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# Collisional Conservation Laws

Consider

 (3.25)

which follows from Equation (3.20). Interchanging primed and unprimed dummy variables of integration on the right-hand side, we obtain

 (3.26)

Hence, making use of Equation (3.16), as well as the fact that , we deduce that

 (3.27)

which implies that

 (3.28)

The previous expression states that collisions with particles of type give rise to zero net rate of change of the number density of particles of type at position and time . In other words, the collisions conserve the number of particles of type . Now, it is easily seen from Equations (3.20) and (3.21) that

 (3.29)

Hence, Equation (3.28) also implies that

 (3.30)

In other words, collisions also conserve the number of particles of type .

Consider

 (3.31)

This integral is obviously zero, as indicated, as a consequence of the conservation law (3.28), 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 (3.29), the previous expression can be rewritten in the form

 (3.32)

This equation states that the rate at which particles of type gain momentum due to collisions with particles of type is equal to the rate at which particles of type lose momentum due to collisions with particles of type . In other words, the collisions conserve momentum.

Finally, consider

 (3.33)

This integral is obviously zero, as indicated, as a consequence of the conservation law (3.28), 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 (3.29) that

 (3.34)

This equation states that the rate at which particles of type gain kinetic energy due to collisions with particles of type is equal to the rate at which particles of type lose kinetic energy due to collisions with particles of type . In other words, the collisions conserve energy.

Next: Boltzmann H-Theorem Up: Collisions Previous: Boltzmann Collision Operator
Richard Fitzpatrick 2016-01-23