Statistical description of nonequilibrium self-gravitating systems

Self-gravitating systems are non-equilibrium a priori. A new approach is proposed, which employs a non-equilibrium statistical operator that takes account inhomogeneous distribution of particles and temperature. The method involves a saddle-point procedure to find the dominant contributions to the partition function, thus obtaining all thermodynamic parameters of the system. Probable peculiar features in the behavior of the self-gravitating systems are considered for various conditions. The equation of state for self-gravitating systems has been determined. A new length of the statistical instability is obtained for a real gravitational system, as are parameters of the spatially inhomogeneous distribution of particles and temperature.     

The quantumstatistical aspects of the independent-particle model for the nucleus

We present a new justification of the independent particle model for the nucleus. It is based on a statistical theory of the short-range correlations in large Fermion systems. The statistical operator of many-body Fermion systems — if averaged over a suitable ensemble — can be written as a product of statistical operators for a one-body system. The statistical operator for the one-body system obeys a Hartree-Fock equation. Physical interpretation and conclusions are discussed.     

A Coupled Hybrid Lattice: Its Related Continuous Equation and Symmetries

The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discrete coupled KdV system, is also found to be discrete form of a coupled mKdV systems. Delayed differential reduction system and pure difference systems are derived from the coupled hybrid system by means of the symmetry reduction approach. Cnoidal wave, positon and negaton solutions for the coupled hybrid system are proposed.     

A generalized Karnahan-Starling approach for molecular systems with a positive definite potential of interaction between particles

A new method for obtaining the well-known Karnahan-Starling equation for systems of hard spheres and its generalization for the case of any number of precisely known virial coefficients is proposed. The efficiency of the method for the construction of the statistical thermodynamics of a system of soft spheres, where an analytical expression for free energy and equations of state that agree well with the data of the computer experiment are found, is shown. The considered approach is generalized for systems with a positive definite potential of interaction between particles.     

弹核碎裂反应中同位旋效应的系统研究

用统计擦碎模型对中能区不同弹靶体系在弹核碎裂反应中的同位旋效应和同位旋标度率现象进行了系统研究. 发现归一的同位素分布峰位差和约化的同位旋标度率参数随(Z_proj－Z)/Z_proj or (N_proj－N)/N_proj呈指数下降, 与反应系统大小无关. 指出约化的同位旋标度率参数可以用来研究中能重离子碰撞中反应系统的激发程度和非对称核物质的状态方程.     

An equation for the quantum dissipative system in statistical mechanics

A formalism for describing quantum dissipative systems in statistical mechanics is developed. A new equation of the Lindblad type with a quadratic superoperator consisting of Hermitian dissipative operators is derived from the Bloch equation for temperature density matrix using the Feynman integral over the trajectories with a modified Menskii weight functional. By way of example, this equation is solved for a one-dimensional quantum harmonic oscillator with linear dissipation. Applying the projection operator technique, an integral-differential equation for a reduced temperature statistical operator is obtained, which is analogous to the Zwanzig equation in statistical mechanics, and its formal solution is found as a convergent series. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 30–34, December, 2006.     

Mass transport theory for the Toda lattices, dispersive and dissipative

To establish mass transport theory on nonlinear lattices, we formulate the Korteweg–deVries (KdV) equation and the Burgers equation using the flow variable representation so as to facilitate comparison with the Boltzmann equation and with the Cahn–Hilliard equation in classical statistical mechanics. We also study Toda lattice microdynamics using the Flaschka representation, and compare with the Liouville equation. Like the linear diffusion equation, the Boltzmann equation and the Liouville equation are to be solved for a distribution function, which is intrinsically probabilistic. Transport theory in linear systems is governed by the isotropic motions of the kinetic equations. In contrast, the KdV perturbation equation derived from the Toda lattice microdynamics expresses hydrodynamic mass transport. The KdV equation in hydrodynamics and the Burgers equation in thermodynamics do not involve a probability distribution function. The nonlinear lattices do not retain isotropy of the mass transport equations. In consequence, it is proposed that in the presence of hydrodynamic flows to the left, KdV wave propagation proceeds to the right. This basic property of the KdV system is extended to thermodynamics in the Burgers system. These features arise because linear systems are driven towards an equilibrium by molecular collisions, whereas the inhomogeneities of the nonlinear lattices are generated by the potential energy of interaction. Diffusion as expressed by the Burgers equation is governed not only by a chemical potential, but also by the Toda lattice potential energy.     

Onsager–Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction IV. Rotating systems

The statistical approach to the nonequilibrium thermodynamics presented in Part I of this paper was generalized to rotating gaseous systems. Basing on the Boltzmann equation written for rotating reference frame the kinetic coefficients satisfying the Onsager–Casimir reciprocity relations are obtained in a general form. An example of the application of this theory is given. Some new phenomena arising only in rotating system are found.     

Application of general transport theories to model system

We consider a simple model, the two-site small-polaron system, and show that its properties can be characterized by a master equation which is in accordance with recent quantum statistical theories of macroscopic observables. It is shown by a combination of formal and numerical analyses that the model is an example of a system describable by the lowest-order term in the expansion of the kernel of the master equation in powers of the ratio of the average polaron-phonon interaction energy to the microscopic phonon energy. We discuss the relevance of the method to actual physical systems.     

