Collisional Conservation Laws
Consider
![$\displaystyle \int C_{ss'}\,d^3{\bf v}_s = \int\!\!\int\!\!\int\!\!\int u_{ss'}...
...'-f_s\,f_{s'})\,d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}',$](img650.png) |
(3.28) |
which follows from Equation (3.23).
Interchanging primed and unprimed dummy variables of integration on the right-hand side, we
obtain
![$\displaystyle \int C_{ss'}\,d^3{\bf v}_s = \int\!\!\int\!\!\int\!\!\int u_{ss'}...
...f_s'\,f_{s'}')\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'\,d^3{\bf v}_s\,d^3{\bf v}_{s'}.$](img651.png) |
(3.29) |
Hence, making use of Equation (3.18), as well as the fact that
, we
deduce that
which implies that
![$\displaystyle \int C_{ss'}\,d^3{\bf v}_s=0.$](img655.png) |
(3.31) |
The previous expression states that collisions with species-
particles give rise to zero net rate of change of the number density of species-
particles
at position
and time
. In other words, the collisions conserve the number of species-
particles.
Now, it is easily seen from Equations (3.23) and (3.24) that
![$\displaystyle C_{ss'}\,d^3{\bf v}_s=C_{s's}\,d^3{\bf v}_{s'}.$](img656.png) |
(3.32) |
Hence, Equation (3.31) also implies that
![$\displaystyle \int C_{s's}\,d^3{\bf v}_{s'}=0.$](img657.png) |
(3.33) |
In other words, collisions also conserve the number of species-
particles.
Consider
![$\displaystyle (m_s+m_{s'}) \int {\bf U}_{ss'}\,C_{ss'}\,d^3{\bf v}_s = {\bf0}.$](img658.png) |
(3.34) |
This integral is obviously zero, as indicated, as a consequence of the conservation law (3.31), as well as the fact that the center
of mass velocity,
, is a constant of the motion. However, making use of Equations (3.10) and (3.32), the
previous expression can be rewritten in the form
![$\displaystyle \int m_s\,{\bf v}_s\,C_{ss'}\,d^3{\bf v}_s = - \int m_{s'}\,{\bf v}_{s'}\,C_{s's}\,d^3{\bf v}_{s'}.$](img659.png) |
(3.35) |
This equation states that the rate at which species-
particles gain momentum due to collisions with species-
particles is equal
to the rate at which species-
particles lose momentum due to collisions with species-
particles. In other words, the collisions
conserve momentum.
Finally, consider
![$\displaystyle \int K_{ss'}\,C_{ss'}\,d^3{\bf v}_s = 0.$](img660.png) |
(3.36) |
This integral is obviously zero, as indicated, as a consequence of the conservation law (3.31), as well as the fact that the kinetic
energy,
, is the same before and after an elastic collision. It follows from Equations (3.15) and (3.32) that
![$\displaystyle \int \frac{1}{2}\,m_s\,v_s^{2}\,C_{ss'}\,d^3{\bf v}_s = - \int\frac{1}{2}\,m_{s'}\,v_{s'}^{2}\,C_{s's}\,d^3{\bf v}_{s'}.$](img662.png) |
(3.37) |
This equation states that the rate at which species-
particles gain kinetic energy due to collisions with species-
particles is equal
to the rate at which species-
particles lose kinetic energy due to collisions with species-
particles.
In other words, the collisions conserve energy.