a derivation of the broadwell equation

Commun. Math. Phys. 135, 443-465 (1991) Communications in Mathematical

Physics �9 Springer-Verlag t991

A Derivation of the Broadwell Equation*

S. Caprino 1, A. DeMasi 1, E. Presutti 2 and M. Pulvirenti 1

1 Dipartimento di Matematica Pura ed Applicata, Universit~ dell'Aquila, 1-67100 L'Aquila, Italy 2 Dipartimento di Matematica, Universitgt di Roma, Tor Vergata, 1-00133 Roma, Italy

Received January 2, 1990; in revised form June 21, 1990

Abstract. We consider a stochastic system of particles in a two dimensional lattice and prove that, under a suitable limit (i.e. N ~ c~, e ~ 0 , Ne 2 --.const, where N is the number of particles and e is the mesh of the lattice) the one-particle distribution function converges to a solution of the two-dimensional Broadwell equation for all times for which the solution (of this equation) exists. Propagation of chaos is also proven.

1. Introduction

The dynamics of the molecules of a rarefied gas could, in principle, be described by Newton's equations, but a reduced description, as provided by the Boltzmann equation, turns out in practice to be more appropriate and useful. On the other hand it is known that in a low density regime, mathematically expressed by the Bol tzmann-Grad limit (B-G limit), the solutions of the Newton equations approximate those of the Boltzmann equation. This is rigorously proven for short times in [6] and globally in time for an expanding cloud of gas in the vacuum, [5]. Both cases are based on the fact that the average number of collisions experienced by any given particle is finite and small. At the present moment no other results concerning more general situations are available.

In this paper we deal with a discrete velocity kinetic equation, namely the two dimensional Broadwell equation. There are pathologies connected with the finite structure of the velocity space: from one side it has been proven, [7], that such an equation does not hold in the B - G limit for a natural class of approximating Hamiltonian systems; on the other side, the recent existence theory, [4], for the Boltzmann equation does not apply to discrete velocity models, essentially for the same reason.

In the present paper we prove that the two-dimensional Broadwell equation holds in the B - G limit for a sequence of stochastic particle systems. The

* Research partially supported by CNR-PS-MMAIT

444 S. Caprino, A. DeMasi, E. Presutti and M. Pulvirenti

stochasticity is absolutely necessary: it delocalizes the particles and, in this way, the pathologies outlined in I-7] are avoided. In fact it is easy to prove (see Sect. 3 below) that the short time analysis presented in [-6] does apply without difficulties to our case, yielding a derivation of the Broadwell equation for short times. In this paper we develop a new technique which allows us to prove convergence up to any time in which the solution to the Broadwell equation stays bounded. Indeed solutions which are bounded for arbitrary long times have been constructed in some cases so that our result provides a validity proof which is global in time, at least in these situations. Our approach to the problem uses probabilistic ideas. We basically prove that a sample of our stochastic process deviates from the correct kinetic behavior with small probability, when the B-G limit is approached. To prove this we introduce and evaluate a sort of truncated correlation functions (i.e. the v-functions defined in Sect. 4) yielding powerful probability estimates, which allow us to iterate the short time result up to arbitrary times. We remark that also in this part of the proof the stochasticity plays a fundamental role: the same strategy does not apply, at least directly, to Hamiltonian systems. The techniques developed in this paper apply to more general discrete velocity equations with minor modifications. We finally mention that other discrete velocity kinetic equations have been derived from stochastic systems (see I-2, 3]) and that techniques similar to those presented here have been used in [-1] to study Reaction-Diffusion equations.

The paper is organized as follows. In Sect. 2 we describe the model and state the main result. Section 3 is devoted to the short time analysis, while in Sects. 4 and 5 we prove the main theorem. The detailed definition of the branching process which describes the evolution of the v-functions is given in Appendix A.

2. The Model and the Results

We consider the unit square A = [0, 1] 2 and its discretization

As = {q = (qx, q y ) ~ A : q x = nle , qr = n2e, ni = 1 . . . . . e - l} , (2.1)

where e-1 is a positive integer to be fixed later on. We then consider a system of N (identical) particles whose configurations are denoted by:

X = {x 1 . . . . . xu}, (2.2a)

Xl = (qi, trl) qi~A~, aiz.Q, (2.2b)

where qi and at are position and velocity of the ith particle and

12 = {(0, 1); (1, 0); (0, - 1); ( - 1, 0)} (2.3)

is the velocity space, consisting of four elements. We denote by F~ = A s • 12 the phase space of a single particle and, accordingly,

F ~ will denote the phase space of our system. In the sequel we shall avoid boundary problems by assuming periodicity

(0, qr) = (1, qr), (qx, 0) = (qx, 1), (2.4)

which makes A a torus.

Derivation of the Broadwell Equation 445

The dynamics of the system is a stochastic process with values in F N described in the following way. The free motion of each particle is a Poisson process of intensity e-1 associated to the transition:

(q,, a i ) ~ (q, +aie, a3. (2.5)

All these Poisson processes are independent. The interaction among the particles is described by N ( N - 1)/2 independent

Poisson processes of intensity one (one for each pair of particles) associated to the transition:

(qi, ai, q j, a j ) ~ (ql, al l , q j, a~-), (2.6) where

cr-t=(-~r2,~h) if ~=(a l , a2 ) , (2.7)

and the transition (2.6) takes place only if

qi---qj and a i = - a i. (2.8)

In terms of differential equations, if #I = #~(X) denotes the probability of finding the configuration X = {x~,..., x~r at the time t, then the following evolution equation holds:

d#~ = e- 1Lo/2~ + L,/~, (2.9) dt

where, denoting by IX[ the cardinality of X, (in this case N)

Lo#~(X) = ~ I-/2*(X,)-/2~(X)], (2.10) i = 1

1 LI/2e(X) = 2 i~=1 [[le(X iZ'J) -- #e(X) 3)~i'j(X)" (2.11)

We have used the following notation: (X = {xl . . . . . xlxl}),

X~ = {x, , . . . , (q~ - a i e , ~,) . . . . . Xlx t }, (2.12)

X~.i = {xl . . . . . (qi, ai ~) . . . . , (qj, a~-) . . . . . xlxl}, (2.13)

I, if q~ = qj and a~ = - aj

)~'J = 0, otherwise (2.14)

We want to investigate the behavior of the system in the limit as e --, 0, N -+ oo, Ne 2 -+). > 0. In such a limit the number of particles per site is expected to be finite so that each particle will undergo a finite number of collisions per unit time. On the other hand the displacement per unit time of each particle is finite since e- * L 0 converges, at least formally, to the generator of the free stream. Therefore we are in a situation which strongly resembles the Boltzmann-Grad limit for Hamiltonian particle systems where the asymptotic behavior of the one-particle distribution function is described by the Boltzmann equation. In our context the kinetic equation which is expected to be valid in this limit is the two dimensional Broadwell equation, as we shall see below by a simple formal argument.

