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 $d\tau$.}

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]] 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): Template:IMP/eq In the same way, mean speed of the medium is defined by: Template:IMP/eq where $dv$ is the sum of the speeds of the $dn$ particles being in the box. Template:IMP/rem Template:IMP/rem 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 $f$ : Template:IMP/eq that is, the average number of particles in volume a volume $d\tau$ is Template:IMP/eq Fluid speed is binded to first moment of $f$ : Template:IMP/eq 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}.