The Maxwellian Velocity Distribution

For the imaginary equilibrium flows we assume that in the neighborhood of any point in the gas, the distribution function is locally Maxwellian, and p, T, and v vary slowly in space and time. It can be shown that the approximation /(r, c,t) /°(r, c,t) is also a solution to (2.223), when /°(r, c, t) is defined by  

It is important to note that this distribution function (2.225), defined so that it resembles (2.224) but with the constant values of n, v and T in (2.224) replaced by the corresponding functions of r and t, remains a solution to (2.223). This distribution function, which is called the local Maxwellian, makes the kinetic theory much more general and practically relevant. 

Both the absolute- and local Maxwellians are termed equilibrium distributions. This result relates to the local and instantaneous equilibrium assumption in continuum mechanics as discussed in chap. 1, showing that the assumption has a probabilistic fundament. It also follows directly from the local equilibrium assumption that the pressure tensor is related to the thermodynamic pressure, as mentioned in sect. 2.3.3. 

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In more refined calculations, 2 is replaced by its average over the Maxwellian velocity distribution and is a slowly varying function of t. 

We shall calculate it from general considerations on ergodicity indeed, we know that any distribution W tends, after a sufficiently long time, toward the Maxwellian velocity distribution ... 

The model yields a set of hydrodynamic equations for the solid phase. For equation closure, additional constitutive relations, which can be obtained by using the kinematic argument of the collision and by assuming the Maxwellian velocity distribution of the solids, are needed. Two examples are given to illustrate the applications of this model in this chapter. 

The physical condition of the kinetic theory of gases can be described by elastic collisions of monodispersed spheres with the Maxwellian velocity distribution in an infinite vacuum space. Therefore, for an analogy between particle-particle interactions and molecular interactions to be directly applicable, the following phenomena in gas-solid flows should not be regarded as significant in comparison to particle-particle interactions the gas-particle... 

From the preceding discussions it is evident that at least four different temperatures have to be considered which under laboratory conditions are all equal the excitation temperature Tex of the molecule, defined by the relative populations of the levels, the kinetic temperature Tk, corresponding to the Maxwellian velocity distribution of the gas particles, the radiation temperature Traa, assuming a (in some cases diluted) black body radiation distribution, and the grain temperature 7, . With no thermodynamic equilibrium established, as is common in interstellar space, none of these temperatures are equal. These non-equilibium conditions are likely to be caused in part by the delicate balance between the various mechanisms of excitation and de-excitation of molecular energy levels by particle collisions and radiative transitions, and in part by the molecule formation process itself. Table 7 summarizes some of the known distribution anomalies. The non-equilibrium between para- and ortho-ammonia, the very low temperature of formaldehyde, and the interstellar OH and H20 masers are some of the more spectacular examples. 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

During the convective transport individual target molecules are dispersed by the presence of small eddies. The random walk motion of small particles suspended in a fluid due to bombardment by molecules obeys the Maxwellian velocity distribution. If a number of particles subject to Brownian motion are present in a given medium and there is no preferred direction for... 

In summary, the statistical tf-theorem of kinetic theory relates to the Maxwellian velocity distribution function and thermodynamics. Most important, the Boltzmann s //-theorem provides a mechanistic or probabilistic prove for the second law of thermodynamics. In this manner, the //-theorem also relates the thermodynamic entropy quantity to probability concepts. Further details can be found in the standard references [97] [39] [12] [100] [47] [28] [61] [85]. 

There are some further steps, which should also be included, and are quite straightforward in practice. The true profiles of the absorption lines are not Lorentzian, as assumed in the simple theory above, but are broadened by the Maxwellian velocity distribution ... 

At any given instance, a number of chain molecules of equal length will have a random distribution of chain end-to-end distances. This information can be obtained from a derivation analogous to that used to derive the Maxwellian velocity distribution of molecules in an ideal gas. 

