Number density function continuous

We represent the NMR relaxation distribution by the continuous number density function, P( t), of characteristic relaxation time t. Our measurements correspond to a series of CPMG echoes, represented by... 

The Ho and White (1994) analysis is based on the idea of a constant spacing 8 1/8 then represents the (very large) number of individual components that are in reality present in the mixture over a unit segment of the label axis. In a strict interpretation of the continuous description, 6 = 0 (this is, essentially, De Donder s 1931 approach) but of course reality is a different matter. One can define a number density function i(x) such that the number of individual compounds in any interval [xi,X2 is... 

To follow the dynamic evolution of PSD in a particulate process, a population balance approach is commonly employed. The distribution of the droplets/particles is considered to be continuous in the volume domain and is usually described by a number density function, (v, t). Thus, n(v, f)dv represents the number of particles per unit volume in the differential volume size range (v, v + dv). For a dynamic particulate system, undergoing simultaneous particle breakage and coalescence, the rate of change of the number density function with respect to time and volume is given by the following non-linear integro-differential population balance equation (PBE) [36] ... 

Consider a continuous crystallizer of volume V, as shown in Figure 6.4.2(a). A feed stream having a particle (crystal) number density function ra/(rp) (which is also the population density function), volumetric flow rate Qf and species i mass concentration enters the crystallizer continuously. Product stream 1, having a particle (crystal) number density function n (tp), volumetric flow rate Qi and species i mass concentration Pf, leaves the crystallizer continuously. The particle (crystal) number density function n rp) in the well-mixed crystallizer is the same throughout the crystallizer. The macroscopic population balance equation for a stirred tank separator may be written using equations (6.2.60) and (6.2.61) as follows ... 

These source term expressions contain the continuous particle number density function, /(, r, t). However, in order to close the equation the source terms must be expressed entirely in terms of the dependent variable j(r, f). The fixed pivot technique is based on the idea of birth modification. The pivot concentrates the particles in the interval at a single representative point. The number density distribution function/i(, r, t) can be approximated as ... 

The constants K depend upon the volume of the solvent molecule (assumed to be spherica in slrape) and the number density of the solvent. ai2 is the average of the diameters of solvent molecule and a spherical solute molecule. This equation may be applied to solute of a more general shape by calculating the contribution of each atom and then scaling thi by the fraction of fhat atom s surface that is actually exposed to the solvent. The dispersioi contribution to the solvation free energy can be modelled as a continuous distributioi function that is integrated over the cavity surface [Floris and Tomasi 1989]. 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

A demand density is by definition a continuous density function defined over the set of all nonnegative numbers, thus demands are always positive or zero. 

The density function of the sum of two independent continuous random variables is computed by the convolution of the two probability densities. Loosely speaking, two random numbers are independent, if they do not influence each other. Unfortunately, convolutions are obviously important but not convenient to calculate. 

On the practical side, we note that nature provides a number of extended systems like solid metals [29, 30], metal clusters [31], and semiconductors [30, 32]. These systems have much in common with the uniform electron gas, and their ground-state properties (lattice constants [29, 30, 32], bulk moduli [29, 30, 32], cohesive energies [29], surface energies [30, 31], etc.) are typically described much better by functionals (including even LSD) which have the right uniform density limit than by those that do not. There is no sharp boundary between quantum chemistry and condensed matter physics. A good density functional should describe all the continuous gradations between localized and delocalized electron densities, and all the combinations of both (such as a molecule bound to a metal surface a situation important for catalysis). 

Both the number and weight basis probability density functions of final product crystals were found to be expressed by a %2-function, under the assumption that the CSD obtained by continuous crystallizer is controlled predominantly by RTD of crystals in crystallizer, and that the CSD thus expressed exhibits the linear relationships on Rosin-Rammler chart in the range of about 10-90 % of the cumulative residue distribution. 

Enlarging the domain of definition of Eqn (1) to all positive n, one could assume the minimum of the density functional E lp], i.e. the ground-state energy Eo n) for a given external potential u(r), to be a continuous and even a differentiable function of the number of electrons n. From the Lagrange multiplier theory, it would further follow that... 

The chemical properties of particles are assumed to correspond to thermodynamic relationships for pure and multicomponent materials. Surface properties may be influenced by microscopic distortions or by molecular layers. Chemical composition as a function of size is a crucial concept, as noted above. Formally the chemical composition can be written in terms of a generalized distribution function. For this case, dN is now the number of particles per unit volume of gas containing molar quantities of each chemical species in the range between ft and ft + / ,-, with i = 1, 2,..., k, where k is the total number of chemical species. Assume that the chemical composition is distributed continuously in each size range. The full size-composition probability density function is... 

Parr and collaborators [8-12] showed how Fukui s frontier-orbital concept could be grounded in a rigorous many-electron theory, density-functional theory (DFT) [8,16-18], They used the ensemble formulation of DFT to introduce the expectation value Jf of the total electron number as a continuous variable. They then defined the Fukui functions... 

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