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I noticed 'Mixing of diffusing particles' research by E. Ben-Naim taking into account Mahonian distribution which tends to normal distribution for very large number of particles (infinite number).

In addition to that there is slightly improved function of Mahonian distribution which depends on the number of particles: $F_N (x)=Φ(x)+\frac{3}{50\sqrt{2π}} e^{-x^2/2} (x^3-3x) \frac {6N^3+21N^2+31N+31} {N(2N+5)^2 (N-1)}+O(\frac{1}{N^2} )$.

So, I just wonder what would be application of that model of Machonian distribution for the finite number of particles in physics? In other words what would be experiments (ways) to prove it on practice or search some new effects?

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