Density basic equation

The basic equation [8] is tlie equation of motion for the density matrix, p, given in equation (B2.4.25), in which H is the Hamiltonian. 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

Basic Equations of Density-functional Perturbation Theory... 

The basic equation for water flow through saturated porous media was developed by Flenry Darcy in 1856 to calculate the flow of water through sand filters. This equation has been found to be valid for the flow of liquids through porous media when adjusted for the viscosity and density of the liquid. The original form of the equation, designed to calculate discharge is as follows ... 

For completeness, we need to point out that the name density functional theory is not solely applied to the type of quantum mechanics calculations we have described in this chapter. The idea of casting problems using functionals of density has also been used in the classical theory of fluid thermodynamics. In this case, the density of interest is the fluid density not the electron density, and the basic equation of interest is not the Schrodinger equation. Realizing that these two distinct scientific communities use the same name for their methods may save you some confusion if you find yourself in a seminar by a researcher from the other community. 

Vacuum pumps are used to reduce the gas pressure in a certain volume and thus the gas density (see equation 1.5). Consequently consider the gas particles need to be removed from the volume. Basically differentiation is made between two classes of vacuum pumps ... 

The basic equations, as recently slightly updated from further experience [51], are now briefly reviewed. Consider a molecule (A) and let p be the electron density in an elementary volume V] centered at point k), with charge qj = For a... 

As a simple but nontrivial example we examine the dynamics of small fluctuations around anisotropic, homogeneous states by linearizing the basic equations (6.26) (6.29). From Eq. (4.39), the force density ) = — LidYlijjdXi is linearized as... 

The basic equations of the Kamlet-Jacobs method for CHNO expls with packing density p0 >1 are ... 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

Chapter 5 deals with derivation of the basic equations of the fluctuation-controlled kinetics, applied mainly to the particular bimolecular A + B 0 reaction. The transition to the simplified treatment of the density fluctuation spectrum is achieved by means of the Kirkwood superposition approximation. Its accuracy is estimated by means of a comparison of analytical results for some test problems of the chemical kinetics with the relevant computer simulations. Their good agreement permits us to establish in the next Chapters the range of the applicability of the traditional Waite-Leibfried approach. 

Equations 9.3-22 and 9.3-26 are the basic equations of the melting model. We note that the solid-bed profile in both cases is a function of one dimensionless group ijj, which in physical terms expresses the ratio of the local rate of melting per unit solid-melt interface JX /X to the local solid mass flux into the interface Vszps, where ps is the local mean solid bed density. The solid-bed velocity at the beginning of melting is obtained from the mass-flow rate... 

Armed with these basic equations, classical partition functions and state densities for various types of motion may be evaluated either directly or via the quantum results. 

Two-phase polymerization is modeled here as a Markov process with random arrival of radicals, continuous polymer (radical) growth, and random termination of radicals by pair-wise combination. The basic equations give the joint probability density of the number and size of the growing polymers in a particle (or droplet). From these equations, suitably averaged, one can obtain the mean polymer size distribution. 

Poisson equation — In mathematics, the Poisson equation is a partial differential equation with broad utility in electrostatics, mechanical engineering, and theoretical physics. It is named after the French mathematician and physicist Simoon-Denis Poisson (1781-1840). In classical electrodynamics the Poisson equation describes the relationship between (electric) charge density and electrostatic potential, while in classical mechanics it describes the relationship between mass density and gravitational field. The Poisson equation in classical electrodynamics is not a basic equation, but follows directly from the Maxwell equations if all time derivatives are zero, i.e., for electrostatic conditions. The corresponding ( first ) Maxwell equation [i] for the electrical field strength E under these conditions is... 

An alternate approach for estimating maximum allowable velocities has been presented by Fair (see reference given in footnote for preceding paragraph) which is based on data obtained with sieve-tray and other types of finite-stage columns and takes into account the effect of surface tension of the liquid in the column, the ratio of the liquid flow rate to the gas flow rate, gas and liquid densities, and dimensions and arrangement of the contactor. In this method, the basic equation for the maximum allowable vapor velocity, equiva-... 

Having discussed the basic equations of the density description and their application to atomic ions we turn now to the much more difficult problem of molecules. Even the simplest density description afforded by the TF theory presents severe computational problems for multicentre problems, as well as some conceptual difficulties on which we shall attempt to throw light in the ensuing discussion. 

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