Momentum average

We will establish a relationship with the Navier-Stokes equation which describes the momentum conserving dynamics of a continuum as a set of differential equations for a space and time dependent velocity field $\overline v=\overline v(x,t)$. So let’s see what we can do with our transport equation and first multiply with $mv$ and integrate over $v$, assuming $m=\text{const}$:

\[\int{\rm d}^3v mv\partial_tf +\int{\rm d}^3v mv(v\cdot\nabla_x)f +\int{\rm d}^3v mv(a\cdot\nabla_v)f =\int{\rm d}^3v mv\left.\partial_tf\right|_{\text{Streu}},\]

in components:

\[\int{\rm d}^3v mv_j\frac{\partial f}{\partial t} +\sum_i \left(\frac\partial{\partial x_i} \int{\rm d}^3v mv_iv_jf + \int{\rm d}^3v ma_iv_j \frac{\partial f}{\partial v_i} \right) =\int{\rm d}^3v mv_j\left.\frac{\partial f}{\partial t}\right|_{\text{Streu}}.\]

We need some definitions:

• mass density
  $\displaystyle\rho(x,t):=\int{\rm d}^3vmf(x,v,t).$

• average velocity
  $\displaystyle\overline v(x,t):=\frac1\rho\int{\rm d}^3vmvf(x,v,t).$

The first term in the integral equation above then acquires the form

\[\int{\rm d}^3v mv_j\frac{\partial f}{\partial t}=\frac\partial{\partial t}(\rho\overline v_j).\]

For the second term, we define $v=\overline v+\delta v$, $\delta v:=v-\overline v$,

\[\begin{eqnarray*} \rho\overline{v_iv_j} &:=& \int{\rm d}^3v mv_iv_jf = \int{\rm d}^3vm(\overline v_i+\delta v_i)(\overline v_j+\delta v_j)f \\ &=& \int{\rm d}^3vm\left(\overline v_i\overline v_j+\overline v_i\delta v_j +\overline v_j\delta v_i+\delta v_i\delta v_j\right)f \\ &=& \rho\overline v_i\overline v_j+\rho\overline v_i\overline{\delta v_j} +\rho\overline v_j\overline{\delta v_i}+\rho\overline{\delta v_i\delta v_j} \\ \Rightarrow\sum_i\frac\partial{\partial x_i}\int{\rm d}^3v mv_iv_jf &=& \sum_i\frac\partial{\partial x_i}\left(\rho\overline v_i\overline v_j +\rho\overline{\delta v_i\delta v_j}\right). \end{eqnarray*}\]

For the third contribution to our integrated Boltzmann equation, we use Gauß’ theorem and assume that $f\to0$ for large enough velocities $v$. With $v_j\partial f/\partial v_i=\partial(v_jf)/\partial v_i-\delta_{ij}f$, we obtain

\[\begin{eqnarray*} \int{\rm d}^3vm\sum_ia_iv_j \frac{\partial f}{\partial v_i} &=& \int{\rm d}^3v m\sum_ia_i \left[ \frac\partial{\partial v_i} \left(v_jf\right) -\delta_{ij}f \right] \\ &=& \int{\rm d}^3v\sum_i\frac\partial{\partial v_i}\left(a_i v_jmf\right) -a_j\int{\rm d}^3v mf \\ &=& \int{\rm d}^3v\nabla\cdot av_jmf-a_j\int{\rm d}^3v mf \\ &=& \int_{\partial V}{\rm d}An\cdot av_jmf-\rho a_j \\ &=& -\rho a_j \end{eqnarray*}\]

for velocity-independent external forces. Averaging over the collision term as well we eventually obtain the momentum-averaged form of Boltzmann’s equation,

\[\fbox{$\displaystyle \frac\partial{\partial t} \rho\overline v_j +\sum_i \frac\partial{\partial x_i} \left( \rho\overline v_i\overline v_j +\rho\overline{\delta v_i\delta v_j} \right) -\rho a_j = \left. \frac\partial{\partial t} \rho\overline v_j \right|_{\text{Streu}}. $}\]

We stress that this is an equation for the mean velocity of a fluid $\overline v=\overline v(x,t)$ and space- and time-dependent fluctuations around this.

Next: 4 The Navier-Stokes equation