Statistical Irreversible Thermodynamics in the Framework of Zubarev’s Nonequilibrium Statistical Operator Method

We describe the formalism of statistical irreversible thermodynamics constructed based on Zubarev’s nonequilibrium statistical operator (NSO) method, which is a powerful and universal tool for investigating the most varied physical phenomena. We present brief overviews of the statistical ensemble formalism and statistical irreversible thermodynamics. The first can be constructed either based on a heuristic approach or in the framework of information theory in the Jeffreys-Jaynes scheme of scientific inference; Zubarev and his school used both approaches in formulating the NSO method. We describe the main characteristics of statistical irreversible thermodynamics and discuss some particular considerations of several authors. We briefly describe how Rosenfeld, Bohr, and Prigogine proposed to derive a thermodynamic uncertainty principle.     

Electrical Conductivity of Charged Particle Systems and Zubarev’s Nonequilibrium Statistical Operator Method

One of the fundamental problems in physics that are not yet rigorously solved is the statistical mechanics of nonequilibrium processes. An important contribution to describing irreversible behavior starting from reversible Hamiltonian dynamics was given by D. N. Zubarev, who invented the method of the nonequilibrium statistical operator. We discuss this approach, in particular, the extended von Neumann equation, and as an example consider the electrical conductivity of a system of charged particles. We consider the selection of the set of relevant observables. We show the relation between kinetic theory and linear response theory. Using thermodynamic Green’s functions, we present a systematic treatment of correlation functions, but the convergence needs investigation. We compare different expressions for the conductivity and list open questions.     

Galerkin’s Method in the Theory of Dipole Antennas

It is proved that an approximate solution of the integro-differential equation for the current at the surface of a dipole antenna obtained by Galerkin’s method converges to the exact solution. A method is proposed for calculating the matrix elements of linear operators that enter the above equation.     

Nonequilibrium Green’s Functions in the Atomic Representation and the Problem of Quantum Transport of Electrons Through Systems With Internal Degrees of Freedom

We develop the theory of quantum transport of electrons through systems with strong correlations between fermionic and internal spin degrees of freedom. The atomic representation for the Hamiltonian of a device and nonequilibrium Green’s functions constructed using the Hubbard operators allow overcoming difficulties in the perturbation theory encountered in the traditional approach because of a larger number of bare scattering amplitudes. Representing the matrix elements of effective interactions as a superposition of terms each of which is split in matrix indices, we obtain a simple method for solving systems of very many equations for nonequilibrium Green’s functions in the atomic representation. As a result, we obtain an expression describing the electron currents in a device one of whose sites is in tunnel coupling with the left contact and the other, with the right contact. We derive closed kinetic equations for the occupation numbers under conditions where the electron flow leads to significant renormalization of them.     

Generalized and Variational Statements of the Riemann Problem with Applications to the Development of Godunov’s Method

Computational Mathematics and Mathematical Physics - This paper is devoted to the numerical method for solving the fluid dynamics equations proposed by Godunov more than 60 years ago. This method...     

Singular Arcs in the Generalized Goddard’s Problem

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control. Support from the French Space Agency CNES (Centre National d’Etudes Spatial) is gratefully acknowledged.     

On One Method of Solving Singularly Perturbed Systems of Tikhonov’s Type

To investigate Tikhonov systems arising in problems of control theory we apply the method of holomorphic regularization, which allows one to obtain solutions to singularly perturbed problems in the form of series converging in the usual sense in powers of a small parameter.     

Modification of Rissanen’s Method in Linear Memory

The problem of solving a linear system with a Hankel or block-Hankel matrix, as well as Rissanen’s algorithm and its generalization to the block case, are considered. Modifications of these algorithms that use less memory (O(n) against O(n²)).     

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub manifolds.     

The Spirallike Mapping of Type α and the Generalized Roper-Suffridge Operator

In this paper, we get the fact that generalized Roper-Suffridge operators act on spirallike mapping of type α on different domains is invariable, these results include some results known.     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

Connectedness of the Set of Efficient Solutions for Generalized Systems

We introduce the concept of positive proper efficient solutions to the generalized system in this paper. We show that, under some suitable conditions, the set of positive proper efficient solutions is dense in the set of efficient solutions to the generalized system. We discuss also the connectedness of the set of efficient solutions for the generalized system with monotone bifunctions in real locally convex Hausdorff topological vector spaces. This research was partially supported by the National Natural Science Foundation of China (10561007), the Natural Science Foundation of Jiangxi Province, China, and a grant from the National Science Council of ROC.