The Maxwell velocity distribution

According to (12 77) the mean number of molecules whose translational energy level is e, irrespective of their particular internal level, is 

The number of molecules in each of the quantum states which is comprised within Cf is obtained by dividing by (o  

Multiplying together the last two expressions and inserting the value offx from (12 48) we find the required number s 

Eliminating de between (12 96) and (12 97) and dividing the resulting expression by N, we find that the fraction of all the N molecules whose magnitude of the x component of velocity lies in the range v ( to v. +d . is 

In this expression v I only be positive, since it refers to a scalar magnitude. By using the standard integrals tabulated at the end of the chapter it is readily confirmed that the integral of (12 98) is unity when taken over the range 0 to oo. On the other hand, if, in place of ii, we consider v itself, its values can range from — oo to + oo. The fraction of the N molecules for which the x component of velocity lies between v and V +is therefore obtained by dividing (12 98) by 2 and isf  

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For a fuller explanation of this usage, see Ref. 5, p. 39. The expression for F in Eq. (18), when multiplied by m3 to convert it from a density in momentum space to one in velocity space, and when a2 is replaced by mkT, coincides in form with the Maxwell velocity distribution. It is,... 

Thirdly it is easy to see that the condition that the X are independent is important. If one takes for all r variables one and the same X the result cannot be true. On the other hand, a sufficiently weak dependence does not harm. This is apparent from the calculation of the Maxwell velocity distribution from the microcanonical ensemble for an ideal gas, see the Exercise in 3. The microcanonical distribution in phase space is a joint distribution that does not factorize, but in the limit r -> oo the velocity distribution of each molecule is Gaussian. The equivalence of the various ensembles in statistical mechanics is based on this fact. 

For H2 molecule, the critical temperature Tt at which freezing-out of rotational modes begins is equal to 90K, in accordance with the classical expression Tt = hr IX it J-kH. where J = mr2 is the rotational moment of inertia for this molecule, m = 3.34 10"27 kg is H2 molecule mass, r = 0.74 10 8 cm means H2 molecule radius, h and kB are Planck s and Boltzmann s constants, respectively. When Tlower temperatures it remains above zero as a consequence of the Maxwell velocity distribution for molecules. 

The calculations in this case are clearly analogous to those required to prove the Bernoulli theorem. In order to show the first part of the statement, all we have to do is to determine the maximum of Eq. (36), i.e., the minimum of Eq. (43), given the auxiliary condition of Eq. (45). Boltzmann makes use of the second half of the statement in all those cases when he calls the Maxwell velocity distribution overwhelmingly the most probable one." A more quantitative formulation and derivation of this part of the statement is sketched by Jeans in [2, 22-26] and in Dynamical Theory, 44-46 and 56. 

When the Maxwell velocity distribution for the molecules takes place,... 

The fusion reaction rate parameter reaction cross section, which depends on the particle energy, and v is the velocity of the ions, averaged over the Maxwell velocity distribution and proportional to the temperature. 

Consider a system of particles moving in a box at thermal equilibrium, under their mutual interactions. In the absence of any external forces the system will be homogenous, characterized by the equilibrium particle density. From the Maxwell velocity distribution for the particles, we can easily calculate the equilibrium flux in any direction inside the box, say in the positive x direction, Jx = p(vx), where p is the density of particles and dvxVx exp(—fimv /2 ). 

The figure shows the Maxwell velocity distribution of the Rb atoms at this temperature. The colors indicate the number of atoms having velocity specified by the two horizontal axes. The blue and white portions represent atoms that have merged to form the BEC. 

The Boltzmann equation is considered valid as long as the density of the gas is sufficiently low and the gas properties are sufficiently uniform in space. Although an exact solution is only achieved for a gas at equilibrium for which the Maxwell velocity distribution is supposed to be valid, one can still obtain approximate solutions for gases near equilibrium states. However, it is evident that the range of densities for which a formal mathematical theory of transport processes can be deduced from Boltzmann s equation is limited to dilute gases, since this relation is reflecting an asymptotic formulation valid in the limit of no coUisional transfer fluxes and restricted to binary collisions only. Hence, this theory cannot without ad hoc modifications be applied to dense gases and liquids. 

Here the angular brackets < > indicate an average in the phase space of a single dumbbell. The first term is the solvent contribution to the stress tensor, the second term is the contribution resulting from the tensions in the springs and the third term is a result of the momentum flux associated with the bead motion. If the Maxwell velocity distribution is assumed, then the last term becomes an isotropic contribution. The (extra) stress tensor can now be written in several equivalent forms ... 

This thermostat reproduces the Maxwell velocity distribution and does not change the viscosity of the fluid. It gives excellent equilibration, and the deviation of the measured kinetic temperature from Tq is smaller than 0.01%. The parameter c controls the rate at which the kinetic temperature relaxes to 7b, and in agreement with experience from MC-simulations, an acceptance rate in the range of 50-65% leads... 

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). 

This function should have the property Cvf—t) = Cv (t), and at r = 0 should agree with the average (V ) predicted by the Maxwell velocity distribution,... 

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