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

Introduction
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
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 $$N$$ particles with the internal energy:

A state of the system is defined by the set of the $$r_i,p_i$$'s. Probability for the system to be in the volume of phase space comprised between hyperplanes $$r_i,p_i$$  and $$r_i+dr_i,p_i+dp_i$$ is:

Probability for one particle to have a speed between $$v$$ and $$v+dv$$ is

$$B$$ is a constant which is determined by the normalization condition $$\int dP =1$$. Probability for one particle to have a speed component on the $$x$$-axis between $$v_x$$ and $$v_x+dv_x$$ is

The distribution is Gausssian. It is known that:

and that

Thus:

This results is in agreement with equipartition energy theorem. Each particle that crosses a surface $$\Sigma$$ increases of $$mv_z$$ the momentum. In the whole box, the number of molecule that have their speed comprised between $$v_z$$ and $$v_z+dv_z$$ is (see figure figboite)

In the volume $$\Delta V$$ it is:

One chooses $$\Delta V =s v_z \Delta t$$. The increasing of momentum is equal to the pressure forces power:

so

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

Kinetic description
Let us introduce

the probability that particle $$1$$ is the phase space volume between hyperplanes $$r_1,p_1$$ and $$r_1 +dr_1,p_1+dp_1$$, particle $$2$$ in the volume between hyperplanes $$r_2,p_2$$ et $$r_2 +dr_2,p_2+dp_2$$,\dots, particle $$n$$ in the volume between hyperplanes $$r_n,p_n$$ and $$r_n +dr_n,p_n+dp_n$$. Since partciles are undiscernable:

is the probability\footnote{At thermodynamical equilibrium, we have seen that$$w(r_1,p_1,\dots,r_n,p_n)$$ csan be written:

} that a particle is in the volume between hyperplanes $$r_1,p_1$$ and $$r_1 +dr_1,p_1+dp_1$$, another particle is in volume between hyperplanes $$r_2,p_2$$ and $$r_2 +dr_2,p_2+dp_2$$, \dots, and one last particle in volume between hyperplanes $$r_n,p_n$$ and $$r_n +dr_n,p_n+dp_n$$. We have the normalization condition:

By differentiation:

If the system is hamiltonian\index{hamiltonian system}, volume element is preserved during the dynamics, and $$w$$ verifies the Liouville equation :

Using $$r$$ and $$p$$ definitions, this equation becomes:

where $$H$$ 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

$$f_k(r_1,p_1,\dots,r_k,p_k,t)$$

defined by:

To close the infinite hierarchy, various closure conditions can be considered. The Vlasov closure condition states that $$f_2$$ can be written:

One then obtains the Vlasov equation \index{Vlasov equation} :

where $$\bar{u}$$ is the mean potential. Vlasov equation can be rewritten by introducing a effective force $$F_e$$ 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).