Particle equations, distinguished

Note that equation (13) shows that there is a hierarchy of differential equations solution of the Smith-Ewart equations provides the boundary conditions for the singly distinguished particle equations these in turn provide the boundary conditions for the doubly distinguished particle equations. 

The quantum analogs of the phase space distribution function and the Lionville equation discussed in Section 1.2.2 are the density operator and the quantum Lionville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles... 

In mechanics, a body is an object made up of several elements and its total mass is the sum of the mass of each part. This is in contradistinction with the notion of particle, which is an elementary entity endowed with an indissociable mass (without losing the nature of the particle). For distinguishing the two systems, the scheme in the case study abstract shows a body which is a cluster of particles the variability of the entity number [the momentum (impulse) which is the sum of all momentum associated with every particle] and, in the equations, the variables featuring the impulse (momentum) and the mass are in uppercase for a body and in lowercase for a particle. 

The continuous and erratic motion of individual particles (i.e., pollen grains) as the result of random collisions with the adjoining molecules of fluid (water) was observed by the botanist Robert Brown in 1827. The term Brownian diffusion is used for colloidal particles to distinguish it from solute molecular diffusion. However, both are end members of a continuum of particle sizes and a fundamental consequence of kinetic theory is that all particles have the same average translational kinetic energy. The average particle velocity increases with decreasing mass (Shaw, 1978). The Stokes-Einstein s equation for particle diffusivity is based on this concept. It is... 

Equation 9.4 yields the probabiUty density of any particular configuration of atoms in space V. Let us momentarily assume the particles are distinguishable. The probability of finding... 

The motion of particles in a fluid is best approached tlirough tire Boltzmaim transport equation, provided that the combination of internal and external perturbations does not substantially disturb the equilibrium. In otlier words, our starting point will be the statistical themiodynamic treatment above, and we will consider the effect of botli the internal and external fields. Let the chemical species in our fluid be distinguished by the Greek subscripts a,(3,.. . and let f (r, c,f)AV A be the number of molecules of type a located m... 

Q is given by Equation (6.4) for a system of identical particles. We shall ignore any normalisation constants in our treatment here to enable us to concentrate on the basics, and so it does not matter whether the system consists of identical or distinguishable particles. We also replace the Hamiltonian by the energy, E. The internal energy is obtained via Equation (6.20) ... 

There have been few discussions of the specific problems inherent in the application of methods of curve matching to solid state reactions. It is probable that a degree of subjectivity frequently enters many decisions concerning identification of a best fit . It is not known, for example, (i) the accuracy with which data must be measured to enable a clear distinction to be made between obedience to alternative rate equations, (ii) the range of a within which results provide the most sensitive tests of possible equations, (iii) the form of test, i.e. f(a)—time, reduced time, etc. plots, which is most appropriate for confirmation of probable kinetic obediences and (iv) the minimum time intervals at which measurements must be made for use in kinetic analyses, the number of (a, t) values required. It is also important to know the influence of experimental errors in oto, t0, particle size distributions, temperature variations, etc., on kinetic analyses and distinguishability. A critical survey of quantitative aspects of curve fitting, concerned particularly with the reactions of solids, has not yet been provided [490]. 

The void fraction should be the total void fraction including the pore volume. We now distinguish Stotai from the superficial void fraction used in the Ergun equation and in the packed-bed correlations of Chapter 9. The pore volume is accessible to gas molecules and can constitute a substantial fraction of the gas-phase volume. It is included in reaction rate calculations through the use of the total void fraction. The superficial void fraction ignores the pore volume. It is the appropriate parameter for the hydrodynamic calculations because fluid velocities go to zero at the external surface of the catalyst particles. The pore volume is accessible by diffusion, not bulk flow. 

If H(, 2) and H(2, 1) were to differ, then the corresponding Schrodinger equations and their solutions would also differ and this difference could be used to distinguish between the two particles. 

It should be pointed out that the n particles have been assumed here to be distinguishable. However, in an ideal gas the molecules cannot be distinguished, as their positions are random in space. Therefore, to be applied to the case of an ideal gas, Eqs. (24) and (25) should be divided by n . Equation (26) becomes simply,... 

In this chapter we first discuss the equations of motion for particles at low Re. Semiempirical extensions beyond the creeping flow regime are then considered. It is useful to distinguish two general kinds of unsteady motion ... 

Microscopic Identification Models. Many different optical and chemical properties of single aerosol particles can be measured by microscopic identification and classification in order to distinguish particles originating in one source type from those originating in another. The microscopic analysis receptor model takes the form of the chemical mass balance equations presented in Equation 1. 

Chemical reactions are classified usually as diffusion-controlled, whose rate is limited by a reactant spatial approach to each other, and reaction-controlled (kinetic stage), whose rate is limited by a reaction elementary event. For systems with ideal reactant mixing considered in Section 2.1.1, there is no mechanism of reactant mutual approach. On the other hand, the kinetic equations (2.1.40) distinguish between reaction in physically infinitesimal volumes and the distant reactant motion in a whole reaction volume. In the absence of reaction particle diffusion is described by equation... 

In our approach [1, 2] termed the dynamic method the complex susceptibility x = x — ix" is determined by a law of undamped motion of a dipole in a given potential well and by dissipation mechanism often described as stosszahlansatz in the underlying kinetic or Boltzmann equation. In this review we shall refer to this (dynamic) method as the ACF method, since it is actually based on calculation of the spectrum of the dipolar autocorrelation function (ACF). Actually we use a one-particle approximation, in which the form of an employed potential well (being in many cases rectangular or close to it) is taken a priori. Correlation of the particles coordinates is characterized implicitly by the Kirkwood correlation factor g, its value being taken from the experimental data. The ACF method is simple and effective, because we do not employ the stochastic equations of motions. This feature distinguishes our method from other well-known approaches—for example, from those described in books [13, 14]. 

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