We can identify a series of forward velocities after \(n\) number of collisions,
\(v\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n \),
\(v\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^2\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n \),
\(v\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^4\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n\)...
The total momentum of this forward particles is,
\(P_{ps}=m_sv\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n lim_{m\rightarrow\infty}\left\{ 1+\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^2+\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^4+...\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^{2m}+... \right\}\)
\(=m_sv \left\{\cfrac { 2}{ N_{ n }(\rho )+1 }\right\}^n lim_{m\rightarrow\infty} \sum^m_0{\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^{2m}}\)
\(P_{ps}=m_sv \left\{\cfrac { 2}{ N_{ n }(\rho )+1 }\right\}^n\cfrac{1}{1-\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^2}\)
\(=m_{ s }v\left\{ \cfrac { 2 }{ N_{ n }(\rho )+1 } \right\} ^{ n }\cfrac { (N_{ n }(\rho )+1)^{ 2 } }{ (N_{ n }(\rho )+1)^{ 2 }-(N_{ n }(\rho )-1)^{ 2 } } \)
\( =m_{ s }v\cfrac { 2^{ n } }{ (N_{ n }(\rho )+1)^{ n-2 } } \cfrac { 1 }{ (N_{ n }(\rho )+1)^{ 2 }-(N_{ n }(\rho )-1)^{ 2 } } \)
\( =m_{ s }v\cfrac { 1 }{ N_{ n }(\rho ) } \left\{ \cfrac { 2 }{ (N_{ n }(\rho )+1) } \right\} ^{ n-2 }\)
If \( N_{ n }(\rho )>1\) the forward momentum dies down eventually as \(n\rightarrow\infty\), after many collisions.
And a series of backward velocities,
\(v\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n \),
\(v\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^3\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n \),
\(v\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^5\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n\)...
The total momentum of this backward particles is,
\(P_{ps\,r}=m_sv\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\} lim_{m\rightarrow\infty}\left\{ 1+\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^2+...\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^{2m}+... \right\}\)
\(=m_sv\left\{\cfrac { 2 }{ N_{ n }(\rho )+1 }\right\}^n\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\} lim_{m\rightarrow\infty} \sum^m_0{\left\{\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }\right\}^{2m}}\)
\(=m_{ s }v\left\{ \cfrac { 2 }{ N_{ n }(\rho )+1 } \right\} ^{ n }\cfrac { N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 } \cfrac { (N_{ n }(\rho )+1)^{ 2 } }{ (N_{ n }(\rho )+1)^{ 2 }-(N_{ n }(\rho )-1)^{ 2 } } \)
\(=m_{ s }v\left\{ \cfrac { 2 }{ N_{ n }(\rho )+1 } \right\} ^{ n-1 }\cfrac { N_{ n }(\rho )-1 }{ 2N_{ n }(\rho ) } \)
\(=m_{ s }v\left\{ \cfrac { 2 }{ N_{ n }(\rho )+1 } \right\} ^{ n-1 }.\cfrac { 1 }{ 2 }\left\{ 1 -\cfrac { 1 }{ N_{ n }(\rho ) } \right\} \)
If \( N_{ n }(\rho )>1\) then the backward momentum also attenuate to zero. When \(N_{ n }(\rho )=1\) there is no backward momentum, the particle has zero velocity.
From the post "Collisions And More Collisions" the last graph shows that when \( N_{ n }(\rho )=2\) velocity the attenuate very quickly within 10 collisions. This would suggest that under normal circumstances \( N_{ n }(\rho )=1\). That the forward momentum is carried from particle to particle without loss.
