Statistical Distributions Boltzmann Distribution Function

Plasma-chemical reaction rates depend on the probability of relevant elementary processes from a fixed quantum-mechaiucal state with fixed energy, which was considered in the previons chapter, and the nnmber density of particles with this energy and in this particular qnantnm-mechanical state, which is to be considered in this chapter. A straightforward determination of the particles distribution in plasma over different energies and different quantum-mechaitical states is related to detailed physical kinetics (see Fridman Keimedy, 

However, wherever possible, the application of quasi-equilibrium statistical distributions is the easiest and clearest way to describe the kinetics and thermodynantics of plasma-chemical systems. 

Assume an isolated system with energy E that consists of N of particles in different states i  

The objective of a statistical approach is to find a distribution function of particles over the different states i, taking into account that the probabihty to find particles in these states is proportional to the number of ways in which the distribution can be arranged. Thermodynamic probability W(ni, ri2, , rii) is the probabihty to have i particles in the state 1, 2 particles in the state 2, etc.  

Here H is a normalizing factor. Let us find the most probable mnnbers of particles, in a state i, when probability (3-2) and its logarithm, 

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Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

In Boltzmann statistics, the orientational distribution function is given by... 

For a gas mixture at rest, the velocity distribution is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity v the thermal diffusivity a and Tick s diffusivity Dy respectively. 

In this expression Ep is the Fenni-energy level, which determines the population statistics. In a non-degenerate semiconductor - Ep and Ep - E are much larger than kT and the Fermi-Dirac distribution can be approximated by the Boltzmann distribution functions for the conduction and valence bands, i.e. 

From statistical mechanics, the Maxwell-Boltzmann distribution function for single particle in equilibrium state can be expressed as follows ... 

The Lattice-Boltzmann method is a numerical scheme for fluid simulations which originated from molecular dynamics models such as the lattice gas automata. In contrast to the prediction of macroscopic properties such as mass, momentum and energy by solving conservation equations, e.g. the Navier-Stokes equations, the LBM describes the fluid behaviour on a so-called mesoscopic scale [7, 19]. The basic parameter in the Boltzmann statistics is the distribution function f = f(x,, 0, which represents the number of fictitious fluid elements having the velocity at the location x and the time t. The temporal and spatial development of the distribution function is described by the Boltzmann equation in consideration of collisions between fluid elements. 

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

The quantum Boltzmann distribution only applies between allowed energy levels of the same family and each type of energy has its own characteristic partition function, that can be established by statistical methods and describes the response of a system to thermal excitation. If the total number of particles N, is known, the Boltzmann distribution may be used to calculate the number, or fraction, of molecules in each of the allowed quantum states. For any state i... 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

At the same time, Prigogine and his co-workers14 15,17 developed a general theory of non-equilibrium statistical mechanics. They derived a non-Markovian evolution equation for the velocity distribution function. Their results contain a generalization of the Boltzmann equation for arbitrary concentration and coupling parameter. This generalization is the long-time limit of their evolution equation. 

We will not prove the Arrhenius relationship here, but it falls out nicely from statistical thermodynamics by considering that all molecules in a reaction must overcome an activation energy before they react and form products. The Boltzmann distribution tells us that the fraction of molecules with the required energy is given by tx (—Ea/RT), which leads to the functional dependence shown in Eq. (3.12). 

The Boltzmann //-theorem generalizes the condition that with a state ol a system represented by its distribution function /. a quantity H. defined as the statistical average of In /, approaches a minimum when equilibrium is reached. This conforms lo the Boltzmann hypothesis of distribution in the above in that S = —kH accounts for equilibrium as a consequence of collisions which change the distribution toward that of equilibrium conditions. 

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 ... 

Entropy is a measure of the degree of randomness in a system. The change in entropy occurring with a phase transition is defined as the change in the system s enthalpy divided by its temperature. This thermodynamic definition, however, does not correlate entropy with molecular structure. For an interpretation of entropy at the molecular level, a statistical definition is useful. Boltzmann (1896) defined entropy in terms of the number of mechanical states that the atoms (or molecules) in a system can achieve. He combined the thermodynamic expression for a change in entropy with the expression for the distribution of energies in a system (i.e., the Boltzman distribution function). The result for one mole is ... 

The Fermi-Dirac and Maxwell-Boltzmann statistical distribution functions are widely used in semiconductor physics, with the latter commonly used as an approximation to the former. The point of this problem is to make you familiar with these distribution functions their forms, their temperature dependencies, and under what conditions they become interchangeable. Throughout this problem, use the energy of silicon s valence band (Evb) as the zero of your energy scale. 

Fig. V-2.—Distribution functions for Fermi-Dirac statistics (a) Maxwell-Boltzmann statistics (b) and Finstoin-Bose statistics (c).

In formula (4.11), as in (2.15), there are certain quantities a and T which are constant as far as the momenta are concerned, but which might vary from point to point of space. We can investigate their variation just as we did for the Boltzmann statistics in Sec. 3. The formula (3.3) for the change of the distribution function with time on account of the action of external forces holds for the Einstein-Bose and Fermi-Dirac statistics just as for the Boltzmann statistics, and leads to a formula very similar to Eq. (3.6) which must be satisfied for equilibrium. The only difference comes on account of the different form in which we have expressed the constants in Eq. (4.11). Demanding as before that the relation like (3.6) must hold independent of momenta, we find that the temperature must be independent of position, and that the constant a of Eq. (4.11) must be given by... 

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