## Abstract

We prove the following asymptotic behaviour for solutions to the generalized Becker–Döring system for general initial data: under a detailed balance assumption and in situations where density is conserved in time, there is a critical density *ρ*_{s} such that solutions with an initial density *ρ*_{0}≤*ρ*_{s} converge strongly to the equilibrium with density *ρ*_{0} and solutions with initial density *ρ*_{0}>*ρ*_{s} converge (in a weak sense) to the equilibrium with density *ρ*_{s}. This extends the previous knowledge that this behaviour happens under more restrictive conditions on the initial data. The main tool is a new estimate on the tail of solutions with density below the critical density.

## 1. Introduction

Coagulation–fragmentation equations are useful as models that describe the dynamics of many physical phenomena in which a large number of particles or units can stick together to form groups of particles, or clusters. A first version of them was initially proposed by Becker & Döring (1935), and a variant by Penrose & Lebowitz (1979); these relatively simple models take into account only processes in which a cluster gains or loses one particle, and describe only the concentration of clusters of a given size at a certain moment, omitting also a description of their spatial distribution. Since then a number of generalizations have been studied which also allow reactions between clusters of more than one particle, the main examples of this being the discrete coagulation–fragmentation equations (for example, Ball & Carr 1990; Carr 1992; Carr & da Costa 1994), their continuous version (Smoluchowski 1917; Stewart 1989; 1990; Laurençot 2000; Escobedo *et al*. 2002, 2003; Laurençot 2002; Mischler & Rodríguez 2003) and the respective versions including a spatial description by means of diffusion (Laurençot & Mischler 2002*a*,*b*). A recent review can be found in Laurençot & Mischler (2004).

The generalized Becker–Döring equations are an intermediate step between the Becker–Döring system and the full discrete coagulation–fragmentation equations in which we allow reactions between clusters of at most a given finite size *N* and other clusters. The system of equations is the following:(1.1)Here, the unknowns are *c*_{j}=*c*_{j}(*t*) for *j*=1, …, positive functions depending on the time *t*, which are intended to represent the density of clusters of size *j* (those formed by *j* elementary particles). The quantities *W*_{jk}, which depend on the *c*_{j}, are given bywhere the numbers *a*_{jk}, *b*_{jk} for *j*, *k*≥1 with min{*j*, *k*}≤*N* are the coagulation and fragmentation coefficients, respectively, which are symmetric in *j*, *k*. As can be seen, this system is a particular case of the coagulation–fragmentation equations when *a*_{jk}=*b*_{jk}=0, if min{*j*,*k*}>*N*.

The study of the long-time behaviour of solutions to these equations is expected to be a model of physical processes such as phase transition. Call , the *density* of a solution {*c*_{j}}_{j≥1}. For the Becker–Döring equations, it was proved in Ball *et al*. (1986) and Ball & Carr (1988) that, under certain general conditions which include a detailed balance (see below), there is a critical density *ρ*_{s}∈[0,∞] such that any solution that initially has density *ρ*_{0}≤*ρ*_{s} (*ρ*_{0}<∞, if *ρ*_{s}=∞) will converge for large times, in a certain strong sense, to an equilibrium solution with density *ρ*_{0}, while any solution with density above *ρ*_{s} will converge (in a weak sense) to the only equilibrium with density *ρ*_{s}. The rate of convergence to equilibrium was studied in Jabin & Niethammer (2003). The mentioned weak convergence can then be interpreted as a phase transition in the physical process modelled by the equation (see below for a precise statement). It is an interesting problem to extend this result to more general models; this has been done for the generalized Becker–Döring equations in Carr & da Costa (1994) under some conditions on the decay of the initial data and in da Costa (1998) for suitably small initial data. The aim of this paper is to prove that this result about the generalized Becker–Döring system is true for general initial data. The corresponding result is expected to hold for the full coagulation–fragmentation equations, but finding a proof of this is still an open problem.

## 2. Statement of the main result

Let us recall some usual definitions and notation from previous works on the coagulation–fragmentation equations. We will make use of the vector spacewith norm

The space *X* is clearly a Banach space (actually, this space is isometric to the space of absolutely summable sequences under the map {*c*_{j}}↦{*jc*_{j}}). In this, we will make use of the notion of convergence associated to the norm ‖.‖, which we will call ‘strong convergence’ following common usage. We will also say that a sequence {*c*^{i}}_{i≥1} of elements of *X* converges weak-* to an element *c*∈*X*, and will denote it by , if

there exists

*M*≥0 such that ‖*c*^{i}‖≤*M*, for all*i*≥1 andwhen

*i*→∞, for all*j*≥1 (where and*c*={*c*_{j}}_{j≥1}).