For a probability measure/2 which is symmetric under the exchange of the


particles, we introduce the rescaled correlation functions:

p~(x 1 . . . . . xj) = (Ng 2)-JN(N - 1)..-(N - j + 1) ~ #~(xl,.. . , xj, xj+ 1 . . . . , XN)" gj+ I ""XN

(2.15)

Proceeding as in the derivation of the BBGKY hierarchy for Hamiltonian systems, we obtain:

d ~ t l J = (8-1L o + L~)p~ + 82NCj,j+ID~+ 1, (2.16)

where

J C j j + I p~+ 1(xl . . . . . x j) = ~ [P;+I (x l , . . . , (q,, a~) . . . . . xj, (qi, - a{))

i = 1

- P~+ 1 ( x l , . . . , (q~, a,) . . . . . x j , (qi, - a i ) ) ] . (2.17) R e m a r k s on the Defini t ion in (2 .15) . Consider the joint distribution densities:

f } ( x 1 , . . . , x j) = ~- 2j 2 #~(X 1 ' ' ' ' ' X j, Xj + 1 . . . . . XN) (2.18) Xj+ i ""XN

(notice therefore that 82Jf~(x1,. . . , x j) is the #*-probability of finding the group of particles 1 .-.j in the states xl - ' . x j). Then the correlation functions are:

u~(x I . . . . . x j ) = U ( U - 1).-.(N - j + 1)f~(xl . . . . . x j). (2.19)

In the B-G limit the correlation functions diverge. Renormalizing by a factor N - j we then get the rescaled correlation functions:

p~(X1,... , X j) = N-JU~(X1 . . . . . X j). (2.20)

These can also be expressed as follows: if xl . . . . . xj are distinct states then

(,0,) p~(xt . . . . . x j) = (N~Z)-~E.0 t/(xi) , (2.21a)

where E.o denotes the expectation with respect to #~ and q(xi) the random variable number of particles in the state x~. If the states are not distinct the expression changes. For instance if x~ . . . . . xj = x then

p~(x , . . . , x) = (N~ 2) 3E~,,(q(x)...(~l(x) - j + 1)). (2.21b)

For the other cases see (4.10) below. The formal limit of (2.16) when ~-~ O, N ~ oo, N e 2 ~ 2 is the following infinite

hierarchy:

~gJ=~ _( J 0 )O.i.~q/ i~=l gj"]- 2 C j , j + l g j + | , (2.22)

The reason why in (2.22) there is not a term corresponding to Llp~ is due to the fact that the flow generated by e- 1Lo gives vanishing probability to configurations in which two tagged particles sit on the same site.


We shall call the set of Eqs (2.22) the Broadwell hierarchy because if g denotes any solution of the two-dimensional BroadweU equation:

~tg(q , a) = -- a '~qg(q, a) + 2[g(q, --a• a • -- g(q, a)g(q, --tr)] (2.23)

then the products: 1

gj,,(xl . . . . . xi) = l-I g,(xi) (2.24) i = 1

solve the Broadwell hierarchy (2.22). Conversely, in case of uniqueness of the solutions of (2.22) and (2.23), any solution of the Broadwell hierarchy which initially factorizes (i.e. satisfies (2.24) at time zero) cannot fail to be of the form (2.24) at all times.

The aim of this paper is to prove rigorously the above limit. We make the following hypothesis at time zero. Consider go ~ C 1 (F) a positive initial value for the Broadwell equation (2.23) satisfying the normalization condition ~ ~ dqgo(q, a)---,1.

r

Let #~ be the probability measure on F ~ defined as follows: N

#~o(Xl . . . . . XN) = l-I ~ dqgo(q, tri) (xi = (qi, tri)), (2.25) i= 1 AlqO

where A(qi) is the atom of the partition into squares induced by the lattice A~ and whose left low corner is qi. Let #~ be the measure evolved according to (2.9), or, what is the same, the distribution of the process {xl . . . . . xn} at time t. Consider the associated rescaled correlation functions Pr (In what follows we shall compare functions on F~ with functions on (A x O) ~ by restricting these latter to F~.) We have the following theorem (the main result of this paper), which is actually valid under more general assumptions on the convergence at time 0.

Theorem 2.1. Let g~Cl([0, T]; CI(F)) be a solution of (2.23) with initial datum go. Let g~,t be given by (2.24) and #~o by (2.25). Then:

lim LI P~,t - gj,t [I oo = 0 (2.26) e-*O,N-' o0,Ns ~

uniformly in tel0, T].

3. Short Time Analysis

In this section we prove Theorem 2.1 for T sufficiently small. This result is not surprising at all: we simply exploit Lanford's technique used for deriving the Boltzmann equation for hard spheres, I-6]. This short time analysis, however, will turn out to be useful in the sequel so that we shall present it here in some detail.

Theorem 3.1. Suppose that go~Cl(F) . Then there exists T(2, [I go LI oo) such that, for t < T(2, Itgo tloo),

II p~,, II ~ <_- (2 tl Vo II ~)J (3.1)

and (2.26) holds.


Proof. We write the solution of (2.21) by means of a perturbation series around V~, t which is the semigroup generated by e- 1L o + L, and acting on functions of j particles. In the sequel we simply write V~ instead of V~. t, making clear the j-dependence from the context. We have:

pj , t=Vtpj , o + (Ne2)n~dtl dt2... ~ dtnVt- t lCLj+lVt~-t2Cj+l, j+2""Vt, Pj+n,O �9 n = l 0 0 0

(3.2)

Since V~ is a contraction semigroup in L~, and

l[ Cj,~+ 1 I[| < 2j, (3.3)

we conclude:

N-j (2tN~2). [[P~,tl[~o --< [[P~,ol[o~ + ~ - - J ( J + 1 ) . . . ( j + n - 1)llPs+.,ol[~. (3.4)

. = 1 rt!

