A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION CHAINS CAMILLE NOÛS SYLVAIN RUBENTHALER

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A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION

CHAINS

CAMILLE NOÛS, SYLVAIN RUBENTHALER,

Abstract. We are interested in a fragmentation process. We observe fragments frozen when

their sizes are less than ε(ε > 0). It is known ([BM05]) that the empirical measure of these

fragments converges in law, under some renormalization. In [HK11], the authors show a bound

for the rate of convergence. Here, we show a central-limit theorem, under some assumptions.

This gives us an exact rate of convergence.

1. Introduction

1.1. Scientiﬁc and economic context. One of the main goals in the mining industry is to

extract blocks of metallic ore and then separate the metal from the valueless material. To do so,

rock is fragmented into smaller and smaller rocks. This is carried out in a series of steps, the ﬁrst

one being blasting, after which the material goes through a sequence of crushers. At each step,

the particles are screened, and if they are smaller than the diameter of the mesh of a classifying

grid, they go to the next crusher. The process stops when the material has a suﬃciently small

size (more precisely, small enough to enable physicochemical processing).

This fragmentation process is energetically costly (each crusher consumes a certain quantity of

energy to crush the material it is fed). One of the problems that faces the mining industry is that

of minimizing the energy used. The optimization parameters are the number of crushers and the

technical speciﬁcations of these crushers.

In [BM05], the authors propose a mathematical model of what happens in a crusher. In this

model, the rock pieces/fragments are fragmented independently of each other, in a random and

auto-similar manner. This is consistent with what is observed in the industry, and this is supported

by the following publications: [PB02, DM98, Wei85, Tur86]. Each fragment has a size s(in

R+) and is then fragmented into smaller fragments of sizes s1,s2, . . . such that the sequence

(s1/s, s2/s, . . . )has a law νwhich does not depend on s(which is why the fragmentation is said to

be auto-similar). This law νis called the dislocation measure (each crusher has its own dislocation

measure). The dynamic of the fragmentation process is thus modeled in a stochastic way.

In each crusher, the rock pieces are fragmented repetitively until they are small enough to

slide through a mesh whose holes have a ﬁxed diameter. So the fragmentation process stops

for each fragment when its size is smaller than the diameter of the mesh, which we denote by

ε(ε > 0). We are interested in the statistical distribution of the fragments coming out of a

crusher. If we renormalize the sizes of these fragments by dividing them by ε, we obtain a measure

γ−log(ε), which we call the empirical measure (the reason for the index −log(ε)instead of εwill

be made clear later). In [BM05], the authors show that the energy consumed by the crusher to

reduce the rock pieces to fragments whose diameters are smaller than εcan be computed as an

integral of a bounded function against the measure γ−log(ε)(they cite [Bon52, Cha57, WLMG67]

on this particular subject). For each crusher, the empirical measure γ−log(ε)is one of the two

only observable variables (the other one being the size of the pieces pushed into the grinder). The

speciﬁcations of a crusher are summarized in εand ν.

Date: 2022.

2000 Mathematics Subject Classiﬁcation. 60J80, 60F05, 60K05.

Key words and phrases. Fragmentation, branching process, renewal theory, central-limit theorems, propagation

of chaos, U-statistics.

arXiv:2210.07791v1 [math.PR] 14 Oct 2022

A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION CHAINS 2

1.2. State of the art. In [BM05], the authors show that the energy consumed by a crusher

to reduce rock pieces of a ﬁxed size into fragments whose diameter are smaller than εbehaves

asymptotically like a power of εwhen εgoes to zero. More precisely, this energy multiplied

by a power of εconverges towards a constant of the form κ=ν(ϕ)(the integral of ν, the

dislocation measure, against a bounded function ϕ). In [BM05], the authors also show a law of

large numbers for the empirical measure γ−log(ε). More precisely, if fis bounded continuous,

γ−log(ε)(f)converges in law, when εgoes to zero, towards an integral of fagainst a measure

related to ν(this result also appears in [HK11], p. 399). We set γ∞(f)to be this limit (check

Equations (5.3), (2.7), (2.2) to get an exact formula). The empirical measure γ−log(ε)thus contains

information relative to νand one could extract from it an estimation of κor of an integral of any

function against ν.

It is worth noting that by studying what happens in various crushers, we could study a family

(νi(fj))i∈I,j∈J(with an index ifor the number of the crusher and the index jfor the j-th test

function in a well-chosen basis). Using statistical learning methods, one could from there make a

prediction for ν(fj) for a new crusher for which we know only the mechanical speciﬁcations (shape,

power, frequencies of the rotating parts . . . ). It would evidently be interesting to know νbefore

even building the crusher.

