Boltzmann s Equation

This equation can be obtained in another way which may be more instructive. Assume that the slow step in the oxidation is the transport of cation vacancies. The positive holes may then be considered to take up their equilibrium distribution, defined by Boltzmann s equation... 

Liouville s equation, derivation of Boltzmann s equation from, 41 Littlewood, J. E., 388 Lobachevskies method, 79,85 Local methods of solution of equations, 78... 

Thirring, W. E., 598,724 Thirteen moment solution of Boltzmann s equation, 40 Thomas, C. J., 313,317 Time averages, 114 of different time functions, 132 "Time development matrix, 410 Time-ordering chronological operator, 416,603 Time reversal... 

Statistical thermodynamics provides the relationships that we need in order to bridge this gap between the macro and the micro. Our most important application will involve the calculation of the thermodynamic properties of the ideal gas, but we will also apply the techniques to solids. The procedure will involve calculating U — Uo, the internal energy above zero Kelvin, from the energy of the individual molecules. Enthalpy differences and heat capacities are then easily calculated from the internal energy. Boltzmann s equation... 

Other than an effect on backbone solvation, side chains could potentially modulate PPII helix-forming propensities in a number of ways. These include contributions due to side chain conformational entropy and, as discussed previously, side chain-to-backbone hydrogen bonds. Given the extended nature of the PPII conformation, one might expect the side chains to possess significant conformational entropy compared to more compact conformations. The side chain conformational entropy, Y.S ppn (T = 298°K), available to each of the residues simulated in the Ac-Ala-Xaa-Ala-NMe peptides above was estimated using methods outlined in Creamer (2000). In essence, conformational entropy Scan be derived from the distribution of side chain conformations using Boltzmann s equation... 

The connection between the multiplicative insensitivity of 12 and thermodynamics is actually rather intuitive classically, we are normally only concerned with entropy differences, not absolute entropy values. Along these lines, if we examine Boltzmann s equation, S = kB In 12, where kB is the Boltzmann constant, we see that a multiplicative uncertainty in the density of states translates to an additive uncertainty in the entropy. From a simulation perspective, this implies that we need not converge to an absolute density of states. Typically, however, one implements a heuristic rule which defines the minimum value of the working density of states to be one. 

As suggested previously, the density of states has a direct connection to the entropy, and, hence, to thermodynamics, via Boltzmann s equation. Alternately, we can consider the free energy analogue, using the Laplace transform of the density of states - the canonical partition function ... 

Define a surface ionization potential, I, as the energy required to convert an adatom to an adion. From Boltzmann s equation it follows that... 

The traditional method of dealing with irreversible processes is, of course, the use of the Boltzmann integro-differential equation and its various extensions. But this method leads to two serious difficulties. The first is that Boltzmann s equation is neither provable nor even meaningful except in the context of molecular encounters, i.e., under the assumption that the intermolecular forces are of such short range in comparison with molecular distances that a molecule spends only a negligible fraction of its time within the influence of others. This drastically restricts the field of applicability, confining the treatment to gases close to the ideal state. But even then the equation can only be established on the basis of an essential assumption of molecular probabilistic independence ( micromolecular chaos ).3... 

At temperatures where the potential minimum, e, is comparable to kT, equation (18) is not valid because the method of treating the thermal velocity distribution is no longer appropriate. When e becomes comparable to kT, orientation effects are believed to become important because in general the depth of the potential minimum is dependent on the configuration of both molecules. The probability of a particular orientation can be derived from Boltzmann s equation. 

A pseudo-steady state distribution function was determined from Boltzmann s equation as f(u) =, ( ) + S v)... 

In the actual CO crystal, represented in Figure 17.1 (b) as a randomly oriented (and therefore disordered) mixture of molecules in which, although aligned, both possible dipole directions are presented so that approximately half point one way and the other half point in the opposite direction. In this (real) case each individual CO molecule therefore, can acquire one, of the two dipole orientations and for 1 mole of CO - which contains L molecules, where L is Avogadro s constant - the number of possible arrangements (see Note 17.2) will then be 2L so that the entropy at 0 K is predicted, by Boltzmann s equation to be ... 

The constant of proportionality, Boltzmann s constant, k, is a fundamental parameter, the same as in the ideal gas law, PV = nkT, when n is the number of molecules. Boltzmann s equation describes what we call a law. It cannot be derived from anything else, but as far as we have ever been able to tell, describes the way the universe works. 

