Maxwell - Boltzmann paradox

Equation (10.10) goes in the limit of X2 different gases and in the limit of xi, identical gases. 

Question 10.1. When two different ideal gases are mixed, the temperature, the volume, and the pressure remain constant, as well as the total energy. However, the total entropy increases. Thus we have 

When we start to mix two pure components, the entropy of mixing remains finite. Thus, we believe that we need finite energy in order to separate the mixture back into the pure compounds. However, in practice we never get them clean. 

If two systems are in equilibrium, thus in thermal equilibrium, they exhibit equal temperatures. On the other hand, a collection of particles are in equilibrium, if there is a certain velocity distribution. The velocity can serve as a measure of the temperature of a particle. If we break down the collection of particles to thermodynamic systems, the concept of thermal equilibrium is violated. According to the classical view of thermodynamics, we could increase the entropy if we pick out two systems of different temperatures and allow equilibrating. Obviously, the application of thermodynamics is restricted to entities that consist of more than one particle. Actually, the variables in classical thermodynamics rely on averages of large ensembles. 

The Maxwell - Boltzmann velocity distribution emerged by theoretical arguments in 1866. However, an experimental proof was given only in 1921 by Stern [9-11], 

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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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