In this section we go back to the classical description of systems of
particles already tackled at section ---secdistclassi---. Henceforth, we are
interested in the presence probability of a particle in an elementary volume
of space phase. A short excursion out of the thermodynamical equilibrium is
also proposed with the introduction of the kinetic evolution equations.
Those equations can be used to prove conservations laws of continuous media
mechanics (mass conservation, momentum conservation, energy
conservation,\dots) as it will be shown at next chapter.
Perfect gas problem can be tackled\index{perfect gas} in the frame of a
kinetic theory\index{kinetic description}. This point of view is much
closer to classical mechanics that statistical physics and has the advantage to
provide more "intuitive" interpretation of results.
Consider a system of particles with the internal energy:
A state of the system is defined by the set of the 's. Probability
for the system to be in the volume of phase space comprised between
hyperplanes and is:
Probability for one particle to have a speed between and is
is a constant which is determined by the normalization condition
.
Probability for one particle to have a speed component on the
-axis between and is
The distribution is Gausssian. It is known that:
and that
Thus:
This results is in agreement with equipartition energy theorem [ph:physt:Diu89].
Each particle that crosses a surface increases of
the momentum.
In the whole box, the number of molecule that have their speed comprised
between and is (see figure figboite)
Momentum transfered by particles in an elementary volume.}
figboite
In the volume it is:
One chooses . The increasing of momentum is equal to
the pressure forces power:
so
We have recovered the perfect gas state equation presented at section secgasparfthe.
the probability that particle is the phase space volume between
hyperplanes and , particle in the volume
between hyperplanes et ,\dots,
particle in the volume between hyperplanes and . Since partciles are undiscernable:
is the probability\footnote{At thermodynamical equilibrium, we have seen that csan be written:
}
that a particle is in the volume between hyperplanes
and
,
another particle is in volume between hyperplanes
and
,
\dots, and one last particle in volume between hyperplanes
and
.
We have the normalization condition:
By differentiation:
If the system is hamiltonian\index{hamiltonian system}, volume element is
preserved during the dynamics, and
verifies the Liouville equation :
Using and
definitions, this equation becomes:
where
is the hamilitonian of the system. One states the following repartition function:
Intergating Liouville equation yields to:
and assuming that
one obtains a hierarchy of equations called
BBGKY hierarchy
\index{BBGKY hierachy}
binding the various functions
defined by:
To close the infinite hierarchy, various closure conditions can be
considered. The Vlasov closure condition states that
can be written:
One then obtains the Vlasov equation \index{Vlasov equation} :
where
is the mean potential. Vlasov equation can be rewritten by introducing a effective force
describing the forces acting on particles in a mean field approximation:
eqvlasov
The various moments of Vlasov equation allow to prove the conservation
equations of mechanics of continuous media (see chapter chapapproxconti).
Remark:
Another dynamical equation close to Vlasov equation is the {\bf Boltzmann equation}
\index{Boltzmann}(see [ph:physt:Diu89]. Difference between both
equation relies on the way to treat collisions.