Using the inequality:

j ( j + 1 ) . . . ( j + n - - 1)<n!2 "+j, (3.5)

we finally obtain:

IIp~,,ll~o ~ IIp~,o I1~o + 2 j ~, (4tNeZ)"llp~+.,oll~o n = l

II go II ~ + (2 II go II o~)J ~ (4tNeZ)" II go [I ~. (3.6) n = l

Therefore (3.1) is verified with t < T(2, II go }1 o~) - 1-122 II go II | x. To prove the convergence we set:

A~(t) = gj,t - P~,,, (3.7)

where 9j,t is given by (2.24). It is easy to see that 9j,, solves the Broadwell hierarchy (2.22) and by using a perturbative expansion as above, we can also prove uniqueness (in a suitable space), j

Denoting by V t the free stream, i.e. the group generated by - ~1 at P~ the semigroup generated by e-1L o, we have: i ~qi and by

t

A~(t) = ~ dsP~_s(L,p~(s)) + (lit - P~)gj,o + P~A~(O) 0

t t

- -~(Vt_s-n~_~)Cj , j+lgj+l , f l s+~P~_sCj , j+lA~+l(s)ds . (3.8) 0 0

To estimate the first term on the right-hand side of (3.8) we develop, in some detail, an elementary argument which will be often used in the sequel. We want to estimate P~X(x, y), where Z is the characteristic function of the event that two particles sit on the same site. We denote by P~(x, ylz, u)=P~(xlz)P~(ylu) the transition probability that two particles, initially in the states z, u, are at time t in the states x, y respectively, when they moved according to the process P~, (the random flights


associated to each particle are independent so that the transit ion probabil i ty factorizes). We have:

e~z(x, y) = Y, e~(x, yiz, u)z(z, u) ZpU

< sup P ~ ( x [ z ) ~ P~(ylu) Z U

< sup P~(xiz). (3.9) X,2

Moreover

P~(xly) < sup e -'/~(t/e)k, (3.10) k k!

and recalling the Stirling formula

we finally get

enn(n+ l/2) _ _ .--+

n!

P~Z(x, y) < const. ~/~. (3.11)

Therefore for t < T(2, [I go I[ o~):

PT-sLIP~,s ][ o~ < const, j 2 , ~ / ~ _(211 go II ~)J. (3.12)

Fur thermore , for a function f ~ C I ( F ) ,

[(V t - P~)f(x)] < ~ P~(x lu ) l f (u ) - f ( x - at)l u

< c o n s t . ~ P~(xiu) lu - x + crtl u

< const. P~(xlu)lu - x + •tl 2

__< const, x / ~ . (3.13)

The third term on the r ight-hand side of (3.8) can be easily bounded by

I[ A3(O) [1 o~ < I[ go I[ ~je( II go II oo) j - 1, (3.14)

so that we obtain the integral inequality:

t

]t A~(t)II ~ < c~(e + V/~tt) + 2 j ~ ds ]l A~+ l(s)N ~. (3.15) 0

From (3.15), since IL A~(s)II < c j for some c, we can obtain a series expansion which converges for short times. It is now easy to conclude that in the B - G limit, (e ~ 0, N --, c~ and Ne E ~ 2)II A~(t)II o~ ~ 0, uniformly in t e [0, T(2, II go II ~)]. �9

Remarks . The only difference between the Lanford argument for hard spheres and the present one is that here we can prove a stronger (uniform) convergence at time t, the same as that assumed at time zero. This is due to the stochasticity


of the free motion. For Hamiltonian systems it is unavoidable to have some weaker convergence at time t because of the reversibility-irreversibility transition occurring in the B-G limit, (see [6] for details).

A general global existence theorem for the two-dimensional Broadwell equation is still lacking. However for special cases, as for space-homogeneous initial data, or when there is a one-dimensional symmetry, a unique, smooth, global solution can be constructed. As an outcome of our analysis we shall see that, in these cases, the short time analysis carried out in this section can be extended to arbitrary times.

We observe that the knowledge of a uniform bound on the limiting solution, say

II gj,, II | < c(T)J, (3.16)

valid up to the time T introduced in Theorem 2.1 does not allow us to extend the convergence result up to such a time. In this section we have established that for t < T(2, Ilgo I1~o)

II A~(t)II ~ -<- c Jr/~t, (3.17)

where the above constant c increases with sup (I1P~a I1~) l/j, It is clear that (3.17) J

combined with (3.16) is not enough to iterate the short time convergence to reach arbitrary times.

4. Proof of Theorem 2.1

According to (3.17) the difference Igj , ,- P~,tl is only bounded by cJx/~; if we had, instead, a bound like cJ(v/-~t) j then using (3.16) we could improve the bound for P~,t and iterate the short time analysis past the time T(2, II go II ~). We notice that if p~,, is going to converge to gj,t then in the limit the p~,t's will factorize as the gj,,'s actually do. With this in mind we introduce the functions:

v,(X) = E (L_l)lX/rlgt(X/Y)p~i(y) ' (4.1) Y c X

where X is a configuration with finitely many particles, ~ denotes the sum over Y c X

all the subsets of X, and p~i(Y) is a shorthand for p~rla(Y). In the sequel we shall denote the restriction of vt on the configurations X with j particles by vja(X ). The v-functions reduce then to:

1

v~,t(X) = 1~ ( g , - Pi,,)(x,) (4.2) i=1

if the p~.{s strictly factorize. Therefore we may hope that, at any fixed time, the v-functions will be bounded by cj~o(e) j, where cj does not depend on r and co(s) is infinitesimal in 5. A control on the size of the v-functions gives a measure of the tendency of the system not only to converge to the limiting kinetic behavior, but also to gain the factorization property. This means that the processes associated to each particle are going to become independent according to the so-called propagation of chaos hypothesis which is used in the heuristic derivation of the Boltzmann equation. However a bound like the above one, i.e. cj~(e) j, is not at


all easy to derive even for short times. Following the definition given in (4.1), one can write down an evolution equation for the v-functions (see Appendix A) which involves not only operators which increase the number of particles, (as Ci, i § 1) but also others which decrease it. A naive bound can be obtained by taking sup norms as in Sect. 3, getting an integral inequality for v j ( t ) - [I vi[I oo which looks like the following one:

t

vj(t) < vj(O) + c S dsjvj+ l(S) + j2[vj(s) + v j_ 1 (s)-l, 0

v~(O) = e j, j = O, 1..., (4.3)

where c is some constant. In addition to (4.3) we have the a priori bound vi(t) <= c j for short times; as follows from (4.1) and the analysis in Sect. 3. If we could improve the a priori bound showing that

vj(t) < (Ce~) j, (4.4)

then we could easily iterate the argument of Sect. 3 by expressing p~ in terms of vt and gt by means of (4.1). However the estimate (4.4) is deafly incompatible with (4.3):'if we just take the contribution coming from the term v j_ ~ we get a bound like

~ [ j ( j - 1)..-(j - k + 1),l 2 j-k,kt. (4.5)

k=0 k!

We could still recover a bound like (4.4) if t ~ e a, fie(O, 1). However this is not a real gain because we would need an estimate like e ~j with ~5 > fi to iterate the procedure to reach arbitrary finite times. The estimate (4.4) with t < e a and fi > fi will be obtained (see Proposition 4.1 below) by exploiting more carefully the real structure of the evolution equation for the v-functions. Unfortunately the best estimate that we have been able to derive gives a bound for vj like c(j)e ~i with c(j) which increases in j faster than exponentially. This new difficulty will be overcome by using probabilistic arguments.