There are various approaches, in addition to Boltzmann s equation, that can be taken to calculate thermodynamic parameters (e.g., through the use of the paitition function). 

Using Boltzmann s equation, 5 = ftlnQ, we can then obtain an expression for the entropy of a chain (with respect to some arbitrary reference state)—Equation 13-39 ... 

The net rate of bubble generation, H, describes redistribution of mass in bubble-bubble interactions. Thus, H is a nonlinear functional of F(x,m,t) and Equations (2) and (3) are a pair of coupled, nonlinear, integro-differential equations in the bubble number density, similar to Boltzmann s equation in the kinetic theory of gases (26,27) or to Payatakes et al (22) equations of oil ganglia dynamics. 

Ludwig Boltzmann s tomb in Vienna. Notice Boltzmann s equation on the monument. 

The kinetic theory of evaporation, first given by Clausius, i is briefly described in 5 and 17.III the quantitative description, which has not so far led to very satisfactory results, is based on Boltzmann s equation (4), 19.111, it being assumed that each molecule leaving the liquid for the vapour phase must overcome an attractive force-field and do the work w=Sp. If n, N are the numbers of molecules per cm. in the vapour and liquid ... 

Boltzmann s equation and latent heat, 335 and thermionics, 292 and vapour pressure, 291... 

Simply by the use of Boltzmann s Equation, one may calculate, fiom an estimation of the bond energy, the number of windows which are broken at any temperature. Guest species may themselves appreciably lower the energy of such bond breaking (by being present as part of the "activated complex"), and in this way, "allow themselves through," or at least contribute substantially to their own passage. 

From the potential Vnem(vt). one can obtain the rod orientation distribution function 1/ (u), and hence the order parameter S, by a self-consistent calculation. At equilibrium, the distribution function V (u) is related to the potential Fnemfu) by Boltzmann s equation ... 

Boltzmann s equation, S = kslnfil, relates the entropy to the number of conformations of the chain Q. Considering an element of constant volume dV, the number of conformations available to the chain is proportional to the probability per unit volume, p(x,y,z), multiplied by the size of the volume element, dV, Eq. (1.12) therefore, the entropy of the chain is given by the expression... 

In the small deformation approximation, it is assumed that the deformations undergone by the material are small, at least in the recent past. Approximations of different orders can be developed. The approximation of first order for an incompressible fluid is given by Boltzmann s equation of linear viscoelasticity,... 

A more general approach to the diffusion problem is needed. The essential concepts behind the development of general relationships regarding diffusion were given more than a century ago, by Maxwell [39] and Stefan [40]. The Maxwell-Stefan approach is an approximation of Boltzmann s equation that was developed for dilute gas mixtures. Thermal diffusion, pressure diffusion, and forced diffusion are all easily included in this theory. Krishna et al. [38] discussed the Maxwell-Stefan diffusion formulation and illustrated its superiority over the Pick s formulation with the aid of several examples. The MaxweU-Stefan formulation, which provides a useful tool for solving practical problems in intraparticle diffusion, is described in several textbooks and in numerous publications [7,41-44]. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

In other words, in approximate accordance with the original paper by Boltzmann [6], we assume that in a given volume element the expected number of collisions between molecules that belong to different velocity ranges can be computed statistically. This assumption is referred to as the Boltzmann Stosszahlansatz (German for Collision number assumption). A result of the Boltzmann H-theorem analysis is that the latter statistical assumption makes Boltzmann s equation irreversible in time (e.g., [28], sect. 4.2). 

Using the relaxation time approximation of Boltzmann s equation, the expression for the properties (electrical conductivity, Seebeck coefficient, and Lorenz number) of the holes and electrons are given by... 

The ratio of equilibrium populations Nn/Nm in the denominator of Equation 2.8 must obey Boltzmann s equation (Equation 2.9) ... 

The second statement of the third law (which bears Planck s name) is that as the temperature goes to zero, AS goes to zero for any process for which a reversible path could be imagined, provided the reactants and products are perfect crystals. Here, perfect crystals are defined as those which are non-degenerate, that is, they have only a single quantum state in which they can exist at absolute zero. This statement follows rigorously from Boltzmann s equation for entropy,... 

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