Handbook of Mathematical Fluid Dynamics (eBook)

2(N-1)Uε(t-t1)C1,2Uε(t1)f2(x1,υ1)=ε2(N-1)∫dυ2∫dn(υ1-υ2)×nf2φ2-t1x1-υ1(t-t1),υ1,x1-εn,υ2.

Repeating the argument, we are going back with particle 1 for the time t - t1, we join a new particle (say 2), with velocity υ2 at distance ε, solve the two-body problem for the time -t1. Finally we integrate with respect to n1, υ2 and t1. There are two possibilities which is convenient to separate. Either (υ1 - υ2) · n is positive (remind that n is -x2|y-y2| where y is given by x1 - υ1 (t - t1)) and then the velocities (υ1, υ2) is a post-collisional pair. This means that the back collision takes place. Or (υ1 - υ2) · n is negative. In this case the velocities are pre-collisional and the argument of f2 is

1-υ1t,υ1,x1-εn-υ2t1,υ2.

In the first case we have to solve the two-body problem. However, we observe that the time in which the pair of particles are really interacting is O(ε) so that the argument of f0.2 is very close to

1-υ1(t,-t1)-υ′1t1,υ′1,x1-υ′2t1,υ′2,

where ′1,υ′2 are the pre-collisional pair associated to υ1, υ2 according to (3.17). They are, as in the previous section, expressed in terms of ω so that, assuming continuity of the initial distributions, we arrive to

2(N-1)Uε(t-t1)C1,2Uε(t1)f2(x1,υ1)→λ-1∫dυ2∫dωB|υ1-υ2|;ω×f0,2x1-υ1(t-t1)-υ′1t1,υ′1,x1-υ′2t1,υ′2-f0,2x1-υ1t,υ1,x1-υ2t1,υ2.

The argument can be extended, using (3.54), with some care, to the general case.

It is convenient to represent the generic term of order m, in the series expansion (3.50) by means of a diagram which is a collection of j binary tree of the form shown in Figure 4, which expresses at what particle of the previous configuration the new particle has to be attached. It is clear that, given a diagram as in Figure 4 and the sequences t1, …, tm, υj+1, …, υj+m, ω1, …, ωm, we can construct the initial configuration (Yj+m, Wj+m) by solving the equation of the motion. Moreover, for almost all (Xj, Vj) and outside a small measure set in dt1 … dtm dυj+l … dυj+m dωl … dωm, if ε is small, the motion is a collection of distinct two-body interactions which are practically instantaneous.

We now observe that the formal limit

N-j)ε2Cj,j+1εfj+1N(x1,υ1,…,xj,υj)→Cj,j+1fj+1N(x1,υ1,…,xj,υj)

defines the operator

j,j+1fj+1(x1,υ1,…,xj,υj)=λ-1∑k=1j∫dυj+1∫S+dωB|υk-υj+1|,ω×fj+1x1,υ1,…,xk,υ′k,…,xk,υ′j+1-fj+1(x1,υ1,…,xk,υk,…,xk,υj+1).

We have also a formal limit hierarchy which is

tfj+∑i=1jυi⋅∇ifj=Cj,j+1fj+1,

because jε→-∑i=1jυi⋅∇i whenever ε → 0, since j is fixed.

Notice that (3.59) are an unbounded set of equations while (3.47) consist in a finite set of equations. Indeed, in spite of their formal similarity, (3.59) and (3.47) are very different, the former being associated to a time reversible Hamiltonian system and the latter to an irreversible (stochastic) system. We shall remark later this important aspects. For the moment we outline a remarkable property of the hierarchy (3.59). Suppose that initially the sequence of distribution functions f0,j}j=1∞ factorizes, namely

0,j(x1,υ1,…,xk,υk)=∏k=1jf0(xk,υk).

Then a solution of (3.59) can be produced by putting

j(x1,υ1,…,xk,υk;t)=∏k=1jf(xk,υk;t),

where f(xk, υk; t) coinciding with f1(xk, υk; t) solves

∂t+υ⋅∇)f=Q(f,f),

where

(f,f)(x,υ)=λ-1∫dυ1∫S+B|υ-υ1|,ω×f(x,υ′)f(x,υ′1)-f(x,υ)f(x,υ1)

is the Boltzmann collision operator. For this reason (3.59) is called the Boltzmann hierarchy.

The factorization property (3.61) (propagation of chaos) states that if initially position and momentum of a given particle are distributed independently of the positions and momenta of all the others, such property is maintained during the time evolution. This is not true for the real particle dynamics which creates correlations. Therefore the propagation of chaos can be verified only in a limiting situation and it will be consequence of the convergence result we are going to illustrate. The physical meaning of this property is now transparent. If we look at the two-particle distribution f2(x1, υ1, x2, υ2, t) at time t, this is a sum of contributions of diagrams obtained by the perturbative expansion (3.50). If f2(x1, υ1, x2, υ2, 0) factorizes this is essentially the product of the sum of the diagrams arising in the expansions of f1 (x1, υ1, t) and f1 (x2, υ2, t) unless the particles of the two diagrams interact. But this is a negligible event (O(ε2) for each pair of particles) so that the factorization holds in the limit.

We now want to show that the solution of the hierarchy (3.47) given by the expansion (3.50) converges to the solutions of the Boltzmann hierarchy (3.59) given by the analogous expansion

j(t)=∑m≥0λ-m∫0tdt1∫0t1dt2…∫0tm-1dtmU(t-t1)Cj,j+1…U(tm-1-tm)×Cj+m-1,j+mU(tm)f0,m+jN,

where

(t)gj(Xj-Vjt,Vj).

We have discussed the term by term convergence a.e. only in the case when j and m equal to 1, however as we said, it holds in general (see [29] for details). Therefore it is enough to show that the sum (3.50) and the series (3.64) are bounded by a converging positive series whose terms are independent of N. To show this we do an unnecessary but simplifying hypotheses, namely that we consider only collision with bounded relative velocity. With this assumption we first establish the obvious estimates

j,j+1εfj+1L∞≤Cj||fj+1||L∞.

Using (3.66), the identity

εfjL∞=||fj||L∞,

the bound on the initial datum

0,jεL∞≤C||f0||L∞j

and the formula

0tdt1…∫0tm-1dtm=tmm!,

we control the mth term in the series expansion (3.50) by

-m||f0||L∞m+jCmj(j+1)⋯(j+m-1)tmm!.

Since

(j+1)⋯(j+m-1)m!≤2j2m-1,

we have estimated the series expansion (3.50) by a convergent series with positive terms not depending on N provided that t is small. The same can be done for the series (3.64) so that the term by term convergence implies the convergence of the solutions.

Finally, we remark once more that if the initial...