Following these considerations we can outline the strategy of our proof. We fix an arbitrary time T and split the interval [0, T] into intervals of length e a, fie(0, 1). On each of these time steps we estimate the v-functions conditioned to a good set of initial configurations (see Proposition 4.1 below). As a consequence we can compare the empirical distribution of the process at the end of the time step with the solution of an approximate Broadwell equation (see Proposition 4.2) and prove that their difference is small in probability. This allows us to say that at this final time the configuration is, with some large probability, good as required for iterating the procedure to a new time step. In this way we prove, see Proposition 4.4, that, typically, the process foUows pieces of trajectories of the approximate Broadwell dynamics. We also prove that these pieces are uniformly close to the Broadwell dynamics so that the proof will be achieved.

From now on we shall put Ne 2 = 1 for notational simplicity and denote by r/ the function which associates to any x e F, the random variable number of particles r/(x) in the state x. The knowledge of r/ uniquely determines a configuration X = {xl "..xs} of particles modulo their labelling. Let ~ = ~(x) be any integer valued function on F~ and p a symmetric function on the configuration space with finitely many particles. We set

p( ~) = p 3(xl ... x i) (4.6)


for any configuration x l . . . x j realizing 4, that is: J

~(x) = ~ 6(x, xi), (4.7a) / = 1

where

6(x, x~) = ~ ~ 1, ifx = x~ (4.7b) ( 0, otherwise

The following functions will be useful in solving combinatorial problems:

D(~, rl) = I] Dr (4.8) x

Dk(j) = j ( j - - 1).-.(j-- k + 1). (4.9)

Let p('; #) be the family of rescaled correlation functions relative to the measure #. Then

p(~; #) = E~,(D(~, ~/)). (4.10)

To understand (4.10) suppose # concentrated on the singleton q. Denote by x l . . . x j a configuration which realizes ~ and by P,(x l . . . xi) the probability of finding the first j particles in the states Xx...xj. Then:

eu(x l ' " xj) = I] rl(x)(rl(X)- 1)... (~/(x)- ~(x) + 1) (4.11) x N (N - 1)...(N - j + 1)

Therefore, according to (2.15) we have:

p(~;/~) = D(~, q). (4.12)

The general statement (4.10) follows by integrating on q. For any configuration q, consider g~(.Iq) the solution of the following initial

value problem:

0 +gt(q ,a I~I)g,(q,-a• ~g,(q,a[tl)=~-aLog~(q,a[tl) ~ •

u g

- g~(q, alq)gT(q, - o-It/) (4.13)

g~o(q, air/) = t/(q, a),

which is an approximate version of the Broadwell equation on the lattice A~. Let P~('Iq) be the rescaled correlation functions of the distribution of the process q(t) which starts from t /a t time zero. Alternatively P~('lq) is the solution of Eq. (2.16) associated to the initial datum D(" I q). We define (compare with (4.1)):

vt(~lt/) = ~ (-- 1)lr -- ~[r/), (4.14)

where [([ denotes the cardinality of the set {~(x), xeF~}, ~ means that we sum over all the set of particles ( contained in the set 4, i.e. ~ ~

= I~ ~ = l-[ ~ D~(~)(~(x)) (4.15)

Derivation of the BroadweU Equation 453

and g~(~lq) = 1--I g~(xlq) ~(x). (4.16)

X

Denoting by E, the conditional expectation that at time zero the process is in the state r/, we have

v , (~lq)=E,(I~I ~ (-1)~(~)g~(xlq)~(X)D(r (4.17) ~(x) c ~(x)

Notice that if ~(x) = 0, 1 for all x, i.e. there is no more than one particle per state, then:

v,( ,iq) = E , ( ~I (q(x) - g:(x,q)). (4.18)

To estimate the v-functions we write, after straightforward but tedious calculations. the following evolution equation:

dv,(~lt/) = 1Lov,(~lq ) +~a,(~,~)v,(~[r/). (4.19) 8- {

The operators a,(~, ~) are nonvanishing only when 141-2 <__ ICI-< Ill + 1. We do not give here their explicit expression referring the reader to Appendix A, where (4.19) is derived.

The main technical point of our result is the following proposition whose proof will be given in Sect. 5:

Proposition 4.1. L e t f l - ~ - i-6-6, 6 = �88 ~ = ao--~. Suppose that for all x rl(x ) < ~-~. Then for all t < eP we have

�9 8 aj

where c(j) is a positive constant depending only on j and j---131.

Remark. The choice of the parameters/7, 6, ~ in Proposition 4.1 is obviously rather arbitrary, but good enough for our purposes.

The above proposition allows us to control the deviation of the process t h from the approximate solution of the Broadwell equation g~(xlt/) by means of the following seminorm:

I l f l l =sup I (4.21)

We remark that P~,/,(xly), x = (q, a), is essentially supported by all y = (q', a) such that q varies in an interval of length 5-3/8 = (s-%~/4)x/z. Therefore (4.21) expresses the supnorm of the one dimensional averages on intervals of size e-3/s. The introduction of these averages is necessary to smooth out the local fluctuations of qt- However the seminorm (4.21) will be sufficient to ensure a macroscopic local control since ~ < 1.

Proposition 4.2. Under the same hypotheses of Proposition 4.1, setting ? =13-g, for


all n > 0 there exists c(n) such that:

Pr,( [I g,p(" I t / ) - r/~p II > ~) < c(n)~", (4.22)

where Prn denotes the law of the process starting from ~l at time zero.

Proof. By the Chebichev inequality, the left-hand side of (4.22) is bounded by:

~-2J~E,(llg~a('l~/)-~/~al[ 2j) < ~-zJ~ Z Y', P~,/,(xlyl)'"P~,/,(xlY2~) X Yt ""Y2j

Consider first the contribution coming from points yl .,.y2i which are mutually distinct. By (4.18) and Proposition 4.1:

~ . ~ , En (ge (yklrl) __ rl,a(yk) ) = VsP(y 1 ..y2jlrl) < c(2j)g2j(1 -fl)6g- 2~j (4.24)

hence the sum in (4.23) over all distinct states is bounded by

c(2j)e2jtl -a)% - 2 e- 2rj e- 2~j. (4.25)

The general case can be recovered by combining the probability estimate that two particles are in the same site, see (3.11), with estimates on the v-functions of lower order. In fact if y~ ""Y2j is a configuration with n, (n < 2j) particles on the same state x, then in the product in (4.23) we have a factor:

(~ - f/)", (4.26)

where r] -= r/,~(x) and ~-g~p(x[r/). The following identity is easily proved by induction:

k

fl k = ~ A(k, h)Dh(fl ), (4.27) h = l

where the A(k, h) are defined by recurrence:

A ( k , h ) = l if k = h or h = l ,

A(k, h) = hA(k - 1, h) + A(k - 1, h - 1). (4.28)

The explicit form of A(k, h) is not relevant at all, we simply remark that A(k, h) may be bounded by (2h) k. We define:

k k! s=O - k-sD (- (4.29) Vk(tT,0)= y" (k_s)!s t (g) ~ tl).

