Applications of the Boltzmann Distribution

Let us first ask to what extent homogeneous stresses influence the mobilities of structure elements. We know that the temperature dependence of mobilities is adequately described by an Arrhenius equation, which reflects the applicability of the Boltzmann distribution for atoms in their activated states (Section 5.1.2). Let us therefore reformulate the question and ask in which way the activated states of mobile SE s are influenced by externally applied stresses and self-stresses. If we take into account the periodicity of the crystal and assume its SE s to reside in harmonic... 

While it is true that large, high-energy deformations are less likely to occur (and be observed) than small, low-energy ones, there is a serious flaw in these arguments. An ensemble of structural parameters obtained from chemically different compounds in a variety of crystal structures does not even remotely resemble a closed system at thermal equilibrium and does not therefore conform to the conditions necessary for the application of the Boltzmann distribution. It is thus misleading to draw an analogy between this distribution and those derived empirically from statistical analysis of observed deformations in crystals [20]. 

Experimental evidence for or against depopulation is not easy to find, but it has long been established that the second resonance doublets for sodium (330-5 nm) and potassium (405 nm) show identical flame reversal temperatures to the first resonance doublets. Since both second resonance doublets lie within 0-75 volt ( a 4 kT) of the ionization limit, it would appear that the depopulation is slight at this distance from the ionization limit. It is true that the conditions in the flame approximate more closely to equilibrium than to the kinetically limited state, but such evidence as there is supports the application of the Boltzmann distribution to the lower levels. 

Beyond the basic applications of the Boltzmann distribution, partieularly in statistical thermodynamics or even in quantum applieations sueh as modeling populations of atoms in laser media, they are fimdamental also in Environmental Physics by the famous modeling of the pressure with the atmosphere known as the barometric formula. 

The second method is to neglect the convection term in Eq. 9 considering that the flow velocity is small for both EOF flow and electrophoretic flow, which decouples the flow field and the EDL potential field. In the second approach, the assumption of a two-species buffer and the application of the Boltzmann distribution are commonly made in order to solve the potential field easily, which yields the well-known Poisson-Boltzmann equation ... 

Example 10.1 illustrates an application of the Boltzmann distribution law. We compute how the atmospheric pressure depends on the altitude above the earth s surface. 

In the statistical discussion of any gas containing identical molecules, cognizance must be taken of the type of statistics applicable. Often, however, we are not primarily interested in the translational motion of the molecules but only in their distribution among various rotational, vibrational, and electronic states. This distribution can usually be calculated by the use of the Boltzmann distribution law, the effect of the symmetry character being ordinarily negligible (except in so far as the sym-... 

Equation (3.18) illustrates the application of the Boltzmann Law to the distribution of molecules in the atmosphere, assuming the temperature is constant throughout. What happens when this is not so is discussed in ... 

Fermi-Dirac distribution - A modification of the Boltzmann distribution which takes into account the Pauli exclusion principle. The number of particles of energy E is proportional to [e > +l] , where p is a normalization constant, k the Boltzmann constant, and T the temperature. The distribution is applicable to a system of fermions. 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

In a plasma, the constituent atoms, ions, and electrons are made to move faster by an electromagnetic field and not by application of heat externally or through combustion processes. Nevertheless, the result is the same as if the plasma had been heated externally the constituent atoms, ions, and electrons are made to move faster and faster, eventually reaching a distribution of kinetic energies that would be characteristic of the Boltzmann equation applied to a gas that had been... 

If the conditions for Forster transfer are not applicable, then the theory must be extended. There is recently experimental evidence that coherent energy transfer participates in photosynthesis [74, 75], In this case, the participating molecules are very close together. The excited state of the donor does not completely relax to the Boltzmann distribution before the energy can be shared with the acceptor, and the transfer can no longer be described by a Forster mechanism. We will not discuss this case. There has been active discussion of coherent transfer and very strong interactions in the literature for a longer time [69], and references can be found in some more recent papers [70-72, 76, 77],... 

When an NMR experiment is performed, the application of a RFpulse orthogonal to the axis of the applied magnetic field perturbs the Boltzmann distribution, thereby producing an observable event that is governed by the Bloch equations [3]. Using a vector representation, the... 

The charge density x on any electrolyte lamina parallel to the electrode and a distance x from it can be obtained by the application of electrostatics (Poisson s equation) and the Boltzmann distribution. Similarly, one can write for the intrinsic semiconductor, Poisson s equation... 

It is convenient and useful to express the Boltzmann distribution law in two forms a quantum form and a classical form. The quantum form of the law, in its application to atoms and molecules, may be expressed as follows The relative probabilities of various quantum states of a system in equilibrium with its environment at absolute temperature T, each state being represented by a complete set of values of the quantum numbers, are proportional to the Boltzmann factor e Wn/kT, in which n represents the set of quantum numbers, Wn is the energy of the quantized state, and k is the Boltzmann constant, with value 1.3804 X 10 16 erg deg 1. The Boltzmann constant k is the gas-law constant R divided by Avogadro s number that is, it is the gas-law constant per molecule. 

All the work just mentioned is rather empirical and there is no general theory of chemical reactions under plasma conditions. The reason for this is, quite obviously, that the ordinary theoretical tools of the chemist, — chemical thermodynamics and Arrhenius-type kinetics - are only applicable to systems near thermodynamic and thermal equilibrium respectively. However, the plasma is far away from thermodynamic equilibrium, and the energy distribution is quite different from the Boltzmann distribution. As a consequence, the chemical reactions can be theoretically considered only as a multichannel transport process between various energy levels of educts and products with a nonequilibrium population20,21. Such a treatment is extremely complicated and - because of the lack of data on the rate constants of elementary processes — is only very rarely feasible at all. Recent calculations of discharge parameters of molecular gas lasers may be recalled as an illustration of the theoretical and the experimental labor required in such a treatment22,23. ... 

The concepts of equilibrium as the most probable state of a very large system, the size of fluctuations about that most probable state, and entropy (randomness) as a driving force in chemical reactions, are very useful and not that difficult. We develop the Boltzmann distribution and use this concept in a variety of applications. 

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