Introduction to Mathematical Physics/N body problems and statistical equilibrium/N body problems and kinetic description

Introduction edit

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.

Gas kinetic theory edit

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




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.}

In the volume   it is:


One chooses  . The increasing of momentum is equal to the pressure forces power:




We have recovered the perfect gas state equation presented at section secgasparfthe.


Kinetic description edit

Let us introduce


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:



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.