This is just the usual weak-* convergence in the space *X* when it is regarded as the dual space of the space of sequences {*c*_{k}}_{k≥1} such that lim_{k→∞} *k*^{−1}*c*_{k}=0, with norm given by ‖{*c*_{k}}‖≔max{*k*^{−1}|*c*_{k}| |*k*≥1} (see Ball *et al*. 1986, p. 672). We also cite a result from Ball *et al*. (1986).

(Ball *et al*. (1986), lemma 3.3). *If {c ^{n}} is a sequence in X such that*

*and ‖c*.

^{n}‖→‖c‖, then c^{n}→c strongly in XThe subset of *X* formed by the sequences of non-negative terms will be referred to as *X*^{+}:

We will ask for any solution {*c*_{j}(*t*)}_{j≥1} to be, for each fixed time *t*, in *X*^{+}; this is natural, given that densities should be positive and that the sum represents the total density of particles at time *t* (or total mass, depending on the interpretation given to the *c*_{j}'s). More precisely, we will use the following concept of solution from Ball & Carr (1990), section 2.

A solution on the interval [0,*T*[ (*for a given T*>0 or *T*=∞) of equation (1.1) is a function *c*:[0,*T*[→*X*^{+} such that, if we put *c*(*t*)={*c*_{j}(*t*)}_{j≥1} for *t*∈[0,*T*[:

is absolutely continuous for all

*j*≥1 and ‖*c*(*t*)‖ is bounded on [0,*T*[,for all

*j*=1, 2, …, the sums and are finite for almost all*t*∈[0,*T*[,and equations (1.1) hold for almost all

*t*∈[0,*T*[.

For convenience, this definition has been slightly changed with respect to that in Ball & Carr (1990): it has been stated for the generalized Becker–Döring system instead of the full coagulation equations, and conditions have been phrased in different terms, but it can easily be checked that if the coefficients *a*_{jk}, *b*_{jk} satisfy hypothesis 2.6 below then this concept of solution is equivalent to that in Ball & Carr (1990). Hence, results from Ball & Carr (1990) are also applicable in our case, a fact that we will use later.

As we do not know of a uniqueness result that can be applied under the above hypotheses, we need to define a concept of admissibility to precise which solutions our result applies to. In Carr & da Costa (1994), this is done by choosing solutions which are limits of solutions to the finite set of equations obtained by truncating system (1.1). We will call these solutions *Carr–da Costa admissible*. Here, we will define a slight modification of this concept: an admissible solution will be one which is the limit of Carr–da Costa admissible solutions with truncated initial data. The concept must, of course, be the same under any set of conditions that ensure uniqueness, but we have not found a sufficiently general uniqueness result and thus the following will be needed.

Take *T*>0 or *T*=+∞. An admissible solution of the generalized Becker–Döring equations (1.1) on [0,*T*[ with initial data is a solution *c* which is a limit in of Carr–da Costa admissible solutions of equation (1.1) with truncated initial data *c*^{0,n} given by

The above convergence is uniform in compact subsets of [0,*T*[, in the sense of the norm ‖.‖ in *X*; in particular, the functions in the definition converge uniformly when *n*→∞ in compact subsets of [0,*T*[ to *c*_{j}.

Below, we state the conditions on the coefficients under which we will prove our result. Though in the equations only the coefficients *a*_{jk}, *b*_{jk} with min{*j*,*k*}≤*N* appear, for convenience we will use coefficients *a*_{jk}, *b*_{jk} defined for all *j*, *k*≥1 and simply set *a*_{jk}=*b*_{jk}=0, if min{*j*,*k*}>*N*. Thus, we have hypothesis 2.6.

There exists an *N*≥2 such that *a*_{jk}=*b*_{jk}=0, if min{*j*,*k*}>*N* and *a*_{jk}, *b*_{jk}>0, otherwise.

Detailed balance is a physical assumption also used, for example, in Ball *et al*. (1986) and Carr & da Costa (1994), which expresses the principle of microscopic reversibility from chemical kinetics; essentially, it states that equilibria of a certain form exist (theorem 2.19).

