Abstract

This article derives the probability density function ψ(ξ;x,x')  of the resulting speed ξ  from the collision of two particles with speeds x  and x' . This function had been left unsolved for about 150 years. Then uses two approaches to obtain the Maxwell speed distribution: (1) Numerical iteration: using the equation   P_new(ξ) =∫_0^∞ ∫_0^∞ψ(ξ;x,x') ∙P_old(x) ∙P_old(x') dxdx'  to get the new speed distribution from the old speed distribution. Also, after 9 iterations, the distribution converges to the Maxwell speed distribution. (2) Analytical integration: using the Maxwell speed distribution as the P_old(x) , and then getting P_new(ξ)  from the above integration. The result of P_new(ξ)  from analytical integration is proved to be exactly the Maxwell speed distribution.

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