Evolution of the q-entropy and energy dissipation during irreversible processes in nonextensive systems

Evolution of the Havrda-Charvat-Daroczy entropy and energy dissipation during irreversible processes in open nonextensive systems is considered. The kinetic equation and statistical criterion of nonextensive system evolution are presented. Nonequilibrium statistical and variational methods of derivation of the kinetic equation with a source being a fluctuation of a physical quantity are described. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 35–41, February, 2006.     

Generating Functionals in Nonequilibrium Statistical Mechanics

The main ideas and methods of calculations within the framework of the generating functional technique are considered in a systematical way. The nonequilibrium generating functionals are defined as functional mappings of the nonequilibrium statistical operator and so appear to be dependent on a certain set of macroscopic variables describing the nonequilibrium state of the system. The boundary conditions and the differential equation of motion for the generating functionals are considered which result in an explicit expression for the nonequilibrium generating functionals in terms of the so-called coarse-grained generating functional being the functional mapping of the quasiequilibrium statistical operator. Various types of integral equations are derived for the generating functionals which are convenient to develop the perturbation theories with respect to either small interaction or small density of particles. The master equation for the coarse-grained generating functionals is obtained and its connection with the generalized kinetic equations for a set of macrovariables is shown. The derivation of the generalized kinetic equations for some physical systems (classical and quantum systems of interacting particles, the Kondo system) is treated in detail, with due regard for the polarization effects as well as the energy and momentum exchange between the colliding particles and the surrounding media.     

On Methods of Finding Bäcklund Transformations in Systems with More than Two Independent Variables

Abstract

Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented.     

Constant pressure ensembles in statistical mechanics

A new statistical procedure is described for obtaining the thermodynamic properties of a molecular system directly as functions of the pressure. This procedure differs in principle from that suggested by Guggenheim [3] in that the members of the representative ensemble are envisaged as being in constant mechanical equilibrium with the exterior. The quantal and classical theories of petit micro-canonical and canonical ensembles of systems at constant pressure are presented, and shown to lead to the established results for perfect and imperfect gases, and for a hypothetical one-dimensional system. The conclusion that the statistical compressibility of a molecular system is essentially positive follows directly from the theory. An alternative procedure which leads to a more satisfactory form of Guggenheim's equation is also described, and its relation to the new approach is shown.     

On a Liouville-master equation formulation of open quantum systems

The dynamics of open quantum systems is formulated in terms of a probability distribution on the underlying Hilbert space. Defining the time-evolution of this probability distribution by means of a Liouvillemaster equation the time-dependent wave function of the system becomes a stochastic Markov process in the sense of classical probability theory. It is shown that the equation of motion for the two-point correlation function of the random wave function yields the quantum master equation for the statistical operator. Stochastic simulations of the Liouville-master equation are performed for a simple example from quantum optics and are shown to be in perfect agreement with the analytical solution of the corresponding equation for the statistical operator.     

General kinetic equation and irreversibility in statistical systems

General kinetic equation for statistical systems is presented. A kinetic equation with source that is fluctuation of physical values was obtained. A new statistical criterion of systems evolution was determined. Nonequilibrium statistical and variational derivations of general kinetic equations are considered. Evolution of nonequilibrium Boltzmann-Gibbs-Shannon entropy, Hamilton function and Hamilton function production are examined.     

Exactly solvable models: The road towards a rigorous treatment of phase transitions in finite systems

The exact analytical solutions of a variety of statistical models recently obtained for finite systems are thoroughly discussed. Among them are a constrained version of the statistical multifragmentation model, the Gas of Bags Model, and the Hills and Dales Model of surface partition. The finite volume analytical solutions of these models were obtained by a novel powerful mathematical method, the Laplace-Fourier transform. Thus, the Laplace-Fourier transform allows one to study the nuclear matter equation of state, the equation of state of hadronic matter and quark gluon plasma, and the surface entropy of large clusters on the same footing. A complete analysis of the isobaric partition singularities of these models is done for finite systems. The developed formalism allows us, for the first time, to exactly define the finite volume analogs of gaseous, liquid, and mixed phases of these models from the first principles of statistical mechanics, and to demonstrate the pitfalls of earlier works. The found solutions may be used for building up a new theoretical apparatus to rigorously study phase transitions in finite systems. The strategic directions of future research opened by these exact results are also discussed. The text was submitted by the author in English.     

Brownian dynamics with constraints

Systems possessing degrees of freedom operating on widely separated timescales, where the effects of those operating on the smaller timescales are relatively unimportant, may be modelled by the use of the Langevin equation. In order to study such systems containing complex polyatomic particles, holonomic constraints may be used. Though there is no lack of published algorithms for the numerical solution of the Langevin equation, few of them have been developed with sufficient rigour to ensure their precision, nor to demonstrate their compatibility with constraints. This study recapitulates an approach based upon Runge-Kutta equations which has the advantage of being perfectible to any desired order in the time-step, and shows how it may be combined with the SHAKE method in order to perform constrained Brownian dynamics simulations. Results are presented for some simple systems with a third order algorithm, and it is found that the correct dynamic and statistical behaviour is recovered.