Kinetic Theory of Cluster Dynamics

0.39 MB·

by  Robert I. A. Patterson

Kinetic Theory of Cluster Dynamics
Robert I. A. Patterson
Weierstrass Institute Berlin
Mohrenstr. 39, 10117 Berlin, Germany
[email protected]
Sergio Simonella
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Zentrum Mathematik, TU Mu¨nchen
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0 Boltzmannstr. 3, 85748 Garching, Germany
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[email protected]
n
u
J Wolfgang Wagner

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Weierstrass Institute Berlin
2
Mohrenstr. 39, 10117 Berlin, Germany
]
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[email protected]
p
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h
t
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InaNewtoniansystemwithlocalizedinteractionsthewholesetofparticlesisnaturallydecomposed
m
into dynamical clusters, defined as finite groups of particles having an influence on each other’s
[

trajectory during a given interval of time. For an ideal gas with short–range intermolecular force,
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v we provide a description of the cluster size distribution in terms of the reduced Boltzmann density.
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3 In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas
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forms a giant component in finite kinetic time. The critical index of this phase transition is in
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0 agreement with previous numerical results on the elastic billiard.
.
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Keywords: low–density gas; Boltzmann equation; cluster dynamics; Maxwell molecules.
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1
:
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i 1. INTRODUCTION
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r
a
As a proposal to gain insight on the statistical properties of large systems in a gaseous
phase, N. Bogolyubov suggested to investigate a simple notion of cluster decomposition
characterizing the collisional dynamics [4]. When the evolution is determined by a sequence
of single, distinct, two–body interactions, a natural partition of the system can be defined
in terms of groups of particles connected by a chain of collisions, so that a “cluster” consists
of elements having affected each other’s trajectory.
This notion has been developed later on, in connection with the problem of the Hamilto-
niandynamicsofaninfinitesystem. InaNewtoniansystem, particleswithrapidlyincreasing
energiesatinfinitymaygenerateinstantaneouscollapsesforspecialinitialconfigurations[13].
Mathematically, one needs to prove that such initial data form a set of measure zero in the
2
phase space of infinitely many particles. In fact, one possible strategy to construct the dy-
namics is to show that, at properly fixed time, the system splits up into an infinite number
of clusters which are moving independently as finite-dimensional dynamical systems. After
some random interval of time, the partition into independent clusters changes, and one iter-
ates the procedure. This dynamics is known as cluster dynamics and its existence has been
proved first in [21] for some one–dimensional models (see [22] for generalizations).
In more recent years, the statistical properties of cluster dynamics of a system obeying
Newton’s law have been studied numerically [10]. In this reference, the authors focus on
the frictionless elastic billiard in a square two–dimensional box with reflecting walls, and
show that the dynamics undergoes a phase transition. This occurs in a way reminiscent
of problems in percolation theory. Namely, the maximal (largest) cluster starts to increase
dramatically at some critical time. At the critical time, the fraction of mass in the maximal
cluster is rather small ( 7% for 5000 disks at small volume density). After the critical time,
∼
it approaches the total mass of the system with exponential rate. Moreover, the transition is
distinguished by a power–law behaviour for the cluster size distribution with exponent 5/2.
Such critical index is believed to be universal, since it has been observed for several different
models (see also [16]).
The cluster dynamics concept, together with the above described statistical behaviour,
appear as well in a number of applied papers, e.g. geophysics, economics, plasma physics:
see [10] and references therein.
Kinetictheoryoftenprovidessuccessfulmethodsforthecomputationofmicroscopicquan-
tities related to properties of the dynamical system, for instance Lyapunov exponents or
Kolmogorov–Sinai entropies [3, 7, 25, 29]. In the present work, we are concerned with the
cluster dynamics of an ideal gas where the kinetic description of Boltzmann based on molec-
ular chaos applies.
