Velocity distribution functions

Let us consider a slanted cylinder of length vdt, constructed on a small surface area dA and containing a large number of gas atoms (Fig. 2.24). 

During the time dt all atoms having velocities in the interval between v and 

V + dv and confined in the cylinder will strike the surface. Their number is given by 

Therefore, the fraction of atoms which strikes the surface with a velocity between v and v - - dv is obtained as 

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The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

The local density nj of species 1 Is related to the velocity distribution function by... 

The spin-displacement density function, c (z, Z), and the normalized displacement distribution function, P(z, Z), can be converted readily into the joint spin-velocity density function, q(z, vn), and the normalized velocity distribution function, P(z, vn), respectively, with the net velocity vn defined as vn = Z/A. Once the velocity density function is determined for each of the volume elements, the superficial average velocity, v, is calculated by [23] ... 

One may also show that MPC dynamics satisfies an H theorem and that any initial velocity distribution will relax to the Maxwell-Boltzmann distribution [11]. Figure 2 shows simulation results for the velocity distribution function that confirm this result. In the simulation, the particles were initially uniformly distributed in the volume and had the same speed v = 1 but different random directions. After a relatively short transient the distribution function adopts the Maxwell-Boltzmann form shown in the figure. 

The motion of activated complexes within the transition state may be analyzed in terms of classical or quantum mechanics. In terms of classical physics, motion along the reaction coordinate may be analyzed in terms of a onedimensional velocity distribution function. In terms of quantum mechanics, motion along the reaction coordinate within the limits of the transition state corresponds to the traditional quantum mechanical problem involving a particle in a box. 

Prove the assertion in the text that the relative velocity of two sets of particles having individual Maxwellian velocity distribution functions also has a Maxwellian distribution with the masses replaced by the reduced mass. 

This paper is organized as follows. Section 2 presents non-trivial properties of the velocity distribution functions for RIG for quasi and ordinary particles in one dimensions. In section 3 we find the state equation for relativistic ideal gas of both types. Section 4 presents the distribution function for the observed frequency radiation generated for quasi and ordinary particles of the relativistic ideal gas, for fluxons under transfer radiation and radiative atoms of the relativistic ideal gas. Section 5 presents a generalization of the theory of the relativistic ideal gas in three dimensions and the distribution function for particles... 

Earlier the velocity distribution function of quasi particles of a relativistic ideal gas for a one dimensional system, for example, fluxons in thermalized Josephson systems and electrons in a high temperature plasma was found. 

As another application of this method of asymptotic integration, we shall consider the problem of the Fourier coefficients pfU(P t) i 1 the limit of long times. As mentioned above, we do not wish to give here a detailed proof of the transport equation for pk] p] t) (see, for instance, Ref. 31). The main result of this analysis is, however, very simple in the limit of long times (t —> oo), the correlations are entirely determined by the velocity distribution function p< p t). One has ... 

We have seen that the zeroth-order velocity distribution function Po p t) tends toward its equilibrium value. We thus have ... 

What we really need is the electrical current (49), and for this it is sufficient to know the one-particle velocity distribution function <5 i(P t) indeed, we may write Eq. (49) as ... 

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. 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

The essential characteristic of the equilibrium correlations is that they originate in a system starting from non-correlated states. We recall also that the correct form of the equilibrium correlations can be obtained if one admits that for long times the velocity distribution function takes a Maxwellian form. 

An early model for slip of fluids in tubes is due to Maxwell [62], wherein the velocity distribution function parallel to the wall is a linear combination of... 

Here, is the mean streaming velocity of particles approaching the wall and (1 — a) is the fraction of fluid particles reflected at the wall, so the first term represents the distribution of particles adsorbed. The velocity distribution functions, /(v), are assumed to be Maxwellian,... 

However, before embarking on this analysis, a brief excursion is of interest [285]. Let us specifically ignore the spatial dependence of the distribution of the N particles. It could have been calculated from the deterministic Newtonian equations of motion. Now considering, in particular, the motion of particles 1 and 2 as above, average the distribution over all position of the N particles to give the velocity distribution function... 

The random velocities of atoms and molecules are described by velocity distribution functions which can often be approximated by a Maxwellian distribution (as in Eq. 2.10). If radiating atoms have such a distribution, the resulting line profile is a Gaussian,... 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

The one-dimensional velocity distribution function will be used in Section 10.1.2 to calculate the frequency of collisions between gas molecules and a container wall. This collision frequency is important, for example, in determining heterogeneous reaction rates, discussed in Chapter 11. It is derived via a change of variables, as above. Equating the translational energy expression 8.9 with the kinetic energy, we have... 

Thus, considering only the positive root for a moment, the one-dimensional velocity distribution function is... 

Molecules throughout a gas have a distribution of velocities and density depending on the temperature, external forces, concentration gradients, chemical reactions, and so on. The properties of a dilute gas are known completely if the velocity distribution function /(r, p, 1) can be found. The Boltzmann equation [38], is an integro-differential equation describing the time evolution of /. The physical derivation of the Boltzmann equation is easy to state, and is presented next. However, its solution is extremely difficult, and relies on varying degrees of approximation. 

For a system composed of N particles, the complete velocity distribution function is denoted f(N> (r(N>, p(/V), t). It is a function of 6N variables, that is, the three vector coordinates for each of the N molecules rW) and the three components of the momentum of each molecule p(-V). Of course, for a macroscopic system, where IV is a very large number, on the order of Avogadro s number A, it is impossible to obtain f(N). One usually attempts to find a less complete description of the system by looking at f(h which depends on the positions and momentum of a smaller number of molecules h and integrates over the effects of the remaining N — h molecules. 

The velocity distribution f(l> is sufficient for calculating most properties of a gas at low density. The distribution function /(1) gives the probability of finding a particular molecule with three coordinates represented by r(1) and three momenta represented by p(l) the locations and velocities of the other N — 1 molecules in the system are not specified. We will not deal with velocity distribution functions of higher order than /(1), and so the superscript will be dropped and implicitly implied from here on (i.e., / = /(1)). We will, however, consider mixtures of gases, and the velocity distribution function for a molecule of type i or type j will be denoted /) (r, p,-, t), /)(r, p , t), and so on. 

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