A short electron-beam pulse (20-50 nsec) produces an ion swarm immediately, a short, high-voltage pulse gives the Maxwellian velocity distribution a constant impulse and this accelerated ion swarm then reacts at constant velocity on its passage through the ionization chamber. The initial Maxwellian velocity distribution imposes an energy dispersion on the reactant ion swarm such that at 400°K, in eV,... 

We point out parenthetically that in the kinetic theory of dilute gases it is just the deviation from the Maxwellian velocity distribution that is of primary interest in the evaluation of the transport properties. In the kinetic theory of polymers, on the other hand, it has been assumed that the deviations from the Maxwellian distribution are of minor importance, and to date few calculations or estimations have been made of the errors introduced by this assumption [16-19], These exploratory efforts indicate that there may be a significant effect on the components of the complex viscosity in high-frequency oscillatory shearing flows. 

Thus, only the potential energy (from the elastic springs) contributes to I/, the kinetic energy making no contribution here because of our definition of T in Eq. (12.9) and the use of the Maxwellian velocity distribution in Eq. (12.11). We now develop the integral in Eq. (C.21) for the Rouse chain model. 

Here is the mass of the light molecule and c g are the velocities of the light molecule before and after the collision, respectively da is the collision cross section element (surface element perpendicular to the relative velocity — c ) /(c ) is the Maxwellian velocity distribution function... 

Aq is the mean free path for energy transfer no change in direction is implied. M is the mass of a rare gas atom. For zero electric field, fo(x) is given as exp(-x) and for f(p) the Maxwellian velocity distribution is obtained. 

In a series of impressive publications. Maxwell [95-98] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. An important introductory result was the Maxwellian velocity distribution function heuristically derived for a gas at equilibrium. It is emphasized that a gas at thermodynamic equilibrium contains no macroscopic gradients, so that the fluid properties like velocity, temperature and density are uniform in space and time. When the gas is out of equilibrium non-uniform spatial distributions of the macroscopic quantities occur, thus additional phenomena arise as a result of the molecular motion. The random movement of molecules from one region to another tend to transport with them the macroscopic properties of the region from which they depart. Therefore, at their destination the molecules find themselves out of equilibrium with the properties of the region in which they arrive. At the continuous macroscopic level the net effect... 

This expression coincides with (2.165) when the speed is calculated from the Maxwellian velocity distribution function. 

To derive the constitutive equations to close the governing transport equations, the Maxwellian velocity distribution function for the particles is required. This distribution function is derived in the sequel [30, 68, 126]. 

Here, n is the number density of the gas atoms and /m is the Maxwellian velocity distribution function, Eq. (2.152). We notice that the spatial dependence of the contribution from the arriving atoms follows the spatial variation of the external field, whereas that is not the case for the contribution of the scattered atoms because of the exponential term in the curly brackets. In other words, the optical response of the gas near the surface is nonlocal. 

If one allows for anisotropic frictional forces by retaining the friction tensor fin equation (51), and allowing for anisotropic Brownian motion by allowing the Maxwellian velocity distribution to be skewed (so that = — (kT/ F)[(5/5ry) f F]), then the diffusion equation and stress tensor expressions become... 

The flux distribution of the neutrons in the beam is obtained by multiplying the Maxwellian velocity distribution of the neutrons in the core, (dn) by v. Thus,... 

If the thickness x of the material is small, the value of transmission T may be close to unity. This would greatly reduce the accuracy of the cross section obtained. It is preferable to use a sample thick enough to attenuate the beam more so that T is much less than 1. But with increasing sample thickness the Maxwell velocity distribution is increasingly distorted due to greater (preferential) absorption of slow neutrons by l/v capture. This change in the Maxwellian velocity distribution is called hardening of the beam (see Fig. 25.4). 

Since the decay actually observed is that of a Maxwellian velocity distribution rather than that of a monoenergetic group of neutrons, the vZj terms in Eqs. (1-7) are actually averaged over the Maxwellian velocity distribution. Thus,... 

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