By inverting (4.29) we have: k k !

Dk(fl) = ~=o (k - s)!s! ok-'v'(fl' O). (4.30)

Using (4.27) we can express f/" in terms of Vs(6, 0) to obtain:

~ ~ n! h! A(s,h)(_l)SOk+,_sV,(fl, tj)" ( # - 4)" = 2=0 h=l r=o (n-s)!s!(h--r)!r!

(4.31)


Since

v,( ~ l rl) = E , [ N V~(x,(rIj(x), g:~(x) ) ] (4.32)

by inserting (4.31) in the expectation in (4.23), we obtain a linear combination of v-functions whose minimum order is given by the number of y{s having only one particle per site. Hence the contribution to the sum ~ relative to the event

Yl ""Y2j with exactly k isolated particles may be estimated by

c ( j )8 - 2 8 - 2j'y~(1 - #)6k~-~ke(2j-k)a/sE-(2j-k)~ ' (4.33)

the first factor c(j) is a suitable constant independent of e, the second factor comes from the sum over xEA~, the third one was already present in (4.23), the fourth and fifth terms are consequences of Proposition 4.1 and the sixth arises from estimate (3.11). Finally the last one bounds the O's in the expansion (4.31). This last bound follows from the short time analysis (recall that fl > () of Sect. 3 which applies as well to the discretized Broadwell equation (4.13). Since:

~ + y < ( 1 - f l ) 6 , ~ + y < ~ , (4.34)

the proof is achieved. �9

In order to iterate the probability estimate given by Proposition 4.2, we introduce a set of trajectories well behaving locally in time, with respect to the approximate Broadwell dynamics.

Definition. We denote by H, the set of all sequences { ~ ( k ) } k = 0 . . . . . in F~, (17 (~ = ~/) such that:

~/(k) ~_ ~ - ~ , (4.35)

[I r / - g~ II < e ~, (4.36)

I[ n (k) - g~J('lr/(k- 1)) [I < e r, k = 1,.. . , n. (4.37)

Let T be an arbitrary fixed time, we set m = integer part of Te-t~ + 1. The next proposition (combined with (4.37)) shows that the solutions of the

Broadwell equation are well approximated by the trajectories in H,,.

Proposition 4.3. In the same hypotheses of Theorem 2.1, if {r/(k)}eHm then there exists a constant c > 0 such that:

i g~p(xjrl(k- 1)) _ gkj(X)l < ckeL (4.38)

Proof. We prove Proposition 4.3 by induction so that we assume the estimate (4.38) true up to k - 1. We set

h(x, t) = g~(xlrl (k- 1)) _ gtk- l)J + ,(X), (4.39)

n(x , t) = g~(xlrl <k- ~) + g(k-1)J +,(X), (4.40) Then

t h(', t) = P~(rl (k- 1) - g(k-1),#)(") "~ P~ -- V,(g(k-1)8 #(') "q- ~ dsP~_~Q(h(', s), n ( ' , s) )

0

t + ~ ds(P~_~ - V~_~)Q(g(k_ 1)~ + s(')), (4.41)

0


where

Q(f, g)(q, a) = �89 - tr x)g(q, a z) - f (q , a)g(q, - a )

+ g(q, - a x)f(q, a i ) _ g(q, a)f(q, - a ) } ,

Q( f ) = Q(f , f ) . (4.42)

We denote by T,, i = 1 . . . . . 4 the four terms in the right-hand side of (4.41). We estimate them separately. For e ~ > t _ e ~/4,

I T11 <-- [I r/r 1) _ g~a(. i~(k- 2})II + II 0~P(" I~ ~k- 2)) _ 0(k- ~a II < e~ + (k -- 1)Ce ~, (4.43)

where the first term arises by the assumption that {r/tk) } ~Hm and the second follows by the induction hypothesis. Moreover:

IT21 < C1 /31 /2 (4.44)

as follows by (3.13) and the hypothesis of the regularity of the solution g. To estimate the third term we split the time interval into two parts: s~[O,e ~/4] and se[e ~/4, t]. In the first we simply use the short time estimate:

Q(h(., s), H(., s)) < C2e- 2r (4.45)

in the second:

I Q(h(', s), H(', s))l < I Q(h(', s))l + 21Q(h(', s), g(k - 1)~ a + s) l

< C3 II h(', s)I1 o~(1 + II h(', s)II ~), (4.46)

so that t

IT31 <= CS2~d/4+ CaSdsx(s>d/4)lLh(',s)lloo(1 + [Ih(',s)l[| (4.47) 0

Finally the last term can be bounded in the same way as the second one. In conclusion we have obtained the following integral inequality: for t~ [e ~/4, eP],

[I h(', t)I[ ~o < C481/2 § C2/~- 2~/~1/4 § FY § (k - 1)Ce v t

+ CaSdsllh(',s)ll| § [Ih(',s)l[oo)~(s>~X/4). (4.48) 0

Because �88 - 2( > V, we can first conclude that the first time gfor which [[ h(', g)I[ | >- 1 is larger than e~; we then solve the linear integral inequality obtained from (4.48) replacing (1 + II h(., s)II oo) by 2. Finally we choose c so large that the estimate

II h(', t)II ~o < cker (4.49)

holds. The case k = 1 can be recovered in a similar fashion so that the proof is

achieved. �9

We now prove that, with large probability, the sequence ~/k,~ obtained by observing the process at times ke g is in Hm. Define the set of trajectories:

H n - {(11t)t>__Ol{rlkea}k=O,...n~nn}. (4.50)

Then


Proposition 4.4. For any n > 0 there exists c(n) such that:

P~(H*) > 1 - c(n)e", (4.51)

where P~,, is the law of the process distributed at time zero accordin# to #~.

