Introduction to Mathematical Physics/Continuous approximation/Introduction

There exist several ways to introduce the matter continuous approximation. They are different approaches of the averaging over particles problem. The first approach consists in starting from classical mechanics and to consider means over elementary volumes called "fluid elements". [[ Image:

 volele   
 | center | frame |Les moyennes sont faite dans la boite \'el\'ementaire de

volume $d\tau$.}

figvolele

]] Let us consider an elementary volume $d\tau$ centred at $r$, at time $t$. Figure figvolele illustrates this averaging method. Quantities associated to continuous approximation are obtained from passage to the limit when box-size tends to zero.

So the particle density is the extrapolation of the limit when box volume tends to zero of the ratio of $dn$ (the number of particles in the box) over $d\tau$ (the volume of the box):

$n(r,t)=\lim _{d\tau \rightarrow 0}{\frac {dn}{d\tau }}$

In the same way, mean speed of the medium is defined by:

$v(r,t)=\lim _{d\tau \rightarrow 0}{\frac {dv}{dn}}$

where $dv$ is the sum of the speeds of the $dn$ particles being in the box.

Remark:

Such a limit passage is difficult to formalize on mathematical point of view. It is a sort of "physicist limit"!

Remark:

Note that the relation between particles characteristics and mean characteristics are not always obvious. It can happen that the speed of the particles are non zero but that their mean is zero. It is the case if particles undergo thermic agitation (if no convection). But it can also happen that the temporal averages of individual particle speed are zero, and that in the same time the speed of the fluid element is non zero! Figure

figvitnonul illustrates this remark in the case of the drift

phenomenon ([#References

Another method consists in considering the repartition function for one particle $f(r,p,t)$ introduced at section

secdesccinet. Let us recall that ${\frac {1}{n!}}f(r,p,t)drdp$ represents

the probability to find at time $t$ a particle in volume of space phase between $r,p$ and $r+dr,p+dp$. The various fluid quantities are then introduced as the moments of $f$ with respect to speed. For instance, particle density is the zeroth order moment of $f$ :

$n(r,t)=\int f(r,p,t)dp$

that is, the average number of particles in volume a volume $d\tau$ is

$dn=n(r,t)d\tau .$

Fluid speed is binded to first moment of $f$ :

$mv(r,t)=\int pf(p,r,t)dp$

The object of this chapter is to present laws governing the dynamics of a continuous system. In general, those laws can be written as {\bf conservation

 laws}.