There exists a positive sequence {*Q*_{j}}_{j≥1} with *Q*_{1}=1 such that for all *j*, *k*≥1:(2.1)

A certain bound on the growth rate of coefficients is known to be necessary to ensure the existence of density-conserving solutions (Ball & Carr 1990; Escobedo *et al*. 2003) (in other situations density is only conserved for a finite time after which density decreases, a phenomenon known as gelation); for our main result to be true (theorem 2.20), it is evidently necessary that density is conserved, so we impose a condition ensuring this.

For some constants *K*>0 and 0≤*α*<1:

In hypothesis 2.9, (2.2) is a physical condition that asserts that any cluster has a lower free energy than its pieces taken separately (Carr & da Costa 1994, remark 5.1); (2.3) will be seen to imply the existence of a critical density *ρ*_{s} (the relationship between the following *z*_{s} and this critical density is given below in (3.4)).

The sequence *Q*_{j} satisfies(2.2)(2.3)

This implies that for *m*≥1 and that .

We also need to assume, as new hypotheses, a certain regularity of the coefficients.

For *j*, *m*=1, …, *N*:

For some constant *K*_{a}, *j*, *m*=1, …, *N* and *k*≥1:

Observe that hypotheses 2.11 and 2.12 are independent; for example, for *j*=1, …, *N* and *k*≥1, *a*_{jk}=exp(−*j*−*k*) satisfies the second one but not the first; and (with [*x*] being the integer part of *x*) satisfies the first, but not the second.

The kind of coefficients allowed by the previous hypotheses are, for example, *a*_{jk}≤*C*(*j*^{α}+*k*^{α}) for *j*=1, …, *N* and *k*≥1, sufficiently regular to fulfil hypotheses 2.11 and 2.12, and *b*_{jk} given by hypothesis 2.7 with any choice of *Q*_{j} satisfying equations (2.2) and (2.3). Note that equation (2.2) implies that *b*_{jk}≤*a*_{jk}, so *b*_{jk}≤*C*(*j*^{α}+*k*^{α}) also. For a concrete example, pick *C*_{1}, *C*_{2}>0 and *α*, *δ*∈[0,1[ and define the following coefficients for min{*j*,*k*}≤*N*:

The coefficients are taken to be zero when min{*j*,*k*}>*N*. These correspond to *Q*_{j}=exp(*C*_{2}(*j*−*j*^{δ})) and have .

We borrow known existence results for the kind of admissible solutions of definition 3 from Ball & Carr (1990).

(Ball & Carr (1990), theorems 2.4, 3.6 and 5.4). *Assume* *hypotheses 2.6 and 2.8**, and take c ^{0}∈X^{+}. Then there exists an admissible solution c to equation*

*(1.1)*

*on*[0,+∞[

*with c*(0)

*=c*

^{0}

*. Furthermore, under*

*hypothesis 2.6*

*all solutions to equation*

*(1.1)*

*are density-conserving*.

Theorem 2.4 in Ball & Carr (1990) gives the existence of a solution (in fact, a Carr–da Costa admissible solution by the method of construction). Theorem 3.6 from Ball & Carr (1990) proves this solution conserves density. Finally, theorem 5.4 in the same paper gives the existence of a solution that can be obtained as the uniform limit in compact sets of [0,*T*[ of Carr–da Costa admissible solutions with truncated initial data, thus giving the existence of an admissible solution in the sense used here.

*Assume* *hypotheses 2.6 and 2.8*. *Take μ>*1 *and suppose that c={c _{j}}_{j≥1} is an admissible solution to equation*

*(1.1)*

*on*[0,

*T*[

*for some T>*0

*with initial data c*(0)

*=c*

^{0}

*such that*.

*Then*

*is finite for all 0≤t<T*.

This is just theorem 3.3 in Carr & da Costa (1994), stated for admissible solutions in the sense we use here. As Carr–da Costa admissible solutions satisfy the estimate given in the proof of the above theorem in Carr & da Costa (1994) (which depends only on ), we can pass to the limit and thus prove that our admissible solutions also satisfy it. ▪

Hypotheses 2.6–2.9 imply those of theorems 5.1 and 5.2 in Carr & da Costa (1994): (1.7) and (H2) in Carr & da Costa (1994) are always fulfilled if we assume hypothesis 2.6; (H1) is our hypothesis 2.8 and (H3), (H4) from Carr & da Costa (1994) are contained in hypotheses 2.6 and 2.9 here, respectively. This enables us to use these theorems here (recall remark 2.3); we will need the following one about the equilibrium solutions of (1.1).

An equilibrium of (1.1) is a solution of (1.1) that does not depend on time. The density of an equilibrium *c* is the norm of *c* in *X*, .