Proof. Denote by Xk the characteristic function of the set H* and by ~/the value of the process at time (k - 1)eP. Then:

E,,(Zk)=E~,(Z~_lPr,{lltlk~B[lo o <e-~; IIr/~a--g~,('lt/)ll < e~'}). (4.52)

If r /< e -~ we already know by Proposit ion 4.2 that:

Pr,{ l[ r/,, - g~,(-lt/)[I > e y} < c(n)e ~ (4.53)

which controls the second event in (4.52). For the first term we have

e# Pr,{ II r/~0 II ~o > e -~ } < Pr,{ 11 g~a('l~/)II | > �89 -~} + Pr,{ II t/,a - g, (. In)II ~ > �89

(4.54)

By Proposit ion 4.3 we know that:

il g24('l'l) - g~(')II ~ < C(e-a T + 1)e r (4.55)

so that, since ~ > fl, we have an uniform bound on lia~('l~)l[~ inherited by the assumed boundedness of the Broadwell solution gk,~" Hence the first probability appearing on the right-hand side of (4.54) is zero. By the Chebichev inequality the second term on the right-hand side of (4.54) is bounded by:

22"e 2~" ~ g~ { (t/~(x) - g~,(x 1 t/)) 2" }. (4.56) x

Now we expand (4.56) by using (4.31) as we did before. We obtain a linear combinat ion of terms with products of v-functions and powers of g~B. Both are known to be bounded uniformly in e so that, for an arbitrary n:

E,~(Zk) > E,,(Z k_ ~)(1 -- c(n)e"). (4.57)

The estimate of E,,(X0) is obvious by the assumption on #' so that by iteration we conclude the proof. �9

We are now ready to conclude the proof of Theorem 2.t.

Proof of Theorem 2.1. By definition, putting t = ken:

p~(r = E(O(~, th) ) = E(Z k_ ~O(~, t/,)) + E([1 - Zk-~ ]O(~, q,)). (4.58)

By using the Cauchy-Schwar tz inequality, the last term is bounded:

E([1 - Zk- ~]D(~, th) ) < E([1 - Zk- a])~/2E(D(~, t/,))2) ~/2 / 'xl/2

= <C(n)en/2E~xl:lt(x)2~(x) )

< C(n)~./Ee- 21~1. (4.59)

Here we have used the conservation of the number of particles for which r/t(x ) < e- 2. For fixed [~t and sufficiently large n the right-hand side of (4.59) vanishes in the


limit e ~ 0 . Finally:

+E(zk_l[I~Ig~,#(xln)r163162 (4.60)

where n denotes, as before, a configuration at time (k - 1)e #. By Proposition 4.3 the last term of the right-hand side of (4.60) is also vanishing. Finally the first term in the right-hand side of (4.60) can be expressed in terms of a linear combination of v-functions, (see formula 4.30) for which, by Proposition 4.1, we can conclude that all terms are vanishing.

From this it is easy to prove convergence uniformly in x and t on the compacts. �9

5. Proof of Proposition 4.1

The analysis at the beginning of Sect. 3 applies as well when the initial measure is supported by a single configuration n: for each ~ > 0 let the configuration n be such that n(x)<e-~, for all x, and denote by P~,r('ln) the correlation functions when the initial measure is supported by 7. By (3.6) (here II 0o II ~ < e-~)

II P~a('lq)II | < 2 ~ - ~ (5.1a)

for t <(122~-~) -1. Hence (5.1a) is satisfied for all times t < e#, those relevant in the proof of Proposition 4.1 (recall that/~ = 1/100 and ~ = 1/1000). Analogously it can be proven that the solution o~('ln) of (4.13) is bounded by

II 0~(" Iv)II | _< 2e-~. (5.1b)

From (4.17) and (5.1) we then derive (4.20) for t < ~. For the remaining values of t a more careful analysis is required, as we shall see hereafter.

We recall that P~ is the semigroup generated by ~-1Lo and denote by P~(X[ Y) its kernel. By (4.19), denoting by a, vt(~ln)= Y', a,(r ()vt((ln), we have:

t

0

r a - 1 t t n - I

= ~ ~dq... ~ dt, P~_t~a,...P~._,_t.a,.P~vo(~ln)+R(t,m), (5.2a) n = O 0 0

where t t r n - 1

. 8 �9 8 p 8 R(t,m) = S d [ 1 . . ~ dtmPt_,a,," .P,.,_,_t a,,. t..vt..(~lq) 0 0

and m = lOOj, j = I~1 = cardinality of ~. We recall that 1 2

a t = ~,al, q = l

(5.2b)


where a~, q = 1, 12 is the finite sequence of the operators C • K +- A • L~ �9 �9 " , 1 , 2 , 1 , 2 ,

defined in Appendix A and given in some order. We shall first estimate:

q 1 ~ q n P , - t l a,~ "'" P, ._ I - , . a t . P t , 1%( ~ I r/)l. (5.3)

The particles appearing in the expansion (5.3) are at most j + n (this value is achieved when only operators of type C are present). Thus we need to estimate the following transition kernel:

8 q l " " . ~ q n " " Pt_,~at, ( t l , j , ) .. P , ._~_, a, . (t,,,j,,)Pt, lVo( ~ ltl)l, (5.4)

where a, q = ~ a~,(i,j), aqt(i,j) is the contribution to the operator a, q coming from the i , j

pair of particles i and j and finally aq(i,j) = 0 if i = j for the operators of type A and Lz. For the operators C and K, a~(i,j) is the contribution due to the particle i and a~t(i,j)= 0 if i vaj. Moreover we use the convention that aqt(i,j)= 0 if the particle i and j are not present at time t.

We denote by F = {ql . . . . . qn; ( i J1 ) . . . . . (in,j,)} the set of all possible choices of operators and pairs of interacting particles�9 The algebraic structure of the process is entirely given by ~r For each y we have the following estimate:

Proposition 5.1. For each f i xed 7 ~ F , n < m - 100j and e < t <- e p,

s q l . �9 " - - d t , . . . S d t n ~ , ( P t - t , at, " aqt2P~.vo)( X} <=c(3)e o+2,)r t "/2. (5.5) o r -t

Since t __< e a the factor e-2n~t"/2--* 0 as e-~ 0, hence Proposition 5.1 will imply Proposition 4.1, once we prove an analogous statement for the remainder term (5.2b).

Proof . Since v 0 vanishes whenever there is a state occupied by a single particle, we have

(5.4) < ~ e - ; I r l /P '~ ~ t-t,aq~tit, ~ ~,J~,'" ~ ~ .a~( i~ , j~)P~)(XIY)~(Y) , , (5.6) r

where X is the set of states realized by ~ and ~(Y) vanishes unless there are at least two particles per site. Therefore

I~l x(Y) = ~ 1-I ;~,(e), (5.7)

i = 1

where ~ = {@ ~ . . . . . ~ }, I ~ I = s, is a partition of the particle labels in Y into atoms ~1 . . . . . ~ , each one containing at least two elements. Finally ~ , ( Y ) = 0, unless all the particles of Y with labels in ~ are on the same site. Since the number of partitions is finite (once j is given) it will be enough to prove the estimate (5.5) for a fixed partition ~ . Hence from now we shall consider a fixed value of ? and We first consider those ?'s where the terms a~ are only of type A and C, i.e. when there is no operator which leaves invariant the number of particles. Then the transition kernel (5.4) has the following structure:

S,_ ~ ~ . . . ~S~,,_1 - ,,,c~S,,,(Xl Y), (5.8)

where z~ . . . . . Zk are the times when a new particle is created and the ~'s are the corresponding creation operators. Each operator S is a string whose transition


kernel is given by: S . . . . _,,(Xl Y) = P~,_, _s~d �9 �9 �9 ~r _..,(X] Y), (5.9)

where f or'tn(r) t~ Ji= 1 are the times when the particles die and the d ' s denote the corresponding destruction operators. It is obviously assumed that S . . . . _,,--P~,_~_,, if no particle is destroyed.