Oursettingisgivenbyadensityfunctionf = f(x,v,t)describingtheamountofmolecules
having position x Λ Rd and velocity v Rd at time t, and evolution ruled by
∈ ⊂ ∈
(cid:90)
(cid:110) (cid:111)
(∂ +v )f = dv dω B(v v ,ω) f(cid:48)f(cid:48) f f , (1.1)
t ·∇x 1 − 1 1 − 1
Rd×Sd−1
where f = f(x,v,t),f = f(x,v ,t),f(cid:48) = f(x,v(cid:48),t),f(cid:48) = f(x,v(cid:48),t), (v,v ) is a pair of
1 1 1 1 1
velocitiesinincomingcollisionconfigurationand(v(cid:48),v(cid:48))isthecorrespondingpairofoutgoing
1
velocities when the scattering vector is ω:

v(cid:48) = v ω[ω (v v )]
1
− · − . (1.2)
v(cid:48) = v +ω[ω (v v )]
1 1 · − 1
Thetime–zerodensityf(x,v,0) = f (x,v)isfixed. Forsimplicity,thegasmovesinthesquare
0
d dimensional box Λ of volume 1, with reflecting boundary conditions. The microscopic
− (cid:82)
potential is assumed to be short–ranged and the cross–section B satisfies dωB(v v ,ω) =
1
−
a( v v ) < (“Grad’s cut–off assumption”).
1
| − | ∞
3
The precise connection with a dynamical system of N particles interacting at mutual
distance ε, such as the one studied in [10], can be established locally in the low–density limit
N , Nεd−1 1 , (1.3)
→ ∞ (cid:39)
(“Boltzmann–Grad regime”) as the convergence of correlation functions to the solution of
the Boltzmann equation [14] (see also [6, 12, 19, 25, 27]). In the regime (1.3), the gas is so
dilute that only two–body collisions are relevant. Furthermore, the collisions are completely
localized in space and time. The limit transition (1.3) explains the microscopic origin of
irreversible behaviour [11].
Our purpose here is to describe how the cluster size distribution is constructed from the
solution to the Boltzmann equation. This is done in Section 2 by means of a suitable tree
graphexpansion, whichisinspiredbypreviouslyknownformulasrepresentingtheBoltzmann
densityasasumovercollisionsequences[28]. InSection3, weindicatehowtoderiveformally
the introduced expressions as the limiting cluster distributions of a system of hard spheres
in the Boltzmann–Grad scaling. Finally, in Section 4, we restrict to the simplest nontrivial
(and paradigmatic) case in kinetic theory, i.e. the model of Maxwellian molecules. We show
that the cluster distribution exhibits a phase transition characterized by a breakdown of
the normalization condition at finite time. This implies that the “percolation” survives in
the Boltzmann–Grad limit, with same qualitative behaviour and same critical index of the
elastic billiard analyzed in [10].
2. CLUSTER DISTRIBUTIONS
A. Bogolyubov clusters
We start with a formal notion of cluster. Let t be a given positive time.
Definition 1.
(i) Two particles are t neighbours if they collided during the time interval [0,t].
−
(ii) A Bogolyubov t cluster is any connected component of the neighbour relation (i).
−
The definition can be generalized to generic time intervals [s,s + t]. However, in what
follows we will study the notion of t cluster only, which is no restriction, and drop often the
−
t-dependence in the nomenclature.
Notice that each particle has collided with at least one other particle of its Bogolyubov
cluster, while it has never collided with particles outside the cluster, within the time interval
[0,t]. In particular, if t = 0, any particle of the gas forms a singleton (cluster of size 1).
At t > 0, the mass of singletons starts to decrease and clusters of size k = 2,3, start
···
to appear. We therefore expect to see (and do observe in the experiments) some “smooth”
exponential distribution in the cluster size.
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B. Backward clusters
In Reference [2], the solution of (1.1) has been expanded in terms of a sum of type
∞
(cid:88)(cid:88)
f = fΓn , (2.1)
n=0 Γn
where Γ = , Γ = (k ,k , ,k ) and k 1 ,k 1,2 , ,k 1,2, ,n . The
0 n 1 2 n 1 2 n
∅ ··· ∈ { } ∈ { } ··· ∈ { ··· }
sequences Γ are in one–to–one correspondence with binary tree graphs, e.g.
n
, ,
···
1 2 1 2 3 1 (2.2)
for n = 1,2,3 respectively. In (2.1), fΓn is interpreted as the contribution to the proba-
···
bility density f due to the event: the backward cluster of 1 has structure Γ . By “backward
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cluster” we mean here the group of particles involved directly or indirectly in the backwards–
in–time dynamics of particle 1. Operatively, in a numerical experiment, we select particle 1
at time t, run the system backwards in time, and collect all the particles which collide with
1 and with “descendants” of 1 in the backwards dynamics, following (2.2). In other words,
(2.1) is an expansion on sequences of real collisions1.