The critical density *ρ*_{s} is defined to be:

(Carr & da Costa (1994), theorem 5.2). *Assume* *hypotheses 2.6–2.9*:

*for 0≤ρ≤ρ*_{s}(and also ρ<+∞, if ρ_{s}=+∞), there exists exactly one equilibrium*of**(1.1)**with density ρ, which is given by**where z is the only positive number such that*.*For ρ*_{s}<ρ<+∞ there is no equilibrium of*(1.1)**with density ρ*.

Observe that when *ρ*_{s} is finite and represents the critical equilibrium (the one with density *ρ*_{s}), *z*_{s} is the single particle density of this equilibrium.

The main result in this paper is the following.

*Assume* *hypotheses 2.6–2.12**, and let c={c _{j}}*

_{j≥1}

*be an admissible solution of the generalized Becker–Döring equation*

*(1.1)*

*(whose existence is given by*

*theorem 2.14*).

*Call*

*, the initial density:*

*if*0*≤ρ*_{0}*≤ρ*_{s}, then c converges strongly in X to the equilibrium with density ρ_{0},*if ρ*_{s}<ρ_{0}*, then c converges in the weak-* topology to the equilibrium with density ρ*._{s}

## 3. Proofs

The following result from Carr & da Costa (1994) already gives part of theorem 2.20. Again, note that hypotheses in Carr & da Costa (1994) are contained in those here.

(Carr & da Costa (1994), theorem 6.1). *Assume* hypotheses 2.6–2.9. *Let c={c _{j}} be a solution of*

*(1.1)*

*on*[0,

*∞*]

*, and call*.

*Then there exists* 0*≤ρ≤*min*{ρ _{0},ρ_{s}} such that*

*, where c*

^{ρ}is the only equilibrium of*(1.1)*

*with density ρ (given by*

*theorem 2.19*

*)*.

With theorem 3.1, the next result will be enough to complete the proof of theorem 2.20.

*Assume* hypotheses 2.6–2.12 *hold. Suppose that c is an admissible solution to the generalized Becker–Döring equations* *(1.1)* *with initial data c*_{0}*∈X _{+} such that c converges weak-* to an equilibrium with density ρ<ρ_{s}. Then, c converges strongly to this equilibrium (and in particular, ρ is the density of the solution c, i.e. ρ=ρ*

_{0}

*)*.

Hence, the aim of the rest of this section will be to prove theorem 3.2. The following key result gives a bound on the solutions that will easily imply the precompactness of the orbits, which in turn implies theorem 2.22. Call, for *i*≥1:

*Let c={c _{j}}*

_{j≥1}

*be an admissible solution of the generalized Becker–Döring equations*

*(1.1)*.

*Assume*hypotheses 2.6–2.12.

*Suppose that for some z<z _{s}:*

*Suppose that {r _{i}}_{i≥1} is a strictly decreasing sequence of positive numbers that satisfy, for some λ with 1<λ<z_{s}/z*

*and such that G*.

_{i}(0)≤r_{i}for all i*Then, there exist a positive integer k _{0} and a constant C>0 such that G_{i}(t)≤Cr_{i} for all i≥k_{0} and all positive times*.

The proof of proposition 3.3, which contains the core of the argument, is a generalization of a method used in unpublished notes by Laurençot Ph. & Mischler, S. This method is inspired by the proof of uniqueness of solutions to the Becker–Döring equation in Laurençot & Mischler (2002*c*). The use of this kind of argument can be traced back to Ball & Carr (1988).

Note that the condition on {*r*_{k}} in proposition 3.3 is not very stringent as the following lemma states.

*Given λ>1 and a positive sequence {g _{k}}_{k≥1}, which tends to zero as k tends to infinity, there exists a strictly decreasing positive sequence {r_{k}}_{k≥1} which converges to zero, such that g_{k}≤r_{k} and*

Define

Then is decreasing, tends to zero and for all *k* we have . Define *s*_{k} recursively as

Then *s*_{k}>0, for all *k* (it is to ensure this that we added 1 to ) and we can see that converges. For this, note *s*_{k+1}≤(*s*_{k}/*λ*)+*h*_{k+1} and write for *m*≥2so we have thatwhich proves the summability of {*s*_{k}}, since *λ*>1. (I thank the referees for suggesting a simpler version of this proof.)