By using (5.1b) we get n(r)

S~.- , - , . (XIY)<(4e-z{) "(') ~ 1--[ y'P d X l Z . ) P , d Z . IZ.) 1"6 *r-- I - - 8 1 ' ' x / $1 - $ 2 x ~t, ,~/

Z! ".Zn(r ) i = 1

. - .Pc (Z,,,,] Y), (5.10) n(r) --*r ~ J

where Xi is the characteristic function of the event that the two particle involved in the interaction must stay at the same site at the time s~. Z* is the set Zi deprived of one or two particles if Ax respectively A 2 acted at time s 7 (one or two particles dying in such a case). Furthermore when the operator A~ acts then the velocity of the particle involved (say i) changes tr~ ~ a~ according to (A.25). By applying the probability estimate (3.11) we easily obtain:

~.xxp~,_~_s~(XlZ~)p~7_,~(Z,~lZ2)<c / e Q~,_,_,~(X(Z2)IZ2), (5.11) Zl ~1 T r - l - - S r l

where the set X(Z2) is the set X deprived of the particle (or particles) which died; Q~(XIY) denotes the transition probability associated to IX[ independent Poisson jump processes to which we add the transition a~-, a~ whenever the operator A~- acts on particle i. Notice that Q~ depends on ~eF.

Iterating the procedure we obtain:

S,,_,_,,(SlY) < (4e- er , - d " *,- l - s'~to

where X(Y) denotes the subset of X which survives in the time interval ~_ x - z,. Before using (5.12) we need to estimate the action of the operators C. For simplicity let us first consider the case when all the operators C are of the form C-. We now denote by Z~'"Zk the single particle states of the newly created particles, cf. (5.8), (they are completely specified by yeF" and by the configurations just before the times of creation). Therefore:

( 5 . 8 ) = Z S t - , t ( X , Z l ) S , t - r Z t - . .Zk

"'" S,k-, -,~(Zk- 1 u Z k_ l, Zk)S,k(Zk u Zk, Y). (5.13)

We introduce the notation: (Z)k, (Z)o (5.14)

to indicate the set of particles in the configuration Z which are present at time tk and at time 0, respectively. By using (5.12) we can bound (5.13) by the following quantity:

F(t, zO.. . F(z k, O)Q~_,,((X) 1 ]Z1)Q~, _,~((z 1 ~2 Zl)2 Iz2)... Q***((Zk U Zk)o ]Y), Zi "..Zk

(5.15)


where

F('c,, z,_ 1) = (4e-2~) -"t') -~- e 1-[ t~,(Y) . (5.16) ~/ z,_ 1 -- s] z,_ 1 - s~(,) ~= 1

I~1

From now on we shall consider only those Y for which I ] ~ , ( Y ) = 1, recall that the partition ~ has been fixed once for all. ~= 1

There are two possibilities. Either the particle created in the state z k dies in the time interval (0, Zk) and therefore

" Z Q~k((ZkUZk)OI Y) -- Q~k(( k)O[ Y) (5.17)

or it survives, so that

Q~((Zk u Zk)o[ Y) ---- Q~((Zk)o[ Yk)P~,~(Zk [Yk), (5.18)

where Yk is the position in Y of this particle and yk = Y/Yk. In the first case, (5.17), we sum over Z k and obtain the same expression as (5.15) with k replaced by k - 1. In the second case, (5.18), we estimate P~k(ZklYk) by (3.11) so that the expression (5.15) is bounded by:

~k F(t, z l ) ' " t ( Z k , O)~ ~ Q~-~((X)I[Z1) yk ZI ""Zk

�9 .. Q;~_ ,((Zk_ I U Zk_ 1)kl Yk) I-I t~(Yk), (5.19) i = i

where k s (5.20) ~ - {~i}/=1

is the partition of the labels of yk obtained from ~ by eliminating Yk" We notice that ~k may have an atom (say ~ ) with a single element (correspondingly X~,~ = 1).

We want to iterate the above procedure but now there is a new possibility, namely that the particle created in Zk-1 belongs, at time 0, to a cluster with a single element. In this ease, as well as in the ease when the particle dies, we do

not gain the factor ~ . In all cases summing first over the particles which are not alive at time 0 and then over Yk- X (if present) we complete this step of the iteration passing from k - 1 to k - 2.

We define yh = yh+ X/y h for h = 1 . . . . . k - 1, with the convention that yh = yh+ 1 whenever the particle Zh is not present at time 0. Analogously we define ~h as the partition derived from ~ by restricting it to the particles in yh. At the end of the iteration we obtain

F(t, zl) . . .F(Zk, e 0)~, Q~((X)ol y1) I-I ~ ( y l ) , (5.21) i = 1 y1 i=I

where al . . . . . ah is the following subset of z I . . . . . Zk: z i e{a l . . . . . ah} if and only if zi is present at time 0 and its label belongs to a cluster in ~i+ i consisting of more than one element, ~k+ i _ ~ . Moreover y1 is the image at time 0 of the particles (X)o in X which are alive at time O. Finally ~1 is the partition of y1 obtained from ~ after subtracting all the new particles alive at time O. We finally bound


(5.21) by

f i F(t, ~x)"'F(~k, 0) 1--[ - , (5.22) i = 1 i = 1

where I ~ I denotes the cardinality of ~ whenever it is larger than 1 and it equals 1 otherwise. We now insert the estimate (5.22) in the left-hand side of (5.5) and perform the time integrations. We get

1.h.s. of ( , . , j = c(jj~ ~ tx/~J . (5.23)

After some thought one can realize that

( l ~ , l - 1)= ~ ( [ ~ : 1 - 1)+h, (5.24a) i = 1 i = 1

M n(r) >- -~, (5.24b) r

where M is the total number of particles which have died. Then

2([~,I_I)+ Zn(r)> IYI +M j+k (5.25) , = 2 2 - - - 2

From (5.23), since g/t < 1 we obtain as a final estimate: . . / ~ \ ( j + k ) / 4

l.h.s, of (5.5)< c(J)~-o+z"'~l-t) (x//t) ". (5.26)

If also the operators C § are present in the expansion (5.8), the same estimate (5.26) can be easily obtained after redefining Q~ to take into account the transitions a ~ a • of the particles which are involved in the interaction.

