Mechanical Proof of the Maxwell Speed Distribution | Lin | International Journal of Statistics and Probability | CCSE

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