Exact Equations

Two exact equations of state can be derived from equation (A2,1.331 and equation (A2,1.341... 

Although the exact equations of state are known only in special cases, there are several usefid approximations collectively described as mean-field theories. The most widely known is van der Waals equation [2]... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

Vugacity Coefficients. An exact equation that is widely used for the calculation of fugacity coefficients and fugacities from experimental pressure—volume—temperature (PVT) data is... 

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

As mentioned above, the numerical solution of exact equations breaks down for low flame speeds, where the strength of the leading shock approaches zero. To complete the entire range of flame speeds, Kuhl et al. (1973) suggested using the acoustic solutions by Taylor (1946) as presented earlier in this section. Taylor (1946) already noted that his acoustic approach is not fully compatible with the exact solution, in the sense that they do not shade into one another smoothly. In particular, the near-piston and the near-shock areas in the flow field, where nonlinear effects play a part, are poorly described by acoustic methods. In addition to these imperfections, the numerical character of Kuhl etal. (1973) method inspired various authors to design approximate solutions. These solutions are briefly reviewed. 

Reid (1976) and many other authors give pure propane a superheat temperature limit of 53 C at atmospheric pressure. The superheat temperature limit calculated from the Van der Waals equation is 38°C, whereas the value calculated from the Redlich-Kwong equation is S8°C. These values indicate that, though an exact equation among P, V, and 7 in the superheat liquid region is not known, the Redlich-Kwong equation of state is a reasonable alternative. 

Altliough tlie e.xact equation for combining risk probabilities includes temis for joint risks, tlie difference between tlie exact equation and tlie approximation described above is negligible for total cancer risks of less tlian 0.1. 

Nielsen suggested a more exact equation relating viscosity and elasticity modulus [27] ... 

There is no doubt, however, that the two so-called constants, a and b, are functions of temperature, and perhaps also of pressure. Numerous attempts have been made to express this dependence, particularly that on temperature, and so to obtain more exact equations. Clausius (1880) wrote ... 

Exact Equations of Motion in Terms of One-particle Functions... 

In this volume dedicated to Yngve Ohm we feel it is particularly appropriate to extend his ideas and merge them with the powerful practical and conceptual tools of Density Functional Theory (6). We extend the formalism used in the TDVP to mixed states and consider the states to be labeled by the densities of electronic space and spin coordinates. (In the treatment presented here we do not explicitly consider the nuclei but consider them to be fixed. Elsewhere we shall show that it is indeed straightforward to extend our treatment in the same way as Ohm et al. and obtain equations that avoid the Bom-Oppenheimer Approximation.) In this article we obtain a formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electtons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, we also obtain exact equations for Aese one-particle functions. 

The objectives in the following talk are to discuss qualitatively the time behaviour of exhalation in closed cans and then compare these qualitative reasonings with the exact results of the time-dependent diffusion theory. The theoretical results are, to a large extent, presented in diagram form only. Readers interested in the exact equations behind these diagrams are referred to the paper by Lamm (Lamm et al., 1983). 

This equation is accurate at low damping (A < 1), but the error becomes large at high damping. More exact equations have been discussed by Struik (II) and Nielsen (4). The standard ASTM test is D2236-69. 

For an anchor compound B (say p-nitroaniline), whose ionization ratios are measurable in dilute acid, we can write the thermodynamically exact equation (14) ... 

When an equation describes a system exactly but the equation cannot be solved, there are two general approaches that are followed. First, if the exact equation cannot be solved exacdy, it may be possible to obtain approximate solutions. Second, the equation that describes the system exactly may be modified to produce a different equation that now describes the system only approximately but which can be solved exactly. These are the approaches to solving the wave equation for the helium atom. 

The relations between the polarization chemical electronic responses (fpn r), 7]p. ..) and the polarizability responses Xn are similar to the exact equations we derived earlier when fp(r) is defined by Equation 24.50 [26]. For instance, the expression of the first nonlinear hardness 17 is obtained by deriving the linear equation (Equation 24.50) relative to A, and by using again the chain mle for functional derivatives ... 

STANJAN The Element Potential Method for Chemical Equilibrium Analysis Implementation in the Interactive Program STANJAN, W.C. Reynolds, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1986. A computer program for IBM PC and compatibles for making chemical equilibrium calculations in an interactive environment. The equilibrium calculations use a version of the method of element potentials in which exact equations for the gas-phase mole fractions are derived in terms of Lagrange multipliers associated with the atomic constraints. The Lagrange multipliers (the element potentials ) and the total number of moles are adjusted to meet the constraints and to render the sum of mole fractions unity. If condensed phases are present, their populations also are adjusted to achieve phase equilibrium. However, the condensed-phase species need not be present in the gas-phase, and this enables the method to deal with problems in which the gas-phase mole fraction of a condensed-phase species is extremely low, as with the formation of carbon particulates. 

The calculations were later reproduced by Kajiwara [97] following a different route. For the exact equation the original paper may be consulted. 

That is, the order of differentiation is immaterial for any function of two variables. Therefore, if dL is exact. Equation (2.23) is correct [8]. 

The starting point of classical statistical mechanics is the exact equation of evolution of the distribution function p in phase space the Liouville equation, which Prigogine always wrote in the form... 

In a first class of approximations, the solution is sought of a simpler set of equations rather than the exact equations. Under the Hartree—Fock (i.e., HE) approximation, - the function of 2>n variables is reduced to n functions, which are referenced as... 

Note that the expansion is done in terms of the tip wavefunctions, which form a complete and orthogonal set of states. Substituting Eq. (2.28) for Eq. (2.24), we obtain the exact equation for Cv(r) ... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

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