Exactly the same estimate may be obtained in the case when in the expression (5.9) there are operators as L~ and K. Consider the following string:

P,o-~, Lap,, _,,L~... Lap~,_` _,,(X, Y), (5.27)

where {s,}~= ~ .... are the times when an operator ~ of type L, or K acts, leaving the number of particles invariant. Removing the characteristic functions of the interaction we bound (5.27) by

2e-*r FIL_,,(X, y), (5.28)

where H r is the transition probability of IX[ independent Poisson jump processes of intensity e- ~ in which the velocity of the particle is changed at time s~ according to the sequence 7. We can replace P by F/~ and obtain the same estimate (5.26) for the general case. The extra factors 2e -'r can be controlled by the time integrals yielding a factor e p*. Thus the proof of Proposition 5.1 is achieved. �9

By Proposition 5.1 we can easily bound the sum in the right-hand side of (5.2a) by the desired estimate.

Finally we estimate the remainder. By definition

Iv, l (~ l r l ) < - : " . k=o ( j - ) ! k i ( I I~174 In) lifo). (5.29)

Derivation of the Broadwell Equation

Using (5.1) and recalling that/8 = 1/100, we get

( J ) ~ IR(t, m)l = < c e-~u+,,)

<- c(j)e~- lOlU

and, with our choice of the parameters, the proof is achieved.

463

(5.30)

Appendix A

In this appendix we derive Eq. (4.19) which describes the time evolution of the v-functions. We recall the hierarchy satisfied by the rescaled correlation functions (Eq. 2.16):

Defining:

d -~p~ = 8-1Lop~ + L~p ~ + Cp ~. (A.1)

J g'(xl . . .x~)= I-I g'(~), (A.2)

i = l

where g' satisfies Eq. (4.13), we have for the set {g'(Xx...xj)}~=l the following hierarchy of equations:

d , 1 ~ a = ~- Log' + Ca'. (A.3)

According to Definition (4.14), the v-functions relative to the correlations p' are implicitly defined by the formula:

p'(X) = ~ g ' ( Y ) v ( X / Y ) ) - (g ' .v)(X) , (A.4) Y c X

where X = (Xx-.. x j) denotes any set of states of single particle and the convolution product �9 is defined by the last equality in (A.4). We first observe that the operator Lo acts as a derivation for the product . :

Lop = (Lov)*g + v*(Log) (A.5)

(from now on we skip the suffix e for notational simplicity). We now express the collision and the interaction operators, C and L~ respectively, in the set formalism. We have:

where

and

C = C + - C- , (A.6)

c * p ( x ) = E -~ ~- p(x, ux_), (A.7) x~X

C-p(X) = ~,, p (Xux_) , (A.8) xc~X

x _ = ( q , - a ) , x • • if x= (q ,a ) , (A.9)

X ~ = (X/x) k; x • (A. 10)


Finally L I = L~- + L~-, (A.11)

and

where

1 = p(Xxy)Xx,r , (A.12) L + P(X) ~ Z .L

(x ,y)~X

1 Li-p(X)=~ E P(X)Xx,y, (A.13)

(x,y)eX

XxXy = (X/{x, y})w {x • y• (A.14)

and Xx.y = Zij (cfr. Eq. (2.14)) whenever the particles i andj are in the states x and y. The following is straightforward algebra:

C-p(X)= Z Z v(Z)o(X x_/Z) x e X Z c X u x -

= Z E v(Twx_)g(X/T)+ E E v(T)o(Xux-/T) T ~ X x e X T c X x~X

=(C-v)*g(X)+ E ~ v(Tux_)o(X/T)+v*(C-o)(X) T c X x e X / T

+ E Z v(T)g(Xux_/T) T c X x a T

= (C-v).g(X) + v*(C-o)(X) + (K[v),g(X) + (K; v), #(X), (A.15)

where

K[v(T) = ~ v(Twx_/x)g(x), (A.16) xgiT

K2 v(T ) = ~, v(T)g(x_). (A.17) x,~T

By analogous computations:

C+ p(X) = (C+ v),g(X) + v,(C+ g)(X) + (K ~ v),g(X) + (K~ v)*g(X),

where

Moreover

(A.18)

1 L;-p(X)=~ ~,, ~ v(T)g(X/T)~x.,. (A.21)

T c X x , y~X

We split the sum x, yeX in the four sums: x, yeT; yeX/T, xeT; xeX/T, yeT; x, yeX/T and, after this, we arrive easily at the following identity:

Lip = Li v,g + A~ v,g + A2v, 0, (A.22) where

A[v(r)= ~ v(T/x)o(X)Zx,y, (A.23) x , y e T

K~-v(T) = ~ v(Tux~_/x)g(x• (A.19) x~T

K~ v(T) = ~ v(Tw xX/x)g(x ~_), (A.20) x E T


1 A2 v(T) =~ ~ v(T/x,y)g(x)g(y)xx,r (A.24)

x,ycT Analogously

Lip = L [ v * g + A~ v*g + A; v*g, (A.25) where

A'~ v(T)= ~_, v(T/{x, y )u x• y, (A.26) x,yeT

1 (A; v)(T) = ~ ~ v(T/{x, y} )g(xi)g(y• (A.27)

x,y~T Defining

K~ = K~ - K[, i = 1, 2, (A.28)

A, = A'+ - A ? , i = 1, 2, (A.29)

we arrive, finally, at the equation:

dv dt = (~- 1L~ + Lt + KI + K2 + C + A 1 + A2)v. (A.30)

References

1. Boldrighini, C., DeMasi, A., Pellegrinotti, A.: Non-equilibrium fluctuation in particle systems modelling Reaction-Diffusion equations. CARR reports in Math. Phys. no. 20/89, July 1989

2. Caprino, S., DeMasi, A., Presutti, E., Pulvirenti, M.: A stochastic particle systems modelling the Carleman equation. J. Stat. Phys. 55, 625-638 (1989)

3. DeMasi, A., Presutti, E.: Lectures on the collective behaviour of particle system. CARR reports in Math. Phys. no. 5/89, November 1989

4. DiPerna, R., Lions, P. L.: On the Cauchy problem for the Boltzmann equations: Global existence and weak stability. Ann. Math. 130, 321-366 (1989) /

5. Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for a two and three- dimensional rare gas in vacuum: Erratum and improved results. Commun. Math. Phys. 121, 143-146 (1989)

6. Lanford, O.: Time evolution of large classical systems. Lecture Notes in Physics, Vol. 38. Moser, E. J. (ed) pp. 1-111. Berlin, Heidelberg, New York: Springer 1975

7. Uchiyama, K.: On the Boltzmann-Grad limit for the Broadwell model of the Boltzmann equation. J. Stat. Phys. 52, 331 (1988)

Communicated by J. L. Lebowitz

a derivation of the broadwell equation

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