Boltzmann H-Theorem

Equation (3.20) can be written

$$\left(\frac{\partial f_1}{\partial t}\right)_2 = \int\!\!\int\!\!... ...\bf v}_2')\,(f_1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.35)

Consider the quantity

$$H = \int f_1\,\ln f_1\,d^3{\bf v}_1.$$ (3.36)

It follows from Equation (3.35) that

$$\frac{dH}{dt} =\int (1+\ln f_1)\,\frac{\partial f_1}{\partial t}\,d^3{\bf v}_1$$

$$=\int\!\! \int\!\!\int\!\!\int u\,\sigma\,(1+\ln f_1)\,(f_1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_1\,d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2',$$ (3.37)

where $\sigma$ is short-hand for $\sigma({\bf v}_1, {\bf v}_2; {\bf v}_1',{\bf v}_2')$ . Suppose that we swap the dummy labels $1$ and $2$ . This process leaves both $u= \vert{\bf v}_1-{\bf v}_2\vert$ 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 $\sigma({\bf v}_1, {\bf v}_2; {\bf v}_1',{\bf v}_2')$ unchanged. Hence, we deduce that

$$\frac{dH}{dt}=\int\!\! \int\!\!\int\!\!\int u\,\sigma\,(1+\ln f_2... ..._1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_1\,d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.38)

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 $u'=u$ , we obtain

$$\frac{dH}{dt}=-\int\!\! \int\!\!\int\!\!\int u\,\sigma\,(1+\ln f_... ..._1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_1\,d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.39)

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

$$\frac{dH}{dt}=-\int\!\! \int\!\!\int\!\!\int u\,\sigma\,(1+\ln f_... ..._1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_1\,d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.40)

The previous four equations can be combined to give

$$\frac{dH}{dt}=\frac{1}{4}\int\!\! \int\!\!\int\!\!\int u\,\sigma\... ..._1'\,f_2'-f_1\,f_2)\, d^3{\bf v}_1\,d^3{\bf v}_2\,d^3{\bf v}_1'\,d^3{\bf v}_2'.$$ (3.41)

Now, $\ln(f_1\,f_2/f_1'\,f_2')$ is positive when $f_1'\,f_2'-f_1\,f_2$ 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,

$$\frac{dH}{dt} \leq 0.$$ (3.42)

This result is known as the Boltzmann H-theorem.

In fact, the quantity $H$ is bounded below (i.e., it cannot take the value minus infinity). Hence, $H$ cannot decrease indefinitely, but must tend to a limit in which $dH/dt =0$ . According to Equation (3.41), the distribution function associated with this limiting state is characterized by

$$f_1\,f_2 = f_1'\,f_2',$$ (3.43)

or, equivalently,

$$\ln f_1 + \ln f_2 - \ln f_1' - \ln f_2'=0.$$ (3.44)

Consider a distribution function that satisfies

$$\ln f_i = a_i + m_i\,{\bf b}\cdot{\bf v}_i+ m_i\,c\,v_i^{\,2},$$ (3.45)

where $i$ is a species label, $m_i$ is the particle mass, and $a_i$ , ${\bf b}$ , and $c$ are constants. It follows that

$$\ln f_1 + \ln f_2 - \ln f_1' - \ln f_2' = {\bf b}\cdot(m_1\,{\bf v}_1+m_2\,{\bf v_2}-m_1\,{\bf v}_1'-m_2\,{\bf v}_2')$$

$$\phantom{=}+ c\,(m_1\,v_1^{\,2} + m_2\,v_2^{\,2}-m_1\,v_1'^{\,2}-m_2\,v_2'^{\,2}).$$ (3.46)

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

$$m_1\,{\bf v}_1 + m_2\,{\bf v}_2 = m_1\,{\bf v}_1'+m_2\,{\bf v}_2',$$ (3.47)

whereas energy conservation yields (see Section 3.3)

$$m_1\,v_1^{\,2} + m_2\,v_2^{\,2} =m_1\,v_1'^{\,2}+m_2\,v_2'^{\,2}.$$ (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

$$a_i = \ln\left[n_i\,\left(\frac{m_i}{2\pi\,T}\right)^{3/2}\right] - \frac{m_i\,V^{\,2}}{2\,T},$$ (3.49)

$${\bf b} = \frac{1}{T}\,{\bf V},$$ (3.50)

$$c = -\frac{1}{2\,T},$$ (3.51)

where $n_i$ , ${\bf V}$ , and $T$ are constants. In this case, Equation (3.45) becomes

$$f_i= n_i\left(\frac{m_i}{2\pi\,T}\right)^{3/2}\,\exp\left[-\frac{m_i\,({\bf v}_i-{\bf V})^{\,2}}{2\,T}\right],$$ (3.52)

which we recognize as a Maxwellian distribution function (Reif 1965). It is easily demonstrated that

$$n_i = \int f_i\,d^3{\bf v}_i,$$ (3.53)

$$n_i\,{\bf V} = \int {\bf v}_i\,f_i\,d^3{\bf v}_i,$$ (3.54)

$$\frac{3}{2}\,n_i\,T = \int \frac{1}{2}\,m_i\,v^2\,f_i\,d^3{\bf v}_i.$$ (3.55)

These relations allow us to identify the constants $n_i$ , ${\bf V}$ , and $T$ with the species-$i$ 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.