Collisional Conservation Laws

Consider

$$\int C_{ss'}\,d^3{\bf v}_s = \int\!\!\int\!\!\int\!\!\int u_{ss'}... ...'-f_s\,f_{s'})\,d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}',$$ (3.28)

which follows from Equation (3.23). Interchanging primed and unprimed dummy variables of integration on the right-hand side, we obtain

$$\int C_{ss'}\,d^3{\bf v}_s = \int\!\!\int\!\!\int\!\!\int u_{ss'}... ...f_s'\,f_{s'}')\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'\,d^3{\bf v}_s\,d^3{\bf v}_{s'}.$$ (3.29)

Hence, making use of Equation (3.18), as well as the fact that $u_{ss'}'=u_{ss'}$, we deduce that

$$\int C_{ss'}\,d^3{\bf v}_s =- \int\!\!\int\!\!\int\!\!\int u_{ss'}\,\sigma'({\bf v}_s, {\bf ... ...}'-f_s\,f_{s'})\,d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'$$

$$= - \int C_{ss'}\,d^3{\bf v}_s,$$ (3.30)

which implies that

$$\int C_{ss'}\,d^3{\bf v}_s=0.$$ (3.31)

The previous expression states that collisions with species-$s'$ particles give rise to zero net rate of change of the number density of species-$s$ particles at position ${\bf r}$ and time $t$. In other words, the collisions conserve the number of species-$s$ particles. Now, it is easily seen from Equations (3.23) and (3.24) that

$$C_{ss'}\,d^3{\bf v}_s=C_{s's}\,d^3{\bf v}_{s'}.$$ (3.32)

Hence, Equation (3.31) also implies that

$$\int C_{s's}\,d^3{\bf v}_{s'}=0.$$ (3.33)

In other words, collisions also conserve the number of species-$s'$ particles.

Consider

$$(m_s+m_{s'}) \int {\bf U}_{ss'}\,C_{ss'}\,d^3{\bf v}_s = {\bf0}.$$ (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, ${\bf U}_{ss'}$, 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

$$\int m_s\,{\bf v}_s\,C_{ss'}\,d^3{\bf v}_s = - \int m_{s'}\,{\bf v}_{s'}\,C_{s's}\,d^3{\bf v}_{s'}.$$ (3.35)

This equation states that the rate at which species-$s$ particles gain momentum due to collisions with species-$s'$ particles is equal to the rate at which species-$s'$ particles lose momentum due to collisions with species-$s$ particles. In other words, the collisions conserve momentum.

Finally, consider

$$\int K_{ss'}\,C_{ss'}\,d^3{\bf v}_s = 0.$$ (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, $K_{ss'}$, is the same before and after an elastic collision. It follows from Equations (3.15) and (3.32) that

$$\int \frac{1}{2}\,m_s\,v_s^{2}\,C_{ss'}\,d^3{\bf v}_s = - \int\frac{1}{2}\,m_{s'}\,v_{s'}^{2}\,C_{s's}\,d^3{\bf v}_{s'}.$$ (3.37)

This equation states that the rate at which species-$s$ particles gain kinetic energy due to collisions with species-$s'$ particles is equal to the rate at which species-$s'$ particles lose kinetic energy due to collisions with species-$s$ particles. In other words, the collisions conserve energy.