Maxwell velocity distribution

This simple calculation gives a result close to that obtained by integrating over the three-dimensional Maxwell velocity distributions for both A and B. In this case, the same expression is obtained with the characteristic velocity of approach between A and B... 

Figure 4.27 Maxwell velocity distribution of ideal gases...

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. 

Gas particles, at a given temperature and pressure, do not all have the same velocities. Instead, the velocities are described by the famous Maxwell velocity distribution, g(vx),... 

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. 

When the system is in so-called thermodynamic equilibrium, the neutrals and the electrons have the same Maxwell velocity distribution and at a temperature T we have ... 

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 ). 

Maxwell velocity distribution of Rb atoms at about 1.7 X I K. The velocity increases from the center (zero) outward along the two axes. The red color represents the lowest number of Rb atoms and the white color the highest. 

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. 

---