Calculation of the Currents

We begin with calculating explicitly the expression for the mass current namely,

$$\vec{J}_{a}=m_{a}\int\vec{c}_{a}f_{a}^{(0)}\varphi_{a}^{(1)}d\vec{c}_{a}$$

which according to Eq. (3.13), omitting $\mathop{\mathrm{grad}}\vec{u}$ terms yields

$$\vec{J}_{a}=m_{a}\int\vec{c}_{a}f_{a}^{(0)}\left\{\vec{\mathbb {A}}_{a}\cdot\mbox{grad }\ln T+\vec{\mathbb{D}}_{a}\cdot\vec {d}_{ab}\right\}d\vec{c}_{a}$$

where $\vec{\mathbb{A}}_{a}$ and $\vec{\mathbb{D}}_{a}$ as well, have the form expressed in Eq. (3.15′).