Clearly, *s*_{k}≥*h*_{k}. Let us finally definewhich is positive, greater than *g*_{k}, strictly decreasing, tends to zero as *k*→∞ and ▪

### (a) Proof of the proposition

We will prove the proposition for solutions whose initial data is a truncation at a sufficiently large finite size of {*c*_{i}(0)}_{i≥1}, with constants *C* and *k*_{0} that do not depend on the size of this truncation; then the proposition follows for general initial data by a standard approximation argument using definition 2.4 of an admissible solution.

Take an *L*≥1 and consider a solution with initial data , for *i*=1, …, *L* and , for *i*>*L*. It is again enough to prove the bound in the result up to a finite time *T*>0, with a constant that does not depend on *T*. So fix *T*>0, and let us find *C* and *k*_{0} (independent of *L* and *T*) such thatBy the admissibility of *c* we know that the functions converge uniformly in [0,*T*] to *c*_{j} as *L*→∞ (see remark 2.5), so the hypotheses of the proposition imply that for sufficiently large *L*

In the following, *L* will always be large enough for this to hold (note that the choice of *L* depends also on *T*).

Furthermore, , for all *i* (where we have denoted , the corresponding to *G*_{i} for the solution ).

From now, to simplify the notation a bit, we will omit the *L* in both and , as the full *c*_{j} and *G*_{j} will not be mentioned anymore. *W*_{jk} will be used to denote .

For any sequence {*Ψ*_{j}}_{j≥1}, it holds formally that

In particular, we can apply the previous relation to *Ψ*_{j}=*j*.1_{j≥i} (*i*≥1) to get(3.1)and this equality is rigorously justified because the solution {*c*_{i}} has finite moments of every order for every positive time *t* (see lemma 2.16), so the sums on both sides of the previous equality converge uniformly and we can obtain the equation by means of standard results on differentiation of uniformly convergent series of functions. One way to obtain the expression on the right hand side is to write the sum over *j*, *k* as a sum over the regions depicted in figure 1, where the value of *Ψ*_{j+k}−*Ψ*_{j}−*Ψ*_{k} is indicated in each of them.

Due to hypothesis 2.6, *W*_{jk}=0, if min{*j*,*k*}>*N*. Hence, for *i*>2*N*, the first sum in (3.1) (which comprises all pairs *j*,*k*<*i* such that *j*+*k*≥*i*) can be broken into those terms where *j*≤*N* and those where *k*≤*N*,where we have changed the order of the double sum and used the symmetry of (*j*+*k*)*W*_{jk}. If *i*>*N*, the second sum in (3.1) is non-zero only if *j*≤*N*, so for *i*>2*N*, we have(3.2)

We rewrite the latter double sum:(3.3)where we have denoted the two double sums as *S*_{1}, *S*_{2} to mention them later.

We know that

Hence, as *z*<*z*_{s}, we see that thanks to hypothesis 2.7 and for *j*∈{1, …, *N*}:Note that the term in parenthesis tends to as *k*→∞ (thanks to hypotheses 2.9 and 2.11), so *S*_{2}≤0, for *t*∈[0,*T*] and *i* sufficiently large. Then, continuing from (3.2), using (3.3) and omitting *S*_{2}, we have for *i* large that(3.4)

Using again *c*_{j}≤*z*_{j}≔*z*^{j}*Q*_{j}, for *j*=1, …, *N* and *c*_{k}=(1/*k*)(*G*_{k} − *G*_{k+1}), for all *k*, rewrite (3.4) as(3.5)where

Now we take any (recall *λ* appears in the condition on *r*_{i}), and note that the following holds for *k* large enough:(3.6)

The proof of this is easy, as (note that we can divide by *a*_{jk} by hypothesis 2.6)(3.7)where the detailed balance hypothesis 2.7 has been used to pass to the second line. Now observe that the term with the negative sign converges to zero, thanks to hypothesis 2.11, and that the other term(3.8)because of hypothesis 2.9. Hence, we have (3.6).

So, thanks to (3.6), we can continue from (3.5) and get, for *i* large enough,(3.9)

It is easy to see from hypothesis 2.11 that for *j*, *m*=1, …, *N*

This means that for small variations of *k*, *A*_{jk} changes little when *k* is large. Take *ϵ* such that(3.10)

We can then find an *i*_{0}>2*N* such that (3.9) holds for *i*>*i*_{0} and we have, also for *i*≥*i*_{0},(3.11)

So for *i*≥*i*_{0}, we can write from (3.9)(3.12)

