Memory Effects and Nonequilibrium Correlations in the Dynamics of Open Quantum Systems

We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.     

Linear Kalman–Bucy Filter with Autoregressive Signal and Noise

In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.     

Heat and mass transfer investigation of rotating hydrocarbons droplet which behaves as a hard sphere

The steady state boundary layer equations around rotating pure hydrocarbon droplet are solved numerically. The droplet is simulated to behave as a hard sphere. The transfer equations are discretized using an implicit finite difference method where Thomas algorithm solves the system of algebraic equations. Moreover, dimensionless parameters of heat and mass transfer phenomena around a rotating hexane droplet concluded. The thickness of the boundary layer is unknown for this model and therefore, it is determined. Further, this work proposes correlations of Nusselt and Sherwood numbers for monocomponent hydrocarbon droplets in evaporation. These correlations consider the rotation phenomena and further, the variation of the thermophysical and transport properties in the vapour phase.     

Many-particle densities

In classical systems with an arbitrary interaction determined by a set of many-particle potentials, the equations for many-particle correlations are reduced to the variational problem for the thermodynamic potential regarded as a functional depending on these correlations. The functional in the variational problem contains not only the total correlations but also the generalized direct correlations determined by the sum of chains composed of total correlations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 136–147 July, 2000.     

Turbulence modeling from a new perspective

In this work we establish a link between a Reynolds averaged turbulence modeling methodology, containing interactions up to the second order correlations between the velocity fluctuations at various scales, and a multi-objective optimization problem with the constraints expressed in terms of equality and inequality, imposed by the given boundary conditions and the positive semi-definiteness of the Reynolds stress tensor, etc. The information unavailability and uncertainty associated with the boundary conditions for the fluctuation correlations of various orders is delineated, and the information from the Navier–Stokes equations is utilized to the extent allowed by the available input data necessary for simulations; turbulence from the perspective of systems simulation is explored and some objective functions are proposed. Finally, the challenges faced by the formulation and the issues yet to be resolved are discussed.     

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.     

Some cases of the solvability in a closed form of singular integral equations and two-dimensional characteristic systems

We consider the characteristic system of singular integral equations for two unknown functions. We obtain certain correlations connecting coefficients of the system which allow one to solve the latter in a closed form. Based on the obtained results, we consider cases of the solvability in a closed form for one class of singular integral equations.     

The Monte-Carlo algorithm for the solving of systems of linear algebraic equations by the Seidel method

The iteration algorithm is used to solve systems of linear algebraic equations by the Monte-Carlo method. Each next iteration is simulated as a random vector such that its expectation coincides with the Seidel approximation of the iteration process. We deduce a system of linear equations such that mutual correlations of components of the limit vector and correlations of two iterations satisfy them. We prove that limit dispersions of the random vector of solutions of the system exist and are finite.     

Nonlocal hydrodynamic equations and BBGKY hierarchy reduction for hard spheres

We discuss the derivation of the kinetic equation for a classical system of hard spheres based on an infinite sequence of equations for distribution functions in the BBGKY hierarchy case. It is well known that the assumption of full synchronization of all distributions leads to certain problems in describing the “tails” of the autocorrelation functions and some other correlation effects with medium or high density. We show how to avoid these difficulties by maintaining the explicit form of time-dependent dynamic correlations in the BBGKY closure scheme. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 3, pp. 394–399, September, 1999.     

A general form of the equations for turbulent flows with a passive scalar

A general tensor form of the first- and second-moment equations for turbulent flow with a passive scalar in a non-Cartesian system is given. For second-order modeling, the unclosed terms (for example, the third moments, pressure correlations, etc.) depend on their model forms. A set of model forms for the various unclosed terms are also suggested and presented with a general tensor form in any system. As an example, we present all the equations and suggested models in cylindrical coordinates, which may be useful for many applications.     

