Introduction to Mathematical Physics/Continuous approximation/Matter conservation

Setting $A_{i}=\rho ={\frac {dm}{d\tau }}$ in equation eqcon one obtains the matter conservation equation:

${\frac {\partial \rho }{\partial t}}+{\mbox{ div }}(\rho u)=0$

$\rho$ is called the volumic mass. This law can be proved by calculations using elementary fluid volumes. So, variation of mass $M$ in volume $d\tau$ per time unit is opposed to the outgoing mass flow:

${\frac {dM}{dt}}=-\int _{d\tau }\rho vdr$

Local form of this equation is thus:

${\frac {\partial \rho }{\partial t}}+{\mbox{ div }}(\rho u)=0$

But this law can also be proved in calculating the first moment of the Vlasov equation (see equation eqvlasov). Volumic mass is then defined as the zeroth order moment of the repartition function times mass $m$ of one particle:

$\rho =m\int dpf(r,p,t)$

Taking the first moment of Vlasov equation, it yields:

${\frac {\partial \rho }{\partial t}}+\nabla (\rho v)=0$

Charge conservation equation is completely similar to mass conservation equation:

${\frac {\partial \rho }{\partial t}}+{\mbox{ div }}{j}=0$

where here $\rho$ is the volumic charge and $j$ the electrical current density:

$j=\rho v$

Flow of $j$ trough an open surface $S$ is usually called electrical current going through surface $S$.