Distribution function, continuous

A continuous distribution function is a mathematical function which gives N as a function of M. This is the most general way of describing the... 

Since we have ended up with a continuous distribution function, it is more appropriate to multiply both sides of Eq. (1.34) by dx and to say that the equation gives the probability of x values between x and x + dx for n steps of length 1. 

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

For a continuous distribution function f(t), the kth moment about the origin is defined by ... 

Equation (8) can be generalized by considering more than two kinds of centers. If there are many different kinds of active centers, their relative proportions may be represented approximately by a continuous distribution function, as has been suggested by Constable (2). In particular, it may be assumed that their relative numbers decrease exponentially with decreasing activation energy A . Thus the number of active centers dn involving activation energies between A and AE -+ dAE can be assumed to be... 

The ASEP/MD method, acronym for Averaged Solvent Electrostatic Potential from Molecular Dynamics, is a theoretical method addressed at the study of solvent effects that is half-way between continuum and quantum mechanics/molecular mechanics (QM/MM) methods. As in continuum or Langevin dipole methods, the solvent perturbation is introduced into the molecular Hamiltonian through a continuous distribution function, i.e. the method uses the mean field approximation (MFA). However, this distribution function is obtained from simulations, i.e., as in QM/MM methods, ASEP/MD combines quantum mechanics (QM) in the description of the solute with molecular dynamics (MD) calculations in the description of the solvent. 

If the states span a continuous, rather than a discrete, energy range, we can introduce a continuous distribution function, f (e) dr., the fraction of molecules that have energy in a range dr around e ... 

The adsorption sites on the metal oxide surface may be described by continuous distribution function, then the fraction of the total adsorption is expressed as follows ... 

This paper is mainly concerned with the excess free energy (A FM)j- y but we shall first give a short discussion of theories of the central force term (A F( 9)t y which has been evaluated by a variety of methods. If the continuous distribution function is replaced by a lattice distribution as in the previous section and if interactions between non-neighbouring sites are neglected, (5.3) becomes... 

Just as for other relaxation processes, the continuous distribution function of T and logarithmic distribution function... 

Introducing now continuous distribution functions for each m-s couple ... 

For certain continuous distribution functions and a Langmuir type local isotherm it is possible to solve Eq. (79) analytically. The thus obtained overall isotherm equations are normalised Freundlich type equations [32, 52, 53,102,107 110], for instance ... 

Although the concepts are somewhat older, the most widely used model for describing adsorption on an energetically heterogeneous surface was first explicidy stated by Ross and Olivier [4, 5]. The model postulates that the surface of a real solid is composed of small patches of different adsorptive potential that adsorb independently of one other. The distribution of adsorptive potentials, Uq, among these patches may be represented by a continuous distribution function ... 

For the continuous distribution function, the collision rate between particles in the size ranges u to u + du and 5 to u + dii Ls given by... 

The determination of the form of ijf is carried out in two steps. First, the special form of the distribution function (7.69) is tested by substitution in the equation of coagulation for the continuous distribution function (7.67) with the appropriate collision frequency function, if the transformarion is consistent with the equation, an ordinary integrodifferential equation for as a function of t) is obtained. The next step is to find a solution of this equation subject to the integral constraints (7.70) and (7,71) and also find the limits on n(u). For some collision kernels, solutions for (tj) that satisfy these constraints may not exist. 

The kinetic equation (10.3) for the continuous distribution function becomes... 

General Dynamic Ecjuaiionfor the Continuous Distribution Function 309... 

GENERAL DYNAMIC EQUATION FOR THE CONTINUOUS DISTRIBUTION FUNCTION... 

As particle size increases (u y ). it becomes convenient to pass from the discrete distribution to the continuous distribution to carry out calculations. The transition to the continuous distribution function requires care. For the growth term, this was shown to be (Chapter 9)... 

Ttirhulem Shear Coagulation 206 Turbulent Inertial Coagulation 206 Limitations on the Analysis 207 Comparison of Collision Mechanisms 208 Equation of Coagulation Continuous Distribution Function 208... 

The constants K depend upon the volume of the solvent molecule (assumed to be spherical in shape) and the number density of the solvent. a-i2 is the average of the diameters of a solvent molecule and a spherical solute molecule. This equation may be applied to solutes of a more general shape by calculating the contribution of each atom and then scaling this by the fraction of that atom s surface that is actually exposed to the solvent. The dispersion contribution to the solvation free energy can be modelled as a continuous distribution function that is integrated over the cavity surface [Floris and Tomasi 1989]. 

Gordiets Vibrational Distribution Function in Non-Thermal Plasma. Compare the discrete Gordiets vibrational distribution with continuous distribution function (3-141). Pay special attention to the exponential decrease of the vibrational distribution functions at high vibrational energies in the case of low translational temperatures (To < ftoS). 

In more realistic terms, there is a distribntion of relaxation times, and a continuous distribution function can be derived, if required. 

Distribution functions can be classified as discontinuous or continuous. Discontinuous distribution functions are subdivided into frequency distributions and cumulative distributions. Continuous distribution functions are further classified as differential and integral distribution functions. 

Discontinuous distribution functions can, of course, be transformed into continuous distribution functions when the difference between two neighboring properties is very small compared with the whole range of property values. Frequency distributions convert to the corresponding differential distributions and cumulative distributions convert to integral distributions,... 

The lattice Boltzmann equation (LBE) is obtained as a dramatic simplification of the Boltzmann equation, (20.1) along with its associated equations, (20.2-20.4). In particular, it was discovered that the roots of the Gauss-Hermite quadrature used to exactly and numerically represent the moment integrals in (20.4), corresponds to a particular set of few discrete particle velocity directions c in the LBE [11]. Eurther-more, the continuous equihbrium distribution (20.3) is expanded in terms of the fluid velocity m as a polynomial, where the continuous particle velocity is replaced by the discrete particle velocity set ga obtained as discussed above, i.e., becomes f 1. After replacing the continuous distribution function / =f x, t) in (20.1) by the discrete distribution function/ =f x, (20.1) is integrated by considering... 

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