Boltzmann H-Theorem

Consider a spatially uniform plasma in the absence of electromagnetic fields. The kinetic equation, (3.9), reduces to

$$\frac{\partial f_s}{\partial t} = \sum_{s'} C_{ss'}(f_s,f_{s'}).$$ (3.123)

Let us investigate the properties of this equation.

Consider the quantity

$$H = \sum_s\int f_s\,\ln f_s\,d^3{\bf v}_s.$$ (3.124)

It follows from Equation (3.123) that

$$\frac{dH}{dt} = \sum_{s,s'}\int (1+\ln f_s)\,C_{ss'}(f_s,f_{s'})\,d^3{\bf v}_s.$$ (3.125)

Making use of the Boltzmann form of the collision operator, (3.23), the previous equation becomes

$$\frac{dH}{dt}=\sum_{s,s'}\int\!\! \int\!\!\int\!\!\int u_{ss'}\,\... ...-f_s\,f_{s'})\, d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}',$$ (3.126)

where $\sigma_{ss'}'$ is short-hand for $\sigma'({\bf v}_s, {\bf v}_{s'};{\bf v}_s',{\bf v}_{s'}')$. Suppose that we swap the dummy species labels $s$ and $s'$. This process leaves both $u_{ss'}= \vert{\bf v}_s-{\bf v}_{s'}\vert$ and the value of the integral unchanged. According to Equation (3.27), it also leaves the quantity $\sigma'({\bf v}_s, {\bf v}_{s'};{\bf v}_s',{\bf v}_{s'}')$ unchanged. Hence, we deduce that

$$\frac{dH}{dt}=\sum_{s,s'}\int\!\! \int\!\!\int\!\!\int u_{ss'}\,\... ...-f_s\,f_{s'})\, d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'.$$ (3.127)

Suppose that we swap primed and unprimed dummy variables of integration in Equation (3.126). This leaves the value of the integral unchanged. Making use of Equation (3.18), as well as the fact that $u_{ss'}'=u_{ss'}$, we obtain

$$\frac{dH}{dt}=-\sum_{s,s'}\int\!\! \int\!\!\int\!\!\int u_{ss'}\,... ...-f_s\,f_{s'})\, d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'.$$ (3.128)

Finally, swapping primed and unprimed variables in Equation (3.127) yields

$$\frac{dH}{dt}=-\sum_{s,s'}\int\!\! \int\!\!\int\!\!\int u_{ss'}\,... ...'-f_s\,f_{s'})\, d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}.$$ (3.129)

The previous four equations can be combined to give

$$\frac{dH}{dt}=\sum_{s,s'}\frac{1}{4}\int\!\! \int\!\!\int\!\!\int... ...-f_s\,f_{s'})\, d^3{\bf v}_s\,d^3{\bf v}_{s'}\,d^3{\bf v}_s'\,d^3{\bf v}_{s'}'.$$ (3.130)

Now, $\ln(f_s\,f_{s'}/f_s'\,f_{s'}')$ is positive when $f_s'\,f_{s'}'-f_s\,f_{s'}$ 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,

$$\frac{dH}{dt} \leq 0.$$ (3.131)

This result is known as the Boltzmann H-theorem (Boltzmann 1995).

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 (3.130), the distribution functions associated with this limiting state are characterized by

$$f_s\,f_{s'} = f_s'\,f_{s'}',$$ (3.132)

or, equivalently,

$$\ln f_s+ \ln f_{s'} - \ln f_s' - \ln f_{s'}'=0.$$ (3.133)

Consider distribution functions that satisfy

$$\ln f_s = a_s + m_s\,{\bf b}\cdot{\bf v}_s+ c\,m_s\,v_s^{2},$$ (3.134)

where $s$ is a species label, $m_s$ is the particle mass, and $a_s$, ${\bf b}$, and $c$ are constants. It follows that

$$\ln f_s + \ln f_{s'} - \ln f_s' - \ln f_{s'}' = {\bf b}\cdot(m_s\,{\bf v}_s+m_{s'}\,{\bf v}_{s'}-m_s\,{\bf v}_s'-m_{s'}\,{\bf v}_{s'}')$$

$$\phantom{=}+ c\,(m_s\,v_s^{2} + m_{s'}\,v_{s'}^{2}-m_s\,v_s'^{2}-m_{s'}\,v_{s'}'^{2}).$$ (3.135)

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

$$m_s\,{\bf v}_s + m_{s'}\,{\bf v}_{s'} = m_s\,{\bf v}_s'+m_{s'}\,{\bf v}_{s'}',$$ (3.136)

whereas energy conservation yields (see Section 3.3)

$$m_s\,v_s^{2} + m_{s'}\,v_{s'}^{2} =m_s\,v_s'^{2}+m_{s'}\,v_{s'}'^{2}.$$ (3.137)

In other words, distribution functions that satisfies Equation (3.134) automatically satisfy Equation (3.133). We, thus, conclude that collisions act to drive the distribution functions for the colliding particles towards particular distribution functions of the form (3.134). [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 (3.134). Hence, in the absence of other conservation laws, we can be sure that Equation (3.134) is the most general expression that satisfies Equation (3.133).]

Without loss of generality, we can set

$$a_s = \ln\left[n_s\,\left(\frac{m_s}{2\pi\,T}\right)^{3/2}\right] - \frac{m_s\,V^{2}}{2\,T},$$ (3.138)

$${\bf b} = \frac{1}{T}\,{\bf V},$$ (3.139)

$$c = -\frac{1}{2\,T},$$ (3.140)

where $n_s$, ${\bf V}$, and $T$ are constants. In this case, Equation (3.134) becomes

$$f_s= n_s\left(\frac{m_s}{2\pi\,T}\right)^{3/2}\,\exp\left[-\frac{m_s\,({\bf v}_s-{\bf V})^{2}}{2\,T}\right],$$ (3.141)

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

$$n_s = \int f_s\,d^3{\bf v}_s,$$ (3.142)

$$n_s\,{\bf V} = \int {\bf v}_s\,f_s\,d^3{\bf v}_s,$$ (3.143)

$$\frac{3}{2}\,n_s\,T = \int \frac{1}{2}\,m_s\,v_s^{2}\,f_s\,d^3{\bf v}_s.$$ (3.144)

These relations allow us to identify the constants $n_s$, ${\bf V}$, and $T$ with the species-$s$ number density, mean flow velocity, and kinetic temperature, respectively. We conclude that collisions tend to relax the distribution functions of the colliding particles toward Maxwellian distributions characterized by a common mean flow velocity and a common temperature.