Philippe Chassaing
email: chassain@iecn.u-nancy.fr

Jean Mairesse
email: mairesse@liafa.jussieu.fr.

A NON-ERGODIC PROBABILISTIC CELLULAR AUTOMATON WITH A UNIQUE INVARIANT MEASURE

Keywords: July 8, 2011. 2000 Mathematics Subject Classification. Primary: 60K35, 60J05, Secondary: 37B15, 68Q80 Probabilistic cellular automaton, interacting particle system, ergodicity

We exhibit a Probabilistic Cellular Automaton (PCA) on {0, 1} Z with a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.

Introduction

Consider a random process on Σ Z d , where Σ is a finite set, with local interactions and a translation invariant dynamic. There are two natural instanciations, one with asynchronous updates of the sites of Z d , and one with synchronous updates. In the first case, the model is a continuous time Markov process, known as a (finite range) Interacting Particle System (IPS). In the second case, the model is a discrete time Markov chain known as a Probabilistic Cellular Automaton (PCA).

The relevance of IPS in statistical mechanics, as well as in many other contexts, is well established. Let us mention a couple of motivations for studying PCA. First, the investigation of fault-tolerant computational models was the motivation for the Russian school [START_REF] Toom | Discrete local Markov systems[END_REF][START_REF] Gács | Reliable cellular automata with self-organization[END_REF]. Second, PCA appear in combinatorial problems related to the enumeration of directed animals [START_REF] Borgne | Directed animals and gas models revisited[END_REF]. Third, in the context of the classification of (deterministic) cellular automata (Wolfram's program), robustness to random errors can be used as a discriminating criterion [START_REF] Fatès | Fully asynchronous behavior of double-quiescent elementary cellular automata[END_REF].

For IPS and PCA, the first question is to study the equilibrium behavior. An equilibrium is characterized by an invariant measure, that is a probability measure on the state space which is left invariant by the dynamic. An invariant measure µ is attractive if, for any initial condition, the state of the system converges (weakly) to µ as time goes on.

By a compactness argument, there always exists at least one invariant measure. Therefore, there are, a priori, three possible situations:

(1) several invariant measures;

(2) a unique invariant measure which is not attractive;

(3) a unique invariant measure which is attractive. In the last case, which corresponds to the nicest possible situation, the model is said to be ergodic. Roughly, an ergodic system completely forgets about its initial condition, while a non-ergodic one remembers something forever.

A classical foundational question is whether the intermediate case exists. In other words, does uniqueness of the invariant measure imply convergence to it ? For monotone systems, the intermediate case does not exist. But in general, the question is open.

For IPS, this question is Open Problem 4 in Chapter 1 of the classical textbook by Liggett [START_REF] Liggett | Interacting Particle Systems[END_REF]. In [START_REF] Mountford | A coupling of infinite particle systems[END_REF], Mountford proves that the intermediate case does not exist for 1dimensional IPS (that is d = 1). Quoting [START_REF] Mountford | A coupling of infinite particle systems[END_REF], "it seems more than plausible that the conclusion (...) is true in higher dimensions". However, the question remains unsettled. For PCA, the same question is Unsolved problem 3.4.3 in Toom [START_REF] Toom | Cellular automata with errors: problems for students of probability[END_REF], or Unsolved problem 5.7 in Toom [START_REF] Toom | Contours, convex sets, and cellular automata[END_REF].

In the present paper, we answer the question for PCA by exhibiting a 1-dimensional PCA, model A, corresponding to the intermediate case (Theorem 3.1). There is a unique invariant measure of the form (µ 0 + µ 1 )/2 and the PCA maps µ 0 to µ 1 and µ 1 to µ 0 . Starting from an initial measure µ 0 , the probability measure of the state of the system is µ 0 at even times and µ 1 at odd times. Therefore there is no convergence.

Observe that the situations for IPS and PCA are different: in 1-d, the intermediate case exists for PCA, and not for IPS. This is consistent with the situation for Markov processes on a finite state space: in discrete time, periodic phenomena may occur which result in the existence of the intermediate case; in continuous time, the intermediate case does not exist.

