Collisions And More Collisions

Consider the particle of space,  $m_{ps}$ with velocity  $v$ in collision with its neighbors.  This collision is elastic and we have,

conservation of momentum,

$m_{ ps }v=N_{ n }(\rho )m_{ ps }v_{ n }+m_{ ps }v_{ s }$

where $N_n(\rho)$ is the number of neighbors the particle collides with,  a number that depends on density,  $\rho$.

$v=N_{ n }(\rho )v_{ n }+v_{ s }$    ---- (1)

and conservation of kinetic energy,

$\frac { 1 }{ 2 } m_{ ps }v^{ 2 }=N_{ n }(\rho )\frac { 1 }{ 2 } m_{ ps }v^{ 2 }_{ n }+\frac { 1 }{ 2 } m_{ ps }v^{ 2 }_{ s }$

$v^{ 2 }=N_{ n }(\rho )v^{ 2 }_{ n }+v^{ 2 }_{ s }$    --- (2)

Squaring equation  (1),

$v^{ 2 }=(N_{ n }(\rho )v_{ n }+v_{ s })^{ 2 }=N^{ 2 }_{ n }(\rho )v^{ 2 }_{ n }+v^{ 2 }_{ s }+2N_{ n }(\rho )v_{ n }.v_{ s }$

minus equation (2)

$0=\left\{N^{ 2 }_{ n }(\rho )-N_{ n }(\rho )\right\}v^{ 2 }_{ n }+2N_{ n }(\rho )v_{ n }.v_{ s }$

$0=v_{ n }N_{ n }(\rho )\left\{ (N_{ n }(\rho )-1)v_{ n }+2v_{ s } \right\}$

Since  $v_n$  and  $N_{ n }(\rho )$  are not zero,

$(N_{ n }(\rho )-1)v_{ n }+2v_{ s }=0$

$-v_{ s }=\cfrac { 1 }{ 2 } \left\{ N_{ n }(\rho )-1 \right\} v_{ n }$

From (1),

$v=N_{ n }(\rho )v_{ n }-\cfrac { 1 }{ 2 } \left\{ N_{ n }(\rho )-1 \right\} v_{ n }$

$v=\cfrac { 1 }{ 2 } \left\{ N_{ n }(\rho )+1 \right\} v_{ n }$

$v_{ n }=\cfrac { 2 }{ N_{ n }(\rho )+1 } v$

and

$v_{ s }=v-\cfrac { 2N_{ n }(\rho ) }{ N_{ n }(\rho )+1 } v=\left\{ 1-\cfrac { 2N_{ n }(\rho ) }{ N_{ n }(\rho )+1 }  \right\} v$

$v_{ s }=-\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }  v$

when  $N_{ n }(\rho )=1$

$v_{ n }=v$    and     $v_{ s }=0$

The moving particle stopped and the once stationary particle moves forward with velocity  $v$.

when  $N_{ n }(\rho )=2$

$v_{ n 1}=\cfrac{2}{3}v$,    $v_{ n 1}=\cfrac{2}{3}v$    and     $v_{ s }=-\cfrac{1}{3}v$

when  $N_{ n }(\rho )=3$

$v_{ n 1}=\cfrac{1}{2}v$,    $v_{ n 2}=\cfrac{1}{2}v$,    $v_{ n 3}=\cfrac{1}{2}v$     and    $v_{ s }=-\cfrac{1}{2}v$

The following diagram shows the cascade of collisions as time progresses,

The envelop of this cascade will always lead the shape of this velocity profile.  Particle collision within the envelop will not gain enough velocity to surpass the envelop.  As such we have a velocity curve as show below for the case of  $N_n(\rho)=2$

The peak of this curve is reduced by subsequent collisions, each time by the collision factor,

$C_c=\cfrac { 2 }{ N_{ n }(\rho )+1 }$

For particles in the reverse direction, they are first reduced by the reverse reduction factor,

$R_c=-\cfrac {N_{ n }(\rho )-1 }{ N_{ n }(\rho )+1 }$

and then by  $C_c$  for each subsequent collision they encounter.

This velocity factor vs collision count curve is not the shape profile of the packet of energy.  For that we need to integrate over time.