Formulas of this kind have been previously studied in the contextof Maxwellian molecules
with cut–off under the name of Wild sums [5, 17, 28], and are written in [2] for a gas of
hard spheres in a homogeneous state. It is not difficult to generalize such a representation
to inhomogeneous states and general interactions. The formula for fΓn reads
fΓn(x1,v1,t) = (cid:90) tdt1(cid:90) t1dt2 (cid:90) tn−1dtn(cid:90) dv2 dv1+ne−(cid:82)tt1dsR1(ζ1(s),s)
··· ···
0 0 0 Rnd
(cid:32) (cid:33)
n (cid:90)
(cid:89) dωrB(ηkrr−1 −v1+r, ωr)e−(cid:82)ttrr+1dsR1+r(ζr+1(s),s) f0⊗(1+n)(ζn+1(0)) ,
r=1 Sd−1
(2.3)
where:
– ζ = (ζ , ,ζ ), ζ = (ξ ,η ) = (position, velocity);
k 1 k i i i
···
1 Observe that recollisions (e.g. the pair (1,2) colliding twice in the backward history), certainly possible
in an experiment, do not affect the notion of backward cluster, which is based on sequences of collisions
involving at least one “new” particle; see [2] for details on the numerical procedure.
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– f is the initial density (and f⊗(1+n)(ζ ) = f (ζ ) f (ζ ));
0 0 n+1 0 1 0 n+1
···
– the “free–flight rate” of k particles R is given by
k
k
(cid:88)
R (ζ ,s) = R(ζ ,s) (2.4)
k k i
i=1
where the function R depends on the solution f(s) of the Boltzmann equation itself:
(cid:90)
R(x,v,t) = dv dω B(v v ,ω)f(x,v ,t) ; (2.5)
∗ ∗ ∗
−
Rd×Sd−1
– the “trajectory of the backward cluster” s ζ (s) is constructed as follows:
r+1
→
(a) fix x ,v ,t,Γ ,t ,v ,ω , ,t ,v ,ω , with t > t > > t > 0 ;
1 1 n 1 2 1 n n+1 n 1 n
··· ···
(b) construct first the sequence of velocities ηr, r = 0, ,n, defined iteratively by:
···
η0 = v = (v , ,v ) , ηr = (ηr−1, ,η(cid:48) , ,η(cid:48) , ,ηr−1) r 1
n+1 1 ··· n+1 1 ··· kr ··· r+1 ··· n+1 ≥
where, at step r, the pair η(cid:48) ,η(cid:48) are the pre–collisional velocities (in the collision with
kr r+1
impact vector ω ) of the pair ηr−1,ηr−1 = v (which are post–collisional, as ensured by the
r kr r+1 r+1
fact that B(v v ,ω) = 0 only for (v v ) ω > 0), according to the transformation (1.2);
1 1
− (cid:54) − ·
(c) construct the trajectory of the backward cluster iteratively by
ξ (s) = x v (t s) , η (s) = v s (t ,t) ,
1 1 1 1 1 1
− − ∈
and, for r 1 and i = 1, ,r+1,
≥ ···
ξ (s) = ξ (t ) ηr(t s) , η (s) = ηr s (t ,t ) ,
i i r − i r − i i ∈ r+1 r
with the convention t = 0.
n+1
The term n = 0 in (2.3) is
f∅(x,v,t) = e−(cid:82)0tdsR(x−v(t−s),v,s)f0(x vt,v) (2.6)
−
(cid:82)
and dxdvf∅(x,v,t) = density of free particles in (0,t). Similarly,
(cid:82) dxdv(cid:80) fΓn(x,v,t) = density of particles having a backward cluster of size n in
Γn
(0,t).
C. Symmetrization
We can see fΓn as an integral over trajectories of a Markov process with n collisions in
specified order. Each trajectory has probability density given by the integrand function,
that is:
f⊗(1+n) = initial distribution of n+1 particles ;
0
B(ηr−1 v , ω ) = transition kernel of the collision (ηr−1,v ) (η(cid:48) ,η(cid:48) ) ;
kr − 1+r r kr r+1 → kr r+1
e−(cid:82)ttrr+1dsR1+r(ζr+1(s),s) = probability of free flight of r+1 particles in (tr+1,tr),
conditioned to the configuration ζ (t ) .
r+1 r
(2.7)
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We remind that, in a backward cluster, the trajectory of particle i is specified only in the
time interval (0,t ), where t > t > > t > 0.
i−1 1 n
···
The density of Bogolyubov clusters can be obtained by “adding” the missing information,
namely the future history of the particles 2,3, in the time intervals (t ,t),(t ,t),
1 2
··· ···
respectively, together with the complete history of the particles with whom they collide.
