Boltzmann H-Theorem

Consider the quantity

It follows from Equation (3.35) that

where is short-hand for . Suppose that we swap the dummy labels and . This process leaves both and the value of the integral unchanged [assuming that there is an implicit summation over different species in Equation (3.36)]. According to Equation (3.24), it also leaves the scattering cross-section unchanged. Hence, we deduce that

Suppose that we swap primed and unprimed dummy variables of integration in Equation (3.37). This leaves the value of the integral unchanged. Making use of Equation (3.16), as well as the fact that , we obtain

(3.39) |

Finally, swapping primed and unprimed variables in Equation (3.38) yields

(3.40) |

The previous four equations can be combined to give

Now, is positive when is negative, and vice versa. We, therefore, deduce that the integral on the right-hand side of the previous expression can never take a positive value. In other words,

(3.42) |

This result is known as the

In fact, the quantity is bounded below (i.e., it cannot take the value minus infinity). Hence, cannot decrease indefinitely, but must tend to a limit in which . According to Equation (3.41), the distribution function associated with this limiting state is characterized by

(3.43) |

or, equivalently,

Consider a distribution function that satisfies

where is a species label, is the particle mass, and , , and are constants. It follows that

(3.46) |

However, for an elastic collision, momentum conservation implies that (see Section 3.3)

(3.47) |

whereas energy conservation yields (see Section 3.3)

(3.48) |

In other words, a distribution function that satisfies Equation (3.45) automatically satisfies Equation (3.44). We, thus, conclude that collisions act to drive the distribution functions for the colliding particles towards particular distribution functions that satisfy Equation (3.45). [Incidentally, elastic collisions generally only conserve particle number, particle momentum, and particle energy. These conservation laws correspond to the three terms appearing on the right-hand side of Equation (3.45). Hence, in the absence of other conservation laws, we can be sure that Equation (3.45) is the most general expression that satisfies Equation (3.44).]

Without loss of generality, we can set

(3.49) | ||

(3.50) | ||

(3.51) |

where , , and are constants. In this case, Equation (3.45) becomes

(3.52) |

which we recognize as a

(3.53) | ||

(3.54) | ||

(3.55) |

These relations allow us to identify the constants , , and with the species- number density, mean flow velocity, and kinetic temperature, respectively. We conclude that collisions tend to relax the distribution functions for the colliding particles toward Maxwellian distributions characterized by a common mean flow velocity and a common temperature.