Statistical distribution laws

Debye and Huckel applied the Boltzmann statistical distribution law and the Poisson equation for electrostatics in the model above (1,6, 10). In the calculations using the model above they considered one particular ion (the reference ion, or central ion) with... 

No theories which liken chemical reactions to processes of hydrodynamic flow, or which introduce conceptions such as friction and lubrication, are of much help. Chemical reactions require a statistical interpretation. Molecules capable of even transient existence represent configurations with a minimum potential energy. The products of a reaction correspond to a lower minimum than the initial substances, and the two minima are separated by a maximum. This maximum corresponds to a transition state, access to which is possible only for those molecules which acquire actimtion energy E). If the activation energy is known, the probability that molecules acquire it by collision or otherwise is calculable from the statistical distribution laws. The need for activation explains the factor occurring in aU expressions for reaction... 

A simple relation will be derived between the probabilities for spontaneous and stimulated emission and absorption of radiation using well-known statistical distribution laws. Consider a system such as that illustrated in Fig.4.3 with two energy levels, E and E2, populated by and N2 atoms, respectively. Three radiative processes can occur between the levels, as discussed above. In the figure the processes are expressed using the so-called Einstein coefficients 6 2, B21 and A21, which are defined such that the rate of change in the population numbers is... 

Case 1. The particles are statistically distributed around the ring. Then, the number of escaping particles will be proportional both to the time interval (opening time) dt and to the total number of particles in the container. The result is a first-order rate law. 

For gas-phase molecules the assumption of electronic adiabaticity leads to the usual Bom-Oppenheimer approximation, in which the electronic wave function is optimized for fixed nuclei. For solutes, the situation is more complicated because there are two types of heavy-body motion, the solute nuclear coordinates, which are treated mechanically, and the solvent, which is treated statistically. The SCRF procedures correspond to optimizing the electronic wave function in the presence of fixed solute nuclei and for a statistical distribution of solvent coordinates, which in turn are in equilibrium with the average electronic structure. The treatment of the solvent as a dielectric material by the laws of classical electrostatics and the treatment of the electronic charge distribution of the solute by the square of its wave function correctly embodies the result of... 

For the statistical copolymer the distribution may follow different statistical laws, for example, Bemoullian (zero-order Markov), first- or second-order Markov, depending on the specific reactants and the method of synthesis. This is discussed further in Secs. 6-2 and 6-5. Many statistical copolymers are produced via Bemoullian processes wherein the various groups are randomly distributed along the copolymer chain such copolymers are random copolymers. The terminology used in this book is that recommended by IUPAC [Ring et al., 1985]. However, most literature references use the term random copolymer independent of the type of statistical distribution (which seldom is known). 

Bills (7) has applied an adaptation of this law to solid propellants and propellant-liner bonds for discrete, constantly imposed stress levels considering U to be the time at the ith stress level and tfi the mean time to failure at the ith stress level. A probability distribution function P was included to account for the statistical distribution of failures. For cyclic stress tests the time is the number of cycles divided by the frequency, and the ith loading is the amplitude. The empirical relationship... 

In the discussion of many properties of substances it is necessary to know the distribution of atoms or molecules among their various quantum states. An example is the theory of the dielectric constant of a gas of molecules with permanent electric dipole moments, as discussed in Appendix IX. The theory of this distribution constitutes the subject of statistical mechanics, which is presented in many good books.1 In the following paragraphs a brief statement is made about the Boltzmann distribution law, which is a basic theorem in statistical mechanics. 

Statistical mechanics (cf. Chapter 13) suggests an alternative way to extract temperature-like properties from molecular energy distributions. According to the classical Boltzmann distribution law, the number N(E) of molecules having energy E is proportional under equilibrium conditions to the Boltzmann factor eE kT,... 

With this bold stroke, Boltzmann escaped the futile attempt to describe microscopic molecular phenomena in terms of then-known Newtonian mechanical laws. Instead, he injected an essential probabilistic element that reduces the description of the microscopic domain to a statistical distribution of microstates, i.e., alternative microscopic ways of partitioning the total macroscopic energy U and volume V among the unknown degrees of freedom of the molecular domain, all such partitionings having equal a priori probability in the absence of definite information to the contrary. 