This amounts, for instance, to extend in the future, by free motion, the trajectories of
2,3,
···
, ,
···
1 2 1 2 3 1 (2.8)
where the dotted lines correspond to free–flight. However, this example is not enough, since
we need to take into account additional trajectories, e.g.
,
···
2 1 2 3 1 (2.9)
In other words, we extend the history of the backward cluster to provide full knowledge
of the trajectory of the particles in the time interval (0,t).
WemakethismorepreciseintherestofthisSection. Ourgoalistowriteaformulaforthe
density of Bogolyubov clusters (Definition 2 below) starting from (2.3) & (2.7). Before that,
we need to introduce notions of ‘collision graph’ and of the associated ‘trajectory of clusters’.
The density of clusters will be indeed expressed as an integral over such trajectories.
Let be a labelled tree with k vertices, i.e. a connected graph with k vertices and k 1
k
G −
edges. For instance for k = 4
The vertices are labelled 1,2, ,k and = (i ,j ), ,(i ,j ) (non ordered set of
k 1 1 k−1 k−1
··· G { ··· }
pairs). Each vertex represents a particle and each link represents a collision. We refer to
k
G
as collision graph of the Bogolyubov cluster.
A trajectory of the Bogolyubov t cluster (z (s)) = (z (s), ,z (s)) , where
k s∈(0,t) 1 k s∈(0,t)
− ···
z (s) = (x (s),v (s)) = (position, velocity), is constructed as follows.
i i i
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(a(cid:48)) Fix x ,t,v , ,v , ,t ,ω , ,t ,ω , with t (0,t);
1 1 k k 1 1 k−1 k−1 i
··· G ··· ∈
(b(cid:48)) let (cid:96) , ,(cid:96) be the permutation of 1, ,k 1 such that t > t > > t ;
1 ··· k−1 ··· − (cid:96)1 (cid:96)2 ··· (cid:96)k−1
construct the sequence of velocities vr, r = 0, ,k 1, defined iteratively by:
··· −
v0 = v = (v , ,v ) , vr = (vr−1, ,v(cid:48) , ,v(cid:48) , ,vr−1) r 1
k 1 ··· k 1 ··· i(cid:96)r ··· j(cid:96)r ··· k ≥
where, at step r, the pair v(cid:48) ,v(cid:48) are the pre–collisional velocities (in the collision with
i j
(cid:96)r (cid:96)r
impact vector ω ) of the pair vr−1,vr−1 (assumed post–collisional), according to the trans-
(cid:96)r i(cid:96)r j(cid:96)r
formation (1.2);
(c(cid:48)) define the trajectory of particle i 1,2, ,k of the Bogolyubov cluster iteratively
∈ { ··· }
by
x (s) = x v (t s) , v (s) = v s (t ,t) ,
i i − i − i i ∈ (cid:96)1
and, for r 1,
≥
x (s) = x (t ) vr(t s) , v (s) = vr s (t ,t ) ,
i i (cid:96)r − i (cid:96)r − i i ∈ (cid:96)r+1 (cid:96)r
with the convention t = 0. Notice that the positions x , ,x at time t are uniquely
(cid:96) 2 k
k ···
determined as soon as we fix x (t) = x .
1 1
With respect to the definition of trajectory used in (2.3), the essential differences are
summarized in the following table.
sequence of collisions times of collisions history of particle i
Backward cluster, size k tree graph Γ t > t > > t > 0 specified in (0,t )
k−1 1 k−1 i−1
···
Bogolyubov cluster, size k tree graph (t , ,t ) (0,t)k−1 specified in (0,t)
k 1 k−1
G ··· ∈
Inbothcases, thereareexactlyk 1collisions, sincerecollisionsareforbiddenintheassumed
−
kinetic regime (see page 11, item (2) for a precise statement). Such collisions are specified,
respectively, by the binary tree graph Γ and by the ordinary tree graph .
k−1 k
G
The Bogolyubov cluster can be described as the time–symmetrized version of the back-
ward cluster. Alternatively, this can be seen as a symmetrization in the labelling of the
particles, since no particle plays a special role anymore.
D. Size distribution of kinetic clusters
Motivated by the previous discussion, we introduce here an explicit f (k) written in terms
t
of the Boltzmann density solving (1.1), which should be interpreted physically as the frac-
tion of particles of the gas belonging to a dynamical cluster of size k. A formal argument
identifying f (k) as the kinetic limit of the corresponding quantity in a system of finitely
t
many particles will be presented below (see Claim 1); a rigorous derivation remains an open
problem.
