METHOD OF THE NONEQUILIBRIUM STATISTICAL OPERATOR

2007 ◽  
Vol 21 (17) ◽  
pp. 2821-2949 ◽  
Author(s):  
A. L. KUZEMSKY
Keyword(s):  
Statistical Mechanics ◽  
Kinetic Equations ◽  
Transport Processes ◽  
Ensemble Method ◽  
Formal Structure ◽  
Nonequilibrium Statistical Operator ◽  
Statistical Characteristics ◽  
Irreversible Processes ◽  
Nonequilibrium Processes ◽  
Statistical Operator

The aim of this paper is to provide better understanding of a few approaches that have been proposed for treating nonequilibrium (time-dependent) processes in statistical mechanics with the emphasis on the interrelation between theories. The ensemble method, as it was formulated by Gibbs, has great generality and broad applicability to equilibrium statistical mechanics. Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. In the present work, the statistical theory of nonequilibrium processes which is based on nonequilibrium ensemble formalism is discussed. We also outline the reasoning leading to some other useful approaches to the description of the irreversible processes. The kinetic approach to dynamic many-body problems, which is important from the point of view of the fundamental theory of irreversibility, is alluded to. Appropriate references are made to papers dealing with similar problems arising in other fields. The emphasis is on the method of the nonequilibrium statistical operator (NSO) developed by Zubarev. The NSO method permits one to generalize the Gibbs ensemble method to the nonequilibrium case and to construct a nonequilibrium statistical operator which enables one to obtain the transport equations and calculate the transport coefficients in terms of correlation functions, and which, in the case of equilibrium, goes over to the Gibbs distribution. Although some space is devoted to the formal structure of the NSO method, the emphasis is on its utility. Applications to specific problems such as the generalized transport and kinetic equations, and a few examples of the relaxation and dissipative processes, which manifest the operational ability of the method, are considered.

Nonequilibrium statistical operator and generalized kinetic equations

1974 ◽  
Vol 49 (3) ◽  
pp. 225-226 ◽  
Author(s):  
K. BiaŁas-BorgieŁ ◽  
A. Pawlikowski ◽  
E. Zipper
Keyword(s):  
Kinetic Equations ◽  
Nonequilibrium Statistical Operator ◽  
Statistical Operator

scholarly journals Social climbing and Amoroso distribution

2020 ◽  
Vol 30 (11) ◽  
pp. 2229-2262
Author(s):  
Giacomo Dimarco ◽  
Giuseppe Toscani
Keyword(s):  
Kinetic Equation ◽  
Value Function ◽  
Kinetic Equations ◽  
Type Equation ◽  
Variable Coefficients ◽  
High Status ◽  
Pareto Tails ◽  
The Social ◽  
The Status ◽  
Fokker Planck

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker–Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann-type kinetic equation is shown to converge towards the solution of a Fokker–Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker–Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a social elite. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing.

β Grain Growth Kinetics of a New Metastable β Titanium Alloy

2013 ◽  
Vol 747-748 ◽  
pp. 844-849 ◽  
Author(s):  
Yue Fei ◽  
Xin Nan Wang ◽  
Zhi Shou Zhu ◽  
Jun Li ◽  
Guo Qiang Shang ◽  
...  
Keyword(s):  
Titanium Alloy ◽  
Grain Growth ◽  
Kinetic Equations ◽  
Type Equation ◽  
Growth Exponent ◽  
Solution Treated ◽  
Growth Activation ◽  
Grain Growth Kinetics ◽  
Kinetics Of ◽  
Strength And Ductility

Ti-Mo-Nb-Cr-Al-Fe-Si alloy is a new metastable β titanium alloy with excellent combination of strength and ductility. The β grain-growth exponent and the activation energies for β grain growth for the investigated alloy at specified temperature were computed by the kinetic equations and the Arrhenius-type equation. The rate of β grain growth decreases with elongating solution treated time and increases with the increasing solution-treated temperature. The β grain-growth exponents, n, are 0.461, 0.464 and 0.469 at 1113, 1133 and 1153K, respectively. The β grain growth activation energy is determined to be 274 KJ/mol.

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