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:

$U=\sum \frac{1}{2}mv_i^2+V(r_1,\dots,r_N)$

A state of the system is defined by the set of the $r_i,p_i$'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:

$dP=\frac{1}{a}e^{-\beta[\sum \frac{1}{2}mv_i^2+V(r_1,\dots,r_N)]}$

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

$dP(v)=\frac{1}{B}e^{-\beta \frac{1}{2}mv^2}dv_xdv_ydv_z$

$B$ 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

$dP(v_x)=N\sqrt{\frac{m}{2\pi k_BT}}e^{-\beta \frac{1}{2}mv_x^2}dv_x$

The distribution is Gausssian. It is known that:

$\overline{v_x}=0$

and that

$\overline{v^2_x}=\frac{k_BT}{m}$

Thus:

$\overline{\frac{1}{2}mv^2_x}=\frac{1}{2}k_BT$

This results is in agreement with equipartition energy theorem [ph:physt:Diu89]. 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)

Momentum transfered by particles in an elementary volume.}

figboite

$dN=NP(v_z)dv_z$

In the volume $\Delta V$ it is:

$\Delta(dN)=N\frac{\Delta V}{V}P(v_z)dv_z$

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

$\int_{-\infty}^{+\infty}m v_z.N\frac{\Delta V}{V}P(v_z)dv_z=p s \Delta t$

so

$pV=Nk_BT$

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

secdesccinet

Kinetic description

Let us introduce

$w(r_1,p_1,\dots,r_n,p_n)dr_1\dots dr_n dp_1\dots dp_n,$

the probability that particle $1$ is the the phase space volume between hyperplanes $r_1,p_1$ and $r_1 +dr_1,p_1+dp_1$, particle $2$ inthe 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:

$\frac{1}{N!}w(r_1,p_1,\dots,r_n,p_n)dr_1\dots dr_n dp_1\dots dp_n$

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

$w(r_1,p_1,\dots,r_n,p_n)=ae^{-\beta[\sum \frac{1}{2}mv_i^2+V(r_1,\dots,r_N)]}$

} that a particle is inthe 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:

$\int w dr_1\dots dr_n dp_1\dots dp_n=1$

By differentiation:

$\int \frac{dw}{dt} dr_1\dots dr_n dp_1\dots dp_n+\int w\frac{d(dr_1\dots dr_n dp_1\dots dp_n)}{dt}=0$

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

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

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

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

where $H$ is the hamilitonian of the system. One states the following repartition function:

$f_1(r,p,t)=\frac{1}{(N-1)!}\int w \Pi_{i=2}^n dr_idp_i$

Intergating Liouville equation yields to:

$\frac{\partial f_1}{\partial t}=\frac{1}{(N-1)!}\int \{H,w\} \Pi_{i=2}^n dr_idp_i$

and assuming that

$H=\sum p_i/2m+\sum u_{ij},$

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:

$f_k(r_1,p_1,\dots,r_k,p_k,t)=\frac{1}{(N-k)!}\int w \Pi_{i=k+1}^ndr_idp_i.$

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

$f_2(r_1,p_1,r_2,p_2)=f_1(r_1,p_1)f_2(r_2,p_2).$

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

$[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{\partial r}+[F-\frac{\partial \bar{u}}{\partial r}]\frac{\partial }{\partial p}]f_1=0$

where $\bar{u}$ 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:

eqvlasov

$[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{\partial r}+F_e\frac{\partial }{\partial p}]f_1=0$

The various momets 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 Boltzman equation} \index{Boltzman}(see [ph:physt:Diu89]. Difference betwen both equation relies on the way to treat collisions.