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Definition 2. Let t [0, ) and k N, then
∈ ∞ ∈
(cid:90) (cid:90) (cid:90) k−1(cid:90)
1 (cid:88) (cid:89)
f (k) := dx dv dt dω B(vr−1 vr−1, ω )
t (k −1)! Gk Λ 1 Rkd k (0,t)k−1 k−1 r=1 Sd−1 (cid:96)r i(cid:96)r − j(cid:96)r (cid:96)r
k(cid:89)−1 −(cid:82)t(cid:96)r dsR (z (s),s)
e t(cid:96)r+1 k k f⊗k(zk(0)) , (2.10)
· 0
r=0
with the conventions t = t,t = 0.
(cid:96)0 (cid:96)k
Compared to fΓk−1, see (2.3) and (2.7), (cid:82)tdt (cid:82)t1dt (cid:82)tk−2dt has been replaced by
(cid:82) 0 1 0 2··· 0 k−1
the symmetric integral 1 dt , and the integrand function is now the probability
(k−1)! (0,t)k−1 k−1
density of a trajectory of a Bogolyubov cluster with collision graph .
k
G
By looking at (2.10) in a simple example, we will show in Section 4 that this distribution
is not normalized for all times. This is due to the development of giant clusters (k = ) at
∞
some critical time t . Therefore, with respect to backward clusters (often associated to the
c
description of correlations [2]), the Bogolyubov clusters exhibit a more interesting statistics.
The following rescaled version is also relevant.
Definition 3. The kinetic fraction of Bogolyubov t clusters with size k is
−
(cid:90) (cid:90) (cid:90) k−1(cid:90)
1 1 (cid:88) (cid:89)
g (k) := dx dv dt dω B(vr−1 vr−1, ω )
t Zt k! G Λ 1 Rkd k (0,t)k−1 k−1 r=1 Sd−1 (cid:96)r i(cid:96)r − j(cid:96)r (cid:96)r
k
k(cid:89)−1 −(cid:82)t(cid:96)r dsR (z (s),s)
e t(cid:96)r+1 k k f⊗k(zk(0)) , (2.11)
· 0
r=0
where
(cid:90) (cid:90) (cid:90) k−1(cid:90)
(cid:88) 1 (cid:88) (cid:89)
Z := dx dv dt dω B(vr−1 vr−1, ω )
t k≥1 k! Gk Λ 1 Rkd k (0,t)k−1 k−1 r=1 Sd−1 (cid:96)r i(cid:96)r − j(cid:96)r (cid:96)r
k(cid:89)−1 −(cid:82)t(cid:96)r dsR (z (s),s)
e t(cid:96)r+1 k k f⊗k(zk(0)) (2.12)
· 0
r=0
is the normalization constant.
The functions just introduced are the kinetic counterparts of the functions f and g
t t
examined in [10]. This connection with the finite dynamical system is clarified in the next
Section.
3. DERIVATION OF (2.10)
The following argument is an heuristic derivation of the formulas given above for the size
distribution of clusters. This is inspired by the papers [23, 24].
9
We consider here, for simplicity, a system of N identical hard spheres. These particles
have unit mass and diameter ε and move inside the box Λ = [0,1]3 with reflecting boundary
conditions. The dynamics T is given by free flow plus collisions at distance ε, which are
N
governedbythelawsofelasticreflection. Welabeltheparticleswithanindexi = 1,2, ,N.
···
The complete configuration of the system is then given by a vector z = (z , ,z ), where
N 1 N
···
z = (x ,v ) collects position x and velocity v of particle i. Let us assign a probability
i i i i i
density WN on the N particle phase space, assuming it symmetric in the exchange of the
0 − (cid:82)
particles. Forj = 1,2, ,N, wecallfN = dz dz WN(z )thej particlemarginal
··· 0,j j+1··· N 0 N −
of WN.
0
Let T ,T(cid:48) be the dynamical flows, considered in isolation, of the groups of particles
k N−k
1, ,k and k +1, ,N respectively. Set T˜ = (T ,T(cid:48) ) (T˜ = T only up to the
{ ··· } { ··· } N k N−k N N
time of the first collision between the two groups). We denote s [0, ], i = 1,2, the
i
∈ ∞ ···
time of the first collision of particle k+i with the set of particles 1, ,k in the dynamics
{ ··· }
T˜ . Furthermore, we set τ = min s , that is the time of the first collision of the group
N m i≥m i
k +m, ,N with the group 1, ,k .