To prove the result for model A, we introduce two auxiliary PCA. The first one, model B, corresponds to independently moving particles annihilating when they meet (p + p → ∅). The second one, model C, corresponds to independently moving particles merging when they meet (p + p → p). We compute exactly the evolution of the onedimensional marginals for model C (Theorem 5.2) and models A and B (Prop. 6.1) starting from a "full" configuration. In particular, it proves that the speed of convergence to the invariant measure is of order 1/ √ n for the three models.

Continuous-time versions of models B and C have been studied in the IPS literature under the names of annihilating random walks and coalescing random walks, respectively, see [START_REF] Arratia | Limiting point processes for rescalings of coalescing and annihilating random walks on Z d[END_REF][START_REF] Bramson | Clustering and dispersion rates for some interacting particle systems on Z[END_REF][START_REF] Griffeath | Additive and cancellative interacting particle systems[END_REF]. The PCA and IPS versions of B and C share the same features: ergodicity with the invariant measure being the "all empty" Dirac measure, and with similar and subexponential speed of convergence. In the IPS setting, the asymptotic speed of convergence was given by Bramson & Griffeath [START_REF] Bramson | Clustering and dispersion rates for some interacting particle systems on Z[END_REF] for model C, and by Arratia [START_REF] Arratia | Limiting point processes for rescalings of coalescing and annihilating random walks on Z d[END_REF] for model B. Also, the coupling between the models B and C, that we use in Section 6, already appears in Griffeath [START_REF] Griffeath | Additive and cancellative interacting particle systems[END_REF]Ch. 3,Sec. 5] and in Arratia [START_REF] Arratia | Limiting point processes for rescalings of coalescing and annihilating random walks on Z d[END_REF] in the continuous-time setting. The novelty is that we get exact computations for the PCA models, as opposed to asymptotic results for the IPS ones. At last, let us mention that IPS versions of models B and C on a finite set of sites have also been studied, see for instance [START_REF] Ayyer | Exact results for an asymmetric annihilation process with open boundaries[END_REF] for B, [START_REF] Momčilović | On throughput in stochastic linear loss networks[END_REF] for C, and the references therein.

Probabilistic Cellular Automaton

Let Σ be a finite set. Denote by M(Σ) the set of probability measures on Σ. Let us equip X = Σ Z with the product topology. Denote by M(X) the set of probability measures on X for the Borelian σ-algebra. Weak convergence of (µ n ) n to µ is denoted by µ n w -→ µ. Let K be a finite subset of Z and consider x ∈ Σ K . The cylinder defined by x is the set *

x * = u ∈ Σ Z , ∀k ∈ K, u k = x k . Given k ∈ Z and V = (v 1 , . . . , v n ) ∈ Z n , we use the notation k + V for (k + v 1 , . . . , k + v n ), and the notation V (K) for {i | ∃k ∈ K, ∃v ∈ V, i = k + v}.
Let us introduce probabilistic cellular automata, restricting ourselves to 1-dimensional models.

Definition 2.1. The alphabet is a finite set Σ; the set of sites is Z. The set of configurations is X = Σ Z . Given V ∈ Z n , a transition function of neighborhood V is a function f : Σ V → M(Σ). The probabilistic cellular automaton (PCA) F of transition function f is the application M(X) → M(X), µ → µF defined on cylinders by: ∀K, ∀y ∈ Σ K ,

µF ( * y * ) = x∈Σ V (K) µ( * x * ) k∈K f ((x i ) i∈k+V )(y k ) .
Let us look at how F acts on a Dirac measure δ x . The value of all the sites are updated. The value x k of the k-th site is changed into the letter a ∈ Σ with probability f ((x i ) i∈k+V )(a), independently of the evolution of the other sites.

By specializing Definition 2.1, we recover two famous models:

• Assume that V = {0}, then all the sites behave independently. The restriction of the PCA to one site is a Markov chain evolving on Σ. Conversely, any Markov chain on a finite state space E can be realized as (a restriction of) a PCA on the alphabet E with neighborhood V = {0}. • Assume that the transition function f is such that: ∀u ∈ Σ V , f (u) is a Dirac probability measure. Then we may view f as a function Σ V → Σ. We obtain a (deterministic) cellular automaton.

