Maxwellian velocity distribution

Doppler broadening arises from the random thermal agitation of the active systems, each of which, in its own test frame, sees the appHed light field at a different frequency. When averaged over a Maxwellian velocity distribution, ie, assuming noninteracting species in thermal equilibrium, this yields a line width (fwhm) in cm C... 

Unlike the initial coordinates, which can be obtained experimentally, the only relevant information available about atomic velocities is the system s temperature T, which determines the velocity distribution. In the absence of a better guideline, initial velocities (v , Vy, Vy) are usually randomly assigned from the standard Maxwellian velocity distribution at a temperature T,... 

If particle i has been selected to undergo a collision, obtain its new velocity from a Maxwellian velocity distribution, defined in Eq. (23), corresponding to the desired temperature Tg. All other particles are unaffected by this collision. 

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. 

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... 

Directional detection will be a fantastic means of recognizing the presence of a dark stream through the Solar system. The recoil distribution due to WIMPs in a stream is very much different from the recoil distribution due a Maxwellian velocity distribution. The corresponding Radon transforms that appear in Eq. (16.44) are for a stream of velocity V,... 

Assuming a weighted Maxwellian velocity distribution for uc, the trapping probability in the hard-cube model can be analytically expressed as [7]... 

Compute the mean free path of a hydrogen molecule in hydrogen at 0°C, using simple theory and then using a maxwellian velocity distribution. 

Here ey is the cross section for a collisionally induced transition and v is the thermal velocity of the colliding particle, < av > is the average value of ov for a Maxwellian velocity distribution. Assuming a typical dipole moment of 1 Debye, a = 1CT15 cm"2, v = 5 x 104 cm/s, one obtains the density n % 103/X3. Thus for the detection of an emission line in the centimeter-wave region (X = 1 cm) the density within the cloud is expected to be of the order of 103 cm-3. On the other hand, a detection of a millimeter-wave transition in emission at X = 1 mm requires densities of the order of 10s to 106 particles/cm3. 

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. 

In the vicinity of 1000°K, u0 10s cm/sec, the second factor is about 0.2. Rapp [83] performed an approximate integration of equation (63) over a Maxwellian velocity distribution to obtain... 

Schematic Maxwellian velocity distribution Up) du is the fraction of the molecules with velocities between v and v + dv. Values of the rms speed u and the mean speed c are shown.

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. 

In later measurements, very slow atoms from the broad Maxwellian velocity distribution were selected in order to allow for even narrower lines and smaller systematic shifts. For these measnrements, the signal detection was enabled only at a (variable) delay time t after blocking the excitation light field with the chopper, such that only atoms with velocities v below iA being the... 

One can give a quite satisfactory interpretation of the energy distribution of Fig. 8.2 by supposing it to result from evaporation of neutrons from the fission product nuclei with a temperature of about 0.5 MeV. Such a Maxwellian velocity distribution is to be relative to the moving fission product nuclei giving rise to a curve like Fig. 8.2. 

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]. 

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