{ ··· } { ··· }
Let us focus on Definition 2 and let
n (k) = number of Bogolyubov t clusters of size k . (3.1)
t
−
Moreover, let
(cid:110) (cid:111)
A := “ 1, ,k forms a t cluster under the reduced flow T ” .
k
{ ··· } −
By the symmetry in the particle labels, the average of n (k) with respect to WN is
t 0
(cid:18) (cid:19)
N (cid:16) (cid:17)
n (k) = P 1,2, ,k is a t cluster
t
(cid:104) (cid:105) k { ··· } −
(cid:18) (cid:19) (cid:90)
N
= dz WN(z )χ(A)χ(τ > t)
k N 0 N 1
(cid:18) (cid:19) (cid:90) (cid:90)
N
= dz χ(A) dz dz WN(z )χ(τ > t)
k k k+1··· N 0 N 1
(cid:18) (cid:19) (cid:90)
N
dz χ(A)fN (z )P (τ > t z ) , (3.2)
≡ k k 0,k k 1 | k
where χ(A) is the characteristic function of the set A, and the last line in (3.2) defines the
conditional probability.
More generally, we indicate by P ( t , ,t ,z ) a conditional probability given z and
i 1 k k
· | ···
s = t , ,s = t . Similarly, p ( t , ,t ,z ) is the conditional probability density of
i i ··· 1 1 si ·| i−1 ··· 1 k
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s , given z and s = t , ,s = t . Using again the symmetry,
i k i−1 i−1 1 1
···
P (τ > t z ) = 1 P (τ < t z )
1 k 1 k
| − |
(cid:90) t
= 1 (N k) dt p (t z ) P(τ > t t ,z )
− − 1 s1 1 | k 2 1 | 1 k
0
(cid:90) t
= 1 (N k) dt p (t z )
− − 1 s1 1 | k
0
(cid:16) (cid:90) t1 (cid:17)
1 (N k 1) dt p (t t ,z ) P(τ > t t ,t ,z )
· − − − 2 s2 2 | 1 k 3 2 | 2 1 k
0
= , (3.3)
···
so that by iteration we obtain
N−k (cid:90) t
(cid:88)
P (τ > t z ) = ( 1)j(N k) dt p (t z )
1 | k − − 1 s1 1| k
0
j=0
(cid:90) t1 (cid:90) tj−1
(N k 1) dt p (t t ,z ) (N k j +1) dt p (t t , ,t ,z ) .
· − − 2 s2 2| 1 k ··· − − j sj j| j−1 ··· 1 k
0 0
(3.4)
The term with j = 0 is defined to be 1.
Next we evaluate p (t t , ,t ,z ). Let v be the velocity of particle i at time s ,
si i| i−1 ··· 1 k ∗ i
k 1, ,k the index of the particle that particle i will hit, and ω S2 the normalized
i
∈ { ··· } ∈
relative displacement of particle i with respect to particle k . Let z(k)(s) = (x(k)(s),v(k)(s))
i ki ki ki
be position and velocity of particle k along the flow T . By definition of s , we must find
i k i
particle i in a cylinder of volume ds dv dωε2B(v(k)(s ) v ,ω) for some k 1, ,k .
i ∗ ki i − ∗ i ∈ { ··· }
We remind that, for hard spheres, the cross–section is B(V,ω) = (V ω)+. It follows that,
·
assuming the Boltzmann approximation fN f and p (t t , ,t ,z ) p (t z )2, the
1 ∼ si i| i−1 ··· 1 k ∼ si i| k
conditional probability dt p (t t , ,t ,z ) will be close to
i si i| i−1 ··· 1 k
k (cid:90)
(cid:88)
ε2 dt dv dωB(v(k)(t ) v ,ω)f(x(k)(t ),v ,t ) .
i ∗ ki i − ∗ ki i ∗ i
ki=1
Hence, in the scaling (1.3) we expect
(cid:88) (cid:89)j (cid:88)k (cid:90) ti−1 (cid:90)
P (τ > t z ) ( 1)j dt dv dωB(v(k)(t ) v ,ω)f(x(k)(t ),v ,t )
1 | k ≈ − i ∗ ki i − ∗ ki i ∗ i
0
j≥0 i=1ki=1
(cid:32) (cid:33)j
(cid:88) ( 1)j (cid:90) t (cid:88)k
= − ds R(x(k)(s),v(k)(s),s)
j! i i
0
j≥0 i=1
= e−(cid:82)0tdsRk(zk(k)(s),s) (3.5)
2 As follows from the statistical independence of particle i from any finite collection of given particles, in
the limit N .
→∞