A PCA F may be viewed as a Markov chain on the state space Σ Z . Thus we borrow the classical terminology of Markov chains.

Definition 2.2. An invariant (probability) measure of F is a probability measure µ ∈ M(X) such that µF = µ. The PCA F is ergodic if it has a unique invariant measure which is attractive, i.e. if (1) (i) ∃!µ ∈ M(X), µF = µ , (ii) ∀ν ∈ M(X), νF n w -→ µ .
Consider for a moment a Markov chain on a finite state space with transition matrix P . Let G(P ) be the graph of the matrix P . Classically, we have

(i) ⇐⇒ G(P ) has a unique terminal component (2) (i)+(ii) ⇐⇒ G(P ) has a unique terminal component which is aperiodic .
In particular, uniqueness of the invariant measure does not imply ergodicity. The simplest example of a non-ergodic Markov chain with a unique invariant measure is the following: the state space is X = {0, 1} and the transition matrix is

(3) P = 0 1 1 0 .
The unique invariant measure is µ = (δ 0 + δ 1 )/2 and for ν = δ 0 , we do not have νP n w -→ µ.

For PCA, it was an open question to know if (i) implies (ii) in [START_REF] Arratia | Limiting point processes for rescalings of coalescing and annihilating random walks on Z d[END_REF]. The purpose of the present paper is to settle the question by proposing a non-ergodic PCA with a unique invariant measure.

To get a hint of the difficulty, consider for instance a PCA F with neighborhood V = {0}. Recall that each site behaves independently and as a finite Markov chain P . As recalled in (2), P may satisfy either [¬(i)], [(i), ¬(ii)], or [(i), (ii)]. We show in Table 1 how this gets reflected on the PCA F .

Markov chain

P PCA F ¬(i) ¬(i) (i), ¬(ii) ¬(i) (i), (ii) (i), (ii)
Table 1. Finite Markov chain versus "neighborhood 0 PCA".

Let us justify the Table . If µ is an invariant measure of P , then the product measure µ ⊗Z is an invariant measure of F . Therefore, if P has several invariant measures, the same holds for F . Assume now that P is ergodic with unique invariant measure µ. One proves easily that µ ⊗Z is attractive, so F is ergodic. Let us concentrate now on the intermediate case for P . If P satisfies [(i), ¬(ii)] then G(P ) has a unique terminal component which is periodic, say of period 2. Let (µ 0 + µ 1 )/2 be the unique invariant measure of P . Then F has an infinite number of invariant measures. Indeed, consider any (u i ) i∈Z ∈ {0, 1} Z , and let (v i ) i∈Z be defined by v i = 1 -u i for all i. Then the probability measure (⊗ i∈Z µ u i + ⊗ i∈Z µ v i )/2 is clearly an invariant measure of F .

a(00)(1) = 1/2, a(01)(1) = 0, a(10)(1) = 1, a(11)(1) = 1/2 .
A realization of the Markov chain is obtained as follows. Consider the function

A : {0, 1} Z × U Z → {0, 1} Z (4) (x i ) i∈Z , (u i ) i∈Z → (x i ) i∈Z , 0 1 1 0 1 0 1 0 0 1 1 proba 1/2 0 proba 1/2 0 proba 1/2 1 proba 1/2
= 0 if x i-1 x i = 01 or (x i-1 x i , u i ) ∈ (00, →), (11, ↑) 1 if x i-1 x i = 10 or (x i-1 x i , u i ) ∈ (00, ↑), (11, →) . Let U = (U i,j ) (i,j
)∈Z×N be a doubly-indexed sequence of i.i.d. r.v.'s with common law

P (U i,j =↑) = P (U i,j =→) = 1/2, called the update process. Set U n = (U i,n ) i∈Z . Given a {0, 1} Z -valued r.v. X 0 = (X i,0 ) i∈Z , such that U ⊥ X 0 , define the sequence of {0, 1} Z -valued r.v.'s (X n ) n∈N as follows: X n+1 = A(X n , U n ). (5)
Then (X n ) n∈N is a realization of model A. The process U is used to randomly update the value of a site, when needed, with → being interpreted as "keep" and ↑ as "switch", and X i,n is the state of site i at time n, so that X n = (X i,n ) i∈Z denotes the state of the system at time n.