In the same spirit, [FKM10] studies the energy eﬃciency of two crushers used after one another.

When the ﬁnal size of the fragments tends to zero, this paper tells us wether it is more eﬃcient

enrgywise to use one crusher or two crushers in a row (another asymptotic is also considered in

the paper).

In [HKK10], the authors prove a convergence result for the empirical measure similar to the

one in [BM05], the convergence in law being replaced by an almost sure convergence. In [HK11],

the authors give a bound on the rate of this convergence, in a L2sense, under the assumption

that the fragmentation is conservative. This assumption means there is no loss of mass due to the

formation of dust during the fragmentation process.

γ−log(ε)(bound on rate)

−→

ε→0

γ∞

lrelation

energy ×(power of ε)∼

ε→0κ=ν(ϕ)

Figure 1.1. State of the art.

So we have convergence results ([BM05, HKK10]) of an empirical quantity towards constants

of interest (a diﬀerent constant for each test function f). Using some transformations, these

constants could be used to estimate the constant κ. Thus it is natural to ask what is the exact

rate of convergence in this estimation, if only to be able to build conﬁdence intervals. In [HK11],

we only have a bound on the rate.

When a sequence of empirical measures converges to some measure, it is natural to study the

ﬂuctuations, which often turn out to be Gaussian. For such results in the case of empirical measures

related to the molliﬁed Boltzmann equation, one can cite [Mel98, Uch88, DZ91]. When interested

in the limit of a n-tuple as in Equation (1.1) below, we say we are looking at the convergence

of a U-statistics. Textbooks deal with the case where the points deﬁning the empirical measure

are independent or with a known correlation (see [dlPG99, DM83, Lee90]). The problem is more

complex when the points deﬁning the empirical measure are in interaction with each other like it

is the case here.

1.3. Goal of the paper. As explained above, we want to obtain the rate of convergence in the

convergence of γ−log(ε)when εgoes to zero. We want to produce a central-limit theorem of the

kind: for a bounded continuous f,εβ(γ−log(ε)(f)−γ∞(f)) converges towards a non-trivial measure

when εgoes to zero (the limiting measure will in fact be Gaussian), for some exponent β. The

technics used will allow us to prove the convergence towards a multivariate Gaussian of a vector

A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION CHAINS 3

of the kind

(1.1) εβ(γ−log(ε)(f1)−γ∞(f1), . . . , γ−log(ε)(fn)−γ∞(fn))

for functions f1, . . . , fn.

More precisely, if by Z1,Z2, . . . , ZNwe denote the fragments sizes that go out from a crusher

(with mesh diameter equal to ε). We would like to show that for a bounded continuous f,

γ−log(ε)(f) :=

i=1

Zif(Zi/ε)−→ γ∞(f), almost surely, when ε→0,

and that for all n, and f1, . . . ,fnbounded continuous function such that γ∞(fi) = 0,

εβ(γ−log(ε)(f1), . . . , γ−log(ε)(fn))

converges in law towards a multivariate Gaussian when εgoes to zero.

The exact results are stated in Proposition 5.1 and Theorem 5.2.

1.4. Outline of the paper. We will state our assumptions along the way (Assumptions A, B,

Ca, D). Assumption D can be found at the beginning of Section 3. We deﬁne our model in Section

2. The main idea is that we want to follow tags during the fragmentation process. Let us imagine

the fragmentation is the process of breaking a stick (modeled by [0,1]) into smaller sticks. We

suppose that the original stick has painted dots and that during the fragmentation process, we

take note of the sizes of the sticks supporting the painted dots. When the sizes of these sticks get

smaller than ε(ε > 0), the fragmentation is stopped for them and we call them the painted sticks.

In Section 3, we make use of classical results on renewal processes and of [Sgi02] to show that the

size of one painted stick has an asymptotic behavior when εgoes to zero and that we have a bound

on the rate with which it reaches this behavior. Section 4 is the most technical. There we study

the asymptotics of symmetric functionals of the sizes of the painted sticks (always when εgoes to

zero). In Section 5, we precisely deﬁne the measure we are interested in (γTwith T=−log(ε)).

Using the results of Section 4, it is then easy to show a law of large numbers for γT(Proposition

5.1) and a central-limit Theorem (Theorem 5.2). Proposition 5.1 and Theorem 5.2 are our two

main results. The proof of Theorem 5.2 is based on a simple computation involving characteristic

functions (the same technique was already used in [DPR09, DPR11a, DPR11b, Rub16]).

