Critical Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution defines the most probable route. Here the Maxwell-Boltzmann distribution is being used in terms of a probability, rather than a fraction of molecules with energy at least a certain critical energy. The probability that a molecule has a given potential energy at any point on the surface is proportional to exp(—s/kT), where e is the PE at the point. 

Although Boltzmann did not fully succeed in proving the tendency of the world to go to a final equilibrium state, there remain after all criticisms the following valuable results first, the derivation of the Maxwell-Boltzmann distribution for equilibrium states, then the kinetic interpretation of the entropy by the //-function, and finally the explanation of the existence of an integrating factor for dU+dA. In thermodynamics the existence of such a factor is always based on an unexplained hypothesis. 

When the temperature T decreases below the critical temperature Tc BEC starts at first only for a small fraction of all atoms in the magnetic trap, i.e. not all atoms are immediately transferred into the coherent state of BEC. With decreasing temperature the fraction of the condensed atoms increases (Fig. 9.35). The BEC-state is separated from the normal state with T > Tc by an energy gap similar to the energy gap between the Cooper pair state in supra-conductivity and the normal state. One therefore has to pump energy into the condensate in order to convert the assemble into the normal state. Also the energy distribution changes from a Maxwell-Boltzmann distribution above to a distribution... 

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. 

Criticism of the Stosszahlansatz and its corollaries arose as soon as it was recognized as paradoxical that the completely reversible gas model of the kinetic theory was apparently able to explain irreversible processes, i.e., phenomena whose development shows a definite direction in time. These nonstationary,51 irreversible processes were brought into the center of interest by the //-theorem of Boltzmann. In order to show that every non-Max-wellian distribution always approaches the Maxwell distribution in time, this theorem synthesizes all the special irreversible processes (like heat conduction and... 

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