3.2. Invariant measure. Let x = (01) Z be the configuration defined by: ∀n ∈ Z, x 2n = 0, x 2n+1 = 1. The configuration (10) Z is defined similarly.

Theorem 3.1. The PCA F A has a unique invariant measure which is µ = (δ (01) Z + δ (10) Z )/2. The PCA is non-ergodic.

On configurations without 00 and 11, the PCA acts as the translation shift. Therefore µ = (δ (01) Z + δ (10) Z )/2 is an invariant measure. Assume that it is the unique one. Then µ is non-attractive, the situation being the same as for (3): consider ν = δ (01) Z , then νF n A = δ (01) Z if n is even, and νF n A = δ (10) Z if n is odd. The purpose of Sections 4 and 5 is to prove Theorem 3.1.

Two auxiliary models

We now define two new PCA, that we call respectively model B and model C. For both models, the alphabet is Σ = {•, •} and the set of sites is Z. Given a configuration u ∈ {•, •} Z , the following interpretation holds: if u i = •, the site i is "empty"; if u i = •, the site i contains a "particle". At a given time step, a particle decides (independently of the others and independently of the past) to remain at its site with probability 1/2, or to jump to the site on the right with probability 1/2. In model B, if two particles collide, then they annihilate. In model C, if two particles collide, they are merged into one particle. Let us define the models more formally.

B : {•, •} Z × U Z → {•, •} Z (6) (y i ) i∈Z , (u i ) i∈Z → (ỹ i ) i∈Z , with U = {↑, →}, and ỹi = • if (y i-1 y i , u i-1 u i ) ∈ {(••, → U ), (••, U ↑), (••, ↑↑), (••, →→)} • otherwise .
Let U be an update process, defined as in Section 3.

1. Given a {•, •} Z -valued r.v. Y 0 , such that U ⊥ Y 0 , define the sequence of {•, •} Z -valued r.v.'s (Y n ) n∈N as follows: Y n+1 = B(Y n , U n ). (7) Then (Y n ) n∈N is a realization of model B. proba 1/4 proba 1/4 proba 1/4 proba 1/4
Step n

Step n+1 Remarks. In the above presentation, model B is a Markov chain with synchronous updates and local interactions, but not stricto sensu a PCA. Indeed, if Y 0 is deterministic, then the r.v.'s Y i,1 and Y i+1,1 are not independent, since they are updated using the non-disjoint r.v.'s {U i-1,0 , U i,0 } and {U i,0 , U i+1,0 }. However, it is possible to give a PCA presentation of model B on a larger alphabet. Define the sequence of {

•, •}×U Z -valued r.v's ( Y n ) n∈N by Y n = (Y n , U n ).
We have: Step n

( Y n+1 ) i = B( Y n ) i , U i,n+1 .
Thus ( Y n ) n is
Step n+1

C : {•, •} Z × U Z → {•, •} Z (z i ) i∈Z , (u i ) i∈Z → (z i ) i∈Z , with zi = • if (z i-1 z i , u i-1 u i ) ∈ {(••, → U ), (••, U ↑), (••, ↑↑), (••, → U )} • otherwise .
Let U be an update process. Given a {•, •} Z -valued r.v. Z 0 , such that U ⊥ Z 0 , define the sequence of {•, •} Z -valued r.v's (Z n ) n∈N as follows:

Z n+1 = C(Z n , U n ) .
Then (Z n ) n∈N is a realization of model C.