Hardy–Sobolev inequalities in half-space and some semilinear elliptic equations with singular coefficients

We study some semilinear elliptic equations with singular coefficients which relate to some Hardy–Sobolev inequalities. We obtain some existence results for these equations and give a theorem for prescribing the Palais–Smale sequence for these equations. Moreover, we find some interesting connections between these equations and some semilinear elliptic equations in hyperbolic space. Using these connections, we obtain many new results for these equations.     

Two-Time Temperature Green's Functions in Kinetic Theory and Molecular Hydrodynamics: III. Taking the Interaction of Hydrodynamic Fluctuations into Account

We describe a systematic method for constructing equations for Green's functions in molecular hydrodynamics and kinetics problems. The method allows consecutively accounting for the contribution to the generalized kinetic coefficients due to the interaction of two, three, and more hydrodynamic modes. In contrast to the standard perturbation theory in the coupling constant, the consecutive approximations are taken with respect to the degree of higher correlations described by irreducible functions.     

Some Results on Correlation Matrices for Interest Rates

In this paper we systematize and develop some theoretical results about shift, slope and curvature for correlations matrices of interest rates. We provide a general investigation on the relations among some standard features of correlation models for interest rates and the existence of shift, slope and curvature. Our results show how their presence, excluding some peculiar behavior strictly related to low dimensions, can not be directly connected to the usual assumptions on the interest rates correlations structure. We provide some estimates of the distance between a shift and the vector, named pure shift, having all entries equal. We prove also that in a two-factor framework shift and slope are sufficient to justify the usual properties of correlations between rates.     

Tensor absolute value equations

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.     

A note on the Cauchy problem of the generalized Davey–Stewartson equations

In this note, we establish some local and global existence results for the Cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear Schrödinger equations and the Davey–Stewartson equations. These results improve some previously obtained results by some other authors when they are restricted to certain special equations.     

Correlation propagation for uncertainty analysis of structures based on a non-probabilistic ellipsoidal model

Traditional non-probabilistic methods for uncertainty propagation problems evaluate only the lower and upper bounds of structural responses, lacking any analysis of the correlations among the structural multi-responses. In this paper, a new non-probabilistic correlation propagation method is proposed to effectively evaluate the intervals and non-probabilistic correlation matrix of the structural responses. The uncertainty propagation process with correlated parameters is first decomposed into an interval propagation problem and a correlation propagation problem. The ellipsoidal model is then utilized to describe the uncertainty domain of the correlated parameters. For the interval propagation problem, a subinterval decomposition analysis method is developed based on the ellipsoidal model to efficiently evaluate the intervals of responses with a low computational cost. More importantly, the non-probabilistic correlation propagation equations are newly derived for theoretically predicting the correlations among the uncertain responses. Finally, the multi-dimensional ellipsoidal model is adopted again to represent both uncertainties and correlations of multi-responses. Three examples are presented to examine the accuracy and effectiveness of the proposed method both numerically and experimentally.     

无限滞后测度泛函微分方程的平均化

本文研究了无限滞后测度泛函微分方程的平均化.利用广义常微分方程的平均化方法,在无限滞后测度泛函微分方程可以转化为广义常微分方程的基础上,获得了这类方程的周期和非周期平均化定理,推广了一些相关的结果.     

Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order

In this paper, we present several methods of judging shape of the solitary wave and solution formulae for some nonlinear evolution equations by means of Lienard equations. Then, using the judgement methods and solution formulae, we obtain solutions of the solitary wave for some of important nonlinear evolution equations, which include generalized modified Boussinesq, generalized nonlinear wave, generalized Fisher, generalized Klein-Gordon and generalized Zakharov equations. Some new solitary-wave solutions are found for the equations.     

Semismooth Newton Methods for Solving Semi-Infinite Programming Problems

In this paper we present some semismooth Newton methods for solving the semi-infinite programming problem. We first reformulate the equations and nonlinear complementarity conditions derived from the problem into a system of semismooth equations by using NCP functions. Under some conditions a solution of the system of semismooth equations is a solution of the problem. Then some semismooth Newton methods are proposed for solving this system of semismooth equations. These methods are globally and superlinearly convergent. Numerical results are also given.