# Editing Maxwell speed distribution

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As a consequence of the former theorems, and by simulation of many examples, the following conjecture can be stated: | As a consequence of the former theorems, and by simulation of many examples, the following conjecture can be stated: | ||

β | For any <math>p</math> with <math>||p||=1</math>, with finite <math> \langle x^2,p \rangle </math> and verifying <math>\lim_{n\rightarrow\infty} ||\mathcal T^np(x)-\mu(x)||=0</math>, the limit <math>\mu(x)</math> is the fixed point <math>p_{\alpha}(x)=\sqrt{\alpha\over\pi}\,e^{-\alpha x^2}</math>, with <math>\alpha=(2\, \langle x^2,p \rangle)^{-1}</math>. That is, the asymptotic steady state is the | + | For any <math>p</math> with <math>||p||=1</math>, with finite <math> \langle x^2,p \rangle </math> and verifying <math>\lim_{n\rightarrow\infty} ||\mathcal T^np(x)-\mu(x)||=0</math>, the limit <math>\mu(x)</math> is the fixed point <math>p_{\alpha}(x)=\sqrt{\alpha\over\pi}\,e^{-\alpha x^2}</math>, with <math>\alpha=(2\, \langle x^2,p \rangle)^{-1}</math>. That is, the asymptotic steady state is the gaussian distribution with the same mean energy than the initial out-of-equilibrium state <math>p</math>. |

===Conclusion=== | ===Conclusion=== |