1.5. Notations. For xin R, we set dxe= inf{n∈Z:n≥x},bxc= sup{n∈Z:n≤x}. The

symbol tmeans “disjoint union”. For nin N∗, we set [n] = {1,2, . . . , n}. For fan application

from a set Eto a set F, we write f:E →Fif fis injective and, for kin N∗, if F=E, we set

f◦k=f◦f◦ ··· ◦ f

| {z }

ktimes

For any set E, we set P(E)to be the set of subsets of E.

2. Statistical model

2.1. Fragmentation chains. Let ε > 0. Like in [HK11], we start with the space

S↓=(s= (s1, s2, . . . ), s1≥s2≥ ··· ≥ 0,

+∞

i=1

si≤1).

A fragmentation chain is a process in S↓characterized by

•a dislocation measure νwhich is a ﬁnite measure on S↓,

•a description of the law of the times between fragmentations.

A fragmentation chain with dislocation measure νis a Markov process X= (X(t), t ≥0) with

values in S↓. Its evolution can be described as follows: a fragment with size xlives for some time

(which may or may not be random) then splits and gives rise to a family of smaller fragments

distributed as xξ, where ξis distributed according to ν(.)/ν(S↓). We suppose the life-time of a

fragment of size xis an exponential time of parameter xαν(S↓), for some α. We could here make

diﬀerent assumptions on the life-time of fragments, but this would not change our results. Indeed,

A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION CHAINS 4

as we are interested in the sizes of the fragments frozen as soon as they are smaller than ε, the

time they need to become this small is not important.

We denote by Pmthe law of Xstarted from the initial conﬁguration (m, 0,0, . . . )with min

(0,1]. The law of Xis entirely determined by αand ν(.)(Theorem 3 of [Ber02]).

We make the same assumption as in [HK11] and we will call it Assumption A.

Assumption A.We have ν(S↓)=1and ν(s1∈]0; 1[) = 1.

Let

U:= {0} ∪

+∞

[

n=1

(N∗)n

denote the inﬁnite genealogical tree. For uin U, we use the conventional notation u= () if

u={0}and u= (u1, . . . , un)if u∈(N∗)nwith n∈N∗.This way, any uin Ucan be denoted by

u= (u1, . . . , un), for some u1, . . . , unand with nin N. Now, for u= (u1, . . . , un)∈ U and i∈N∗,

we say that uis in the n-th generation and we write |u|=n, and we write ui = (u1, . . . , un, i),

u(k) = (u1, . . . , uk)for all k∈[n]. For any u= (u1, . . . , un)and v=ui (i∈N∗), we say that u

is the mother of v. For any uin U\{0}(Udeprived of its root), uhas exactly one mother and we

denote it by m(u). The set Uis ordered alphanumerically :

•If uand vare in Uand |u|<|v|then u<v.

•If uand vare in Uand |u|=|v|=nand u= (u1, . . . , un),v= (v1, . . . , vn)with u1=v1,

. . . , uk=vk,uk+1 < vk+1 then u<v.

Suppose we have a process Xwhich has the law Pm. For all ω, we can index the fragments that

are formed by the process Xwith elements of Uin a recursive way.

•We start with a fragment of size mindexed by u= ().

•If a fragment x, with a birth-time t1and a split-time t2, is indexed by uin U. At time t2,

this fragment splits into smaller fragments of sizes (xs1, xs2, . . . )with (s1, s2, . . . )of law

ν(.)/ν(S↓). We index the fragment of size xs1by u1, we index the fragment of size xs2

by u2, and so on.

A mark is an application from Uto some other set. We associate a mark ξ... on the tree Uto each

path of the process X. The mark at node uis ξu, where ξuis the size of the fragment indexed by

u. The distribution of this random mark can be described recursively as follows.

Proposition 2.1. (Consequence of Proposition 1.3, p. 25, [Ber06]) There exists a family of i.i.d.

variables indexed by the nodes of the genealogical tree, ((e

ξui)i∈N∗, u ∈ U), where each (e

ξui)i∈N∗is

distributed according to the law ν(.)/ν(S↓), and such that the following holds:

Given the marks (ξv,|v| ≤ n)of the ﬁrst ngenerations, the marks at generation n+ 1 are given

ξui =e

ξuiξu,

where u= (u1,...,un)and ui = (u1, . . . , un, i)is the i−th child of u.

2.2. Tagged fragments. From now on, we suppose that we start with a block of size m= 1. We

assume that the total mass of the fragments remains constant through time, as follows.

Assumption B.(Conservative property).

We have ν(P+∞

i=1 si= 1) = 1.

This assumption was already present in [HK11]. We observe that the Malthusian exponent of

[Ber06] (p. 27) is equal to 1under our assumptions. Without this assumption, the link between

the empirical measure γ−log(ε)and the tagged fragments (Equation (5.2)) vanishes and our proofs

of Proposition 5.1 and Theorem 5.2 fail.

We can now deﬁne tagged fragments. We use the representation of fragmentation chains as

random inﬁnite marked tree to deﬁne a fragmentation chain with qtags. Suppose we have a

fragmentation process Xof law P1. We take (Y1, Y2, . . . , Yq)to be qi.i.d. variables of law U([0,1]).

We set, for all uin U,

(ξu, Au, Iu)

A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION CHAINS 5

with ξudeﬁned as above. The random variables Autake values in the subsets of [q]. The random

variables Iuare intervals. These variables are deﬁned as follows.

•We set A{0}= [q],I{0}= (0,1] (I{0}is of length ξ{0}= 1)

•For all n∈N. Suppose we are given the marks of the ﬁrst ngenerations. Suppose that,

for uin the n-th generation, Iu= (au, au+ξu]for some au∈R(it is of length ξu). Then

the marks at generation n+ 1 are given by Proposition 2.1 (concerning ξ.) and, for all u

such that |u|=nand for all iin N∗

Iui = (au+ξu(e

ξu1+··· +e

ξu(i−1)), au+ξu(e

ξu1+··· +e

ξui)] ,

k∈Aui if and only if Yk∈Iui ,

(Iui is then of length ξui). We observe that for all j∈[q],u∈ U,i∈N∗,

(2.1) P(j∈Aui|j∈Au,e

ξui) = e

ξui .

In this deﬁnition, we imagine having qdots on the interval [0,1] and we impose that the dot jhas

the position Yj(for all jin [q]). During the fragmentation process, if we know that the dot jis

in the interval Iuof length ξu, then the probability that this dot is on Iui (for some iin N∗,Iui

of length ξui) is equal to ξui/ξu=e

ξui.

In the case q= 1, the branch {u∈ U :Au6=∅} has the same law as the randomly tagged

branch of Section 1.2.3 of [Ber06]. The presentation is simpler in our case because the Malthusian

exponent is 1under Assumption B.

2.3. Observation scheme. We freeze the process when the fragments become smaller than a

given threshold ε > 0. That is, we have the following data

(ξu)u∈Uε,

where

Uε={u∈ U, ξm(u)≥ε, ξu< ε}.

We now look at qtagged fragments (q∈N∗). For each iin [q], we call L(i)

0= 1,L(i)

1,L(i)

2. . .

the successive sizes of the fragment having the tag i. More precisely, for each n∈N∗, there is

almost surely exactly one u∈ U such that |u|=nand i∈Au; and so, L(i)

n=ξu. For each i, the

process S(i)

0=−log(L(i)

0) = 0 ≤S(i)

1=−log(L(i)

1)≤. . . is a renewal process without delay, with

waiting-time following a law π(see [Asm03], Chapter V for an introduction to renewal processes).

The waiting times are (for iin [q]): S(i)

0,S(i)

1−S(i)

0,S(i)

2−S(i)

1, . . . The renewal times are (for i

in [q]): S(i)

0,S(i)

1,S(i)

2, . . . The law πis deﬁned by the following.

(2.2)

For all bounded measurable f: [0,1] →[0,+∞),ZS↓

+∞

i=1

sif(si)ν(ds) = Z+∞

f(e−x)π(dx),

(see Proposition 1.6, p. 34 of [Ber06], or Equations (3), (4), p. 398 of [HK11]). Under Assumption

A and Assumption B, Proposition 1.6 of [Ber06] is true, even without the Malthusian Hypothesis

of [Ber06].

We make the following assumption on π.

Assumption Ca.There exist aand b(0< a < b < +∞) such that the support of πis [a, b]. We

set δ=e−b.

We added a comment about the above Assumption in Remark 4.3. We believe that we could

replace the above Assumption by the following.

Assumption Cb.The support of πis (0,+∞).

However, this would lead to diﬃcult computations.

We set

(2.3) T=−log(ε).

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ACENTRAL-LIMITTHEOREMFORCONSERVATIVEFRAGMENTATIONCHAINSCAMILLENOÛS,SYLVAINRUBENTHALER,Abstract.Weareinterestedinafragmentationprocess.Weobservefragmentsfrozenwhentheirsizesarelessthan"(">0).Itisknown([BM05])thattheempiricalmeasureofthesefragmentsconvergesinlaw,undersomerenormalization.In[HK11],theau...

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A CENTRAL-LIMIT THEOREM FOR CONSERVATIVE FRAGMENTATION CHAINS CAMILLE NOÛS SYLVAIN RUBENTHALER

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