Boltzman distribution, equation

So far we have discussed mainly stable configurations that have reached an equilibrium. What about the evolution of a system from an arbitrary initial state In particular, what do we need to know in order to be assured of reaching an equilibrium state that is described by the Boltzman distribution (equation 7.1) from an arbitrary initial state It turns out that it is not enough to know just the energies H ct) of the different states a. We also need to know the set of transition probabilities between ail pairs of states of the system. 

Euler s equation (equation 9.7) may be recovered from Boltzman s equation as a consequence of the conservation of momentum, but only in the zeroth-order approximation to the full distribution function. Setting k — mvi in equation 9.52 gives, in component form. 

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Using the fact that we have a well defined energy function (equation 10.9), we know from statistical mechanics that when the system has reached equilibrium, the probability that it is in some state S = Si, S, , Sm) is given by the Boltzman distribution ... 

The Gouy—Chapman treatment combines the Poisson equation in onedimensional form, which relates the electrical potential (x) to the charge density at x, and the Boltzman distribution of ions in thermal motion [6,18]... 

This equation describes an equilibrium distribution of the electroinactive ion in an electrostatic field. This is a typical situation to which one may apply the Boltzman distribution law, which states... 

The Poisson-Boltzman (PB) equation relates the electric displacement to the charge density (see Equation (4.28)). The total charge distribution includes the solute charge inside the solute cavity (pint) and that generated by the ion atmosphere outside the cavity (Pext) The external charge density can be represented as shown in Equation (4.30), which leads to the expanded form of the PB equation (Equation (4.31)), which can be simplified for low (Equation (4.32)) and zero (Equation (4.33)) ionic strengths. 

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 =... 

This distribution equation is known as the Maxwell-Boltzman distribution. 

In classical molecular dynamics simulations, atoms are generally considered to be points which interact with other atoms by some predehned potential form. The forms of the potential can be, for example, Lennard-Jones potentials or Coulomb potentials. The atoms are given velocities in random directions with magnitudes selected from a Maxwell-Boltzman distribution, and then they are allowed to propagate via Newton s equations of motion according to a finite-difference approximation. See the following references for much more detailed discussions Allen and Tildesley (1987) and Frenkel... 

In spite of the difference in the underlying concepts and the forms of equations, Eqs. (3.3) and (3.4), both descriptions reflect the statistical sense of the rate constant. The latter statement is crucially important for better understanding of the problem existing in heterogeneous kinetics. Indeed, the above-mentioned theories are based on gas statistics and the given equations assume an equilibrium Maxwell-Boltzman distribution for gas species, which in the absence of reaction interact only via elastic collisions. If this can be considered as a satisfactory approximation for gas reactions at moderate temperatures and pressures discussed here (with some exceptions—see Section III.D), its applicability to the processes involving surface sites (i.e., elements of solid lattice) or adsorbed species is not so obvious. 

The catalytic effect for reactions involving an ionic reactant usually shows a strong dependence on the total amphiphile concentration. The maximal effective rate constant is attained at concentrations just over the CMC. Romsted284 showed that this occurs due to the competition between the ion binding of the reactive ions (OH- in the example above) and the counterions of the amphiphile. Recently, Diekman and Frahm285 286 showed that it is possible to rationalize the kinetic data by describing the ion distribution through a solution of the Poisson-Boltzman equation. (See Fig. 5.1). 

According to the law of distribution of molecular velocities (Glasstone and Lewis, 1960), molecules in two different phases, at equilibrium, are related in translation through the Boltzman equation, stated as... 

Let us first outline the theoretical background of the evaluation of both the charge and potential of two interacting diffuse electric layers. It is well known that the charge and potential distribution in the diffuse layer can be represented with a sufficient degree of accuracy using the Poisson-Boltzman (PB) approximation [e.g. 246]. For a planar film from aqueous symmetrical electrical electrolyte of valence z, the respective equation can be written in dimensionless form as... 

To describe polydispersed multiphase systems the Boltzmann equation can be extended by including the dependency of the internal property coordinates such as the particle size and shape in the definition of the distribution function. In this way a statistical balance formulation can be obtained by means of a distribution function on the form p, r,v, c,t)d dr dv dc, defined as the probable number of particles with internal properties in the range about with a velocity range in property space dv about v, located in the spatial range dr about the position r, with a velocity range dc about c, at time t. The particular Boltzman type of equation is given by ... 

Boltzman equation because it does not distinguish between ions of like charge and therefore cannot account for the specificity of the distribution. In order to reflect the experimental reality, the Leodidis and Hatton model takes into account three characteristics of ions their charge, size and electrostatic free energy of hydration. 

We start with the simpler model, for the electron gas. In a metal, the electron density is so high that the Pauli exclusion principle must be taken into account, i.e., there can be only one electron in each quantum state. The result is that the electrons obey the Fermi distribution rather than Boltzman s, and may be considered strongly degenerate (8). The equation of state is... 

Zeitsch [1977] has pressed an analysis of capillaiy drainage involving a Boltzman-type distribution of pore diatneters to produce a method for the calculation of drainage times for filter cakes of known permeabiBty k or specific resistance. The equations take the form reported below in vriiich H is the cake thickness ... 

All these experimental results have been recently complemented by a very useful theoretical study by Kharkats and Ulstrup," who calculated analytically the electrostatic Gibbs energy profile of an ion between two dielectric phases separated by a planar boundary, incorporating both the ionic finite size and the dielectric image interactions. The profile obtained, illustrated in Fig. 4, shows that there is no discontinuity as the ion traverses the boundary and that cation and anion concentration distribution will differ if they have different ionic radii, as they will penetrate the boundary to a different extent. This has important repercussions on the Poisson-Boltzman equation as the work term is not only the electrical energy, —but also an electrostatic contribution to the Gibbs energy of solvation as the ion... 

A chapter introducing the Bose-Einstein, Maxwell-Boltzman, Planck, and Fermi-Dirac distribution fimctions follows before discussing the thermal, electronic, magnetic, and optical properties for the benefit of students who have not been exposed to quantum statistical mechanics. This chapter is a logical beginning for the second half of this book since these concepts are essential to an imderstanding of these properties. Similarly, the Maxwell equations are used to derive the equations for absorption and normal reflection of electromagnetic waves in the chapter on optical properties. The band structure of metals... 

The Monte-Carlo simulation of these models reproduces well the charge carrier transport properties in the mesophases, i.e., the mobility independence of both temperature and electric field in the temperature above room temperature, if a small sigma (40 60 meV) is taken for the Gaussian width of the distribution of localized states in Eq. (2.2). In this equation p, is the mobility, a is the Gaussian width of the distributed energy states for hopping sites, E is an index of the positional disorder, k is the Boltzman constant, T is the temperature, E is the electric field and C is a constant. The constants a and n depend on the type of mesophase, e.g., 0.8 and 2 for the SmB phase and 0.78 and 1.5 for the SmE phase, respectively. This value of cr (40-60 meV) is half that for typical amorphous solids [61]. 

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