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

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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 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[edit]

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 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:

\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 thatw(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 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:

\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.