A typical processive enzyme is terminal transferase. It adds on deoxynucleo-side monophosphates randomly to exposed 3 -hydroxyl termini so that the final products are formed in a statistical distribution. The distribution follows Poisson s law.5 Suppose that the enzyme adds on an average of x residues per chain. Then the probability of a particular chain having k residues added [i.e., p(k)] is given by... 

This fundamental relation is called Boltzmann s distribution law after the creator of statistical mechanics, Ludwig Boltzmann (1844-1906), Professor of Physics in Leipzig, and k is called Boltzmann s constant. 

In conclusion we must mention that a necessary condition for the validity of Eq, (3), and consequently of other formulas derived from Eq. (3) is that Ni < 1 for the state (or states) of lowest energy and a fortiori for all other states, When this inequality does not hold. Boltzmann s distribution law must be replaced by a more general and more precise distribution law, either that of Fermi and Dirac or that of Bose and Einstein according to the nature of the molecules. See also Statistical Mechanics. 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

Expressions for a number of main moments of the spectrum may be utilized to develop a new version of the semi-empirical method. Evaluation of the statistical characteristics of spectra with the help of their moments is also useful for studying various statistical peculiarities of the distribution of atomic levels, deviations from normal distribution law, etc. Such a statistical approach is also efficient when considering separate groups of levels in a spectrum (e.g. averaging the energy levels with respect to all quantum numbers but spin), when studying natural widths or lifetimes of excited levels, etc. 

Many interesting and useful concepts follow from classical statistical con side rations (eg, the Boltzmann distribution law) and their later modifications to take into account quantum mechanical effects (Bose-Einstein and Fermi-Dirac statistics). These concepts are quite beyond the scope of the present article, and the reader should consult Refs 14 16. A brief excursion into this area is appropriate, however. A very useful concept is the so-called partition function, Z, which is defined as ... 

In the result of experiments the values of time delays were obtained and analyzed by the tools of mathematical statistics on accordance to the normal distributive law, they were tested on homogeneity and their belonging to the same universal set. 

We now proceed to develop a specific expression for the rate constant for reactants where the velocity distributions /a( )(va) and /B(J)(vB) for the translational motion are independent of the internal quantum state (i and j) and correspond to thermal equilibrium.4 Then, according to the kinetic theory of gases or statistical mechanics, see Appendix A.2.1, Eq. (A.65), the velocity distributions associated with the center-of-mass motion of molecules are the Maxwell-Boltzmann distribution, a special case of the general Boltzmann distribution law ... 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

We can show, as we did with the Fermi-Dirac statistics, that the distribution (6.5) approaches the Maxwell-Boltzmann distribution law at high temperatures. It is no easier to make detailed calculations with the Einstein-Bose law than with the Fermi-Dirac distribution, and on account of its smaller practical importance we shall not carry through a detailed... 

The challenge is therefore to find a theoretical expression for these scaling laws. It will in any case depend upon scaling laws for the statistical distribution of fundamental geometrical reservoir properties. It will also depend upon these hidden processes that arise because of the nonlinear nature of movable boundary flows (quite apart from nonlinearities intrinsic to the continuum relations themselves). There have been some remarkable pioneering attempts to predict continuum properties of porous media from fundamental parameters, mainly by chemical engineers (of whom I wish to single out Howard Brenner and co-workers) and physicists, but they have as yet made little impact on the oil industry. 

The volumetric charge density is of interest in the study of ionic solutions, in which one can calculate the charge density around a specific ion. This is done by using the Poisson equation, based on electrostatic electric fields or by Boltzman distribution law of classical statistic mechanics. For the simpler case of dilute solutions this approach yields the expression p =... 

Many of the known laws of statistical distribution (of Gauss, Maxwell, Pirson, etc.) can be derived from this formula substituting the constants with the appropriate values. 

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