# 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 ${\displaystyle N}$  particles with the internal energy:

${\displaystyle U=\sum {\frac {1}{2}}mv_{i}^{2}+V(r_{1},\dots ,r_{N})}$

A state of the system is defined by the set of the ${\displaystyle r_{i},p_{i}}$ 's. Probability for the system to be in the volume of phase space comprised between hyperplanes ${\displaystyle r_{i},p_{i}}$  and ${\displaystyle r_{i}+dr_{i},p_{i}+dp_{i}}$  is:

${\displaystyle dP={\frac {1}{a}}e^{-\beta [\sum {\frac {1}{2}}mv_{i}^{2}+V(r_{1},\dots ,r_{N})]}}$

Probability for one particle to have a speed between ${\displaystyle v}$  and ${\displaystyle v+dv}$  is

${\displaystyle dP(v)={\frac {1}{B}}e^{-\beta {\frac {1}{2}}mv^{2}}dv_{x}dv_{y}dv_{z}}$

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

${\displaystyle dP(v_{x})=N{\sqrt {\frac {m}{2\pi k_{B}T}}}e^{-\beta {\frac {1}{2}}mv_{x}^{2}}dv_{x}}$

The distribution is Gausssian. It is known that:

${\displaystyle {\overline {v_{x}}}=0}$

and that

${\displaystyle {\overline {v_{x}^{2}}}={\frac {k_{B}T}{m}}}$

Thus:

${\displaystyle {\overline {{\frac {1}{2}}mv_{x}^{2}}}={\frac {1}{2}}k_{B}T}$

This results is in agreement with equipartition energy theorem [ph:physt:Diu89]. Each particle that crosses a surface ${\displaystyle \Sigma }$  increases of ${\displaystyle mv_{z}}$  the momentum. In the whole box, the number of molecule that have their speed comprised between ${\displaystyle v_{z}}$  and ${\displaystyle v_{z}+dv_{z}}$  is (see figure figboite)

Momentum transfered by particles in an elementary volume.}
figboite
${\displaystyle dN=NP(v_{z})dv_{z}}$

In the volume ${\displaystyle \Delta V}$  it is:

${\displaystyle \Delta (dN)=N{\frac {\Delta V}{V}}P(v_{z})dv_{z}}$

One chooses ${\displaystyle \Delta V=sv_{z}\Delta t}$ . The increasing of momentum is equal to the pressure forces power:

${\displaystyle \int _{-\infty }^{+\infty }mv_{z}.N{\frac {\Delta V}{V}}P(v_{z})dv_{z}=ps\Delta t}$

so

${\displaystyle pV=Nk_{B}T}$

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

secdesccinet

## Kinetic description

Let us introduce

${\displaystyle w(r_{1},p_{1},\dots ,r_{n},p_{n})dr_{1}\dots dr_{n}dp_{1}\dots dp_{n},}$

the probability that particle ${\displaystyle 1}$  is the phase space volume between hyperplanes ${\displaystyle r_{1},p_{1}}$  and ${\displaystyle r_{1}+dr_{1},p_{1}+dp_{1}}$ , particle ${\displaystyle 2}$  in the volume between hyperplanes ${\displaystyle r_{2},p_{2}}$  et ${\displaystyle r_{2}+dr_{2},p_{2}+dp_{2}}$ ,\dots, particle ${\displaystyle n}$  in the volume between hyperplanes ${\displaystyle r_{n},p_{n}}$  and ${\displaystyle r_{n}+dr_{n},p_{n}+dp_{n}}$ . Since partciles are undiscernable:

${\displaystyle {\frac {1}{N!}}w(r_{1},p_{1},\dots ,r_{n},p_{n})dr_{1}\dots dr_{n}dp_{1}\dots dp_{n}}$

is the probability\footnote{At thermodynamical equilibrium, we have seen that${\displaystyle w(r_{1},p_{1},\dots ,r_{n},p_{n})}$  csan be written:

${\displaystyle w(r_{1},p_{1},\dots ,r_{n},p_{n})=ae^{-\beta [\sum {\frac {1}{2}}mv_{i}^{2}+V(r_{1},\dots ,r_{N})]}}$

} that a particle is in the volume between hyperplanes ${\displaystyle r_{1},p_{1}}$  and ${\displaystyle r_{1}+dr_{1},p_{1}+dp_{1}}$ , another particle is in volume between hyperplanes ${\displaystyle r_{2},p_{2}}$  and ${\displaystyle r_{2}+dr_{2},p_{2}+dp_{2}}$ , \dots, and one last particle in volume between hyperplanes ${\displaystyle r_{n},p_{n}}$  and ${\displaystyle r_{n}+dr_{n},p_{n}+dp_{n}}$ . We have the normalization condition:

${\displaystyle \int wdr_{1}\dots dr_{n}dp_{1}\dots dp_{n}=1}$

By differentiation:

${\displaystyle \int {\frac {dw}{dt}}dr_{1}\dots dr_{n}dp_{1}\dots dp_{n}+\int w{\frac {d(dr_{1}\dots dr_{n}dp_{1}\dots dp_{n})}{dt}}=0}$

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

${\displaystyle {\frac {dw}{dt}}=0.}$

Using ${\displaystyle r}$  and ${\displaystyle p}$  definitions, this equation becomes:

${\displaystyle {\frac {\partial w}{\partial t}}+\{w,H\}=0}$

where ${\displaystyle H}$  is the hamilitonian of the system. One states the following repartition function:

${\displaystyle f_{1}(r,p,t)={\frac {1}{(N-1)!}}\int w\Pi _{i=2}^{n}dr_{i}dp_{i}}$

Intergating Liouville equation yields to:

${\displaystyle {\frac {\partial f_{1}}{\partial t}}={\frac {1}{(N-1)!}}\int \{H,w\}\Pi _{i=2}^{n}dr_{i}dp_{i}}$

and assuming that

${\displaystyle H=\sum p_{i}/2m+\sum u_{ij},}$

one obtains a hierarchy of equations called BBGKY hierarchy \index{BBGKY hierachy} binding the various functions

${\displaystyle f_{k}(r_{1},p_{1},\dots ,r_{k},p_{k},t)}$

defined by:

${\displaystyle f_{k}(r_{1},p_{1},\dots ,r_{k},p_{k},t)={\frac {1}{(N-k)!}}\int w\Pi _{i=k+1}^{n}dr_{i}dp_{i}.}$

To close the infinite hierarchy, various closure conditions can be considered. The Vlasov closure condition states that ${\displaystyle f_{2}}$  can be written:

${\displaystyle f_{2}(r_{1},p_{1},r_{2},p_{2})=f_{1}(r_{1},p_{1})f_{2}(r_{2},p_{2}).}$

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

${\displaystyle [{\frac {\partial }{\partial t}}+{\frac {p}{m}}{\frac {\partial }{\partial r}}+[F-{\frac {\partial {\bar {u}}}{\partial r}}]{\frac {\partial }{\partial p}}]f_{1}=0}$

where ${\displaystyle {\bar {u}}}$  is the mean potential. Vlasov equation can be rewritten by introducing a effective force ${\displaystyle F_{e}}$  describing the forces acting on particles in a mean field approximation:

eqvlasov

${\displaystyle [{\frac {\partial }{\partial t}}+{\frac {p}{m}}{\frac {\partial }{\partial r}}+F_{e}{\frac {\partial }{\partial p}}]f_{1}=0}$

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.