Again, the continuous time version of model C, with exponential holding times, is called a coalescing random walk (cf. [6, Ch. 2, Sec. 9]). 4.3. Links between models A, B, and C. One-step transition of the model B, resp. C, defines the mapping

F B : M({•, •} Z ) -→ M({•, •} Z ) µ -→ µF B ,
respectively,

F C : M({•, •} Z ) -→ M({•, •} Z ) µ -→ µF C . Define ϕ : {0, 1} Z -→ {•, •} Z (x i ) i∈Z -→ (y i ) i∈Z ,
with

y i = • if x i x i+1 ∈ {00, 11} • if x i x i+1 ∈ {01, 10}
By extension, define ϕ :

M({0, 1} Z ) → M({•, •} Z ).
Lemma 4.1. The diagram below is commutative:

M({0, 1} Z ) ϕ F A / / M({0, 1} Z ) ϕ M({•, •} Z ) F B / / M({•, •} Z ) If (X n ) n∈N is a realization of the Markov chain A, then (ϕ(X n )) n∈N is a realization of the Markov chain B.
Proof. Recall that A and B are defined in ( 4) and ( 6) respectively. We are going to prove

(8) ϕ • A = B • (ϕ, Id) .
The statement of the lemma follows. Set

(x i ) i , (u i ) i A -→ (x i ) i ϕ -→ (ỹ i ) i , (x i ) i , (u i ) i ϕ,Id -→ (y i ) i , (u i ) i B -→ (ŷ i ) i .
To obtain [START_REF] Liggett | Interacting Particle Systems[END_REF], it is enough to check that ỹ0 = ŷ0 . This is done by systematic inspection in Table 2. Each one of the 32 cases mimicks the commutative diagram: in the first line, from left to right, (x -2 , x -1 , x 0 ), (u -1 , u 0 ), and (x -1 , x0 ); in the second line, from left to right, (y -1 , y 0 ), (u -1 , u 0 ), and ỹ0 = ŷ0 . If the process X is defined by [START_REF] Gács | Reliable cellular automata with self-organization[END_REF], relation [START_REF] Liggett | Interacting Particle Systems[END_REF] entails that the process Y , defined by Y n = ϕ(X n ), satisfies relation [START_REF] Borgne | Directed animals and gas models revisited[END_REF].

Lemma 4.2. Model B is dominated by model

C: for x, u ∈ {•, •} Z × U Z , B(x, u) ≤ C(x, u) ,
where ≤ is the coordinate-wise product ordering on {•, •} Z , with • ≤ •.

Proof. This can be checked directly on the definitions of B and C. Intuitively, particles are merged in model C, and annihilate in model B.

Lemma 4.3. The following implications hold:

[C is ergodic with invariant measure δ • Z ] =⇒ [B is ergodic with invariant measure δ • Z ] ⇐⇒ [A is non-ergodic with invariant measure (δ (01) Z + δ (10) Z )/2] .
Proof. This is a direct consequence of Lemmas 4.1 and 4.2.

Therefore, in order to prove Theorem 3.1, it is sufficient to prove that model C is ergodic with invariant measure δ • Z . This is the purpose of next section.

111 or 000 →→ 11 or 00 101 or 010 →→ 10 or 01

•• ↑↑ • •• ↑↑ • Table 2. The 32 possible cases. 5. Model C is ergodic Lemma 5.1. Model C is monotone, that is: for z ∈ {•, •} Z , z ∈ {•, •} Z , u ∈ U Z , z ≤ z =⇒ C(z, u) ≤ C(z, u) ,
where ≤ is the coordinate-wise product ordering.

Proof. It can be checked directly on the definition of C.

With this monotonicity, to get the ergodicity, it is enough to prove that δ • Z F n C → δ • Z . Indeed, consider two realizations of model C, one, say Z = (Z n ) n , that starts with all sites occupied, the other, say Z = ( Zn ) n , that starts with an arbitrary initial condition, their evolution using the same update process U . According to Lemma 5.1, at any time n ∈ N, Zn ≤ Z n .

From now on, we focus on the process Z. Recall that for each n, Z n = (Z k,n ) k∈Z is the state of the system at time n. The process Z n is stationary, i.e. invariant by translation, since Z 0 , U , and C are invariant too. Define ( 9)

d n = P (Z k,n = •) = P (