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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:
  
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>.
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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===

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