From the hypothesis on *r*_{i}, for *j*=1, …, *N* and *i*>*j*

Apply this *j* times to get(3.13)where we used (3.10) together with to say that

If the sequence {*r*_{i}} satisfies (3.13), then {*Cr*_{i}} also satisfies it, for any positive *C*. Take *C*>1 sufficiently large so that(3.14)where by *M*_{0}, we mean the density of the full initial data with no truncation. Now define(3.15)

(3.16)We know *H*_{i}(*t*)=0, for *i*<*i*_{0} and *t*<*T* because of (3.14). As the *Cr*_{i} satisfy (3.13) we can write, continuing from (3.12), for *i*≥*i*_{0}(3.17)

Then, the same inequality holds for *H*_{i}: note that most of the previous reorganization was done in order to have the term in *M*_{i} as *the only term with negative sign* in (3.17). Otherwise, we cannot justify writing the inequality in terms of *H*_{i} as is done next:(3.18)(We have used and for any *i*, *k*.) Now, we can sum this from *i*=*i*_{0} to infinity (note again that the sums are all convergent, as the solution {*c*_{j}} with truncated initial data has finite moments of all orders) and reorganize the terms:(3.19)where the *T*_{i} (*i*=1–4) are the sums above. Observe that *T*_{4} is negative and *T*_{1} only contains terms in *H*_{i} for *i*<*i*_{0}, so it is directly zero (recall (3.14)). Also, note that for *j*=1, …, *N* and *i*≥*i*_{0} we have, by using hypothesis 2.12, thatwhich is easily seen to be bounded by a certain constant *A*′, for *j*=1, …, *N* and *i*>*N*. Hence, the coefficient of *H*_{i} in *T*_{2} and *T*_{3} is bounded by a certain constant *A* independent of *j* and *i* and then

Gronwall's lemma then shows that *H*_{i}(*t*)=0, for *i*≥*i*_{0} and *t*∈[0,*T*]; that is to say *G*_{i}(*t*)≤*Cr*_{i}. This proves our claim.

### (b) Proof of the main theorem

Finally, we arrive at the proof of theorems 3.2 and 2.20, which is not difficult once the proposition of the previous section has been established.

Let *c* be an admissible solution that converges weak-* in *X* to an equilibrium of mass *ρ*<*ρ*_{s}, which must be given by for some (see theorem 2.19). We will prove that the orbit of any such solution must be relatively compact in *X* and hence the convergence must be strong.

Pick . As we know that when *t*→∞, for all *j*, we can find a *t*_{0}>0 so that

As lemma 3.4 ensures, we can always find a sequence {*r*_{i}} tending to zero as *i*→∞ that satisfies the conditions in proposition 3.3 with *G*_{i}(*t*_{0}) instead of *G*_{i}(0). We can apply the proposition, with the *z* we have chosen, to {*c*_{j}(*t*+*t*_{0})}_{j≥1} (which is a translation in time of the solution *c* and thus is a solution itself) and deduce that for some *C*>0, *k*_{0}≥1 and all *t*>*t*_{0}As {*r*_{i}} tends to zero, this bound says that the solution *c* is relatively compact in *X*_{+}, and we have finished. ▪

Suppose that *c* is an admissible solution of (1.1) in [0,+∞[ with initial data *c*(0)=*c*^{0}∈*X*_{+}. Theorem 3.1 shows that *c* converges weak-* in *X* to an equilibrium of mass *ρ* for some 0≤*ρ*≤*ρ*_{0}.

If *ρ*_{0}<*ρ*_{s}, then this convergence is also strong (by theorem 3.2) and hence *ρ*=*ρ*_{0}.

If *ρ*_{0}=*ρ*_{s} (with *ρ*_{s}<∞), then *ρ*≤*ρ*_{s}. But if *ρ*<*ρ*_{s} then again the convergence must be strong and *ρ*=*ρ*_{s}, which is a contradiction. Hence, *ρ*=*ρ*_{0}=*ρ*_{s} and we see the convergence is strong because of lemma 2.1.

Finally, if *ρ*_{0}>*ρ*_{s}, then *ρ*≤*ρ*_{s} and it must be *ρ*=*ρ*_{s} or otherwise the convergence is strong, which is not possible given that *ρ*_{0}>*ρ*. ▪

## Acknowledgements

These results were obtained under the supervision and help of Stéphane Mischler. I wish to thank him for his explanations and suggestions. I would also like to thank the anonymous referees for their valuable corrections and comments. The author was supported by FPU grant AP2001-3940 and by EU financed network no. HPRN-CT-2002-00282.

## Footnotes

- Received December 9, 2004.
- Accepted May 31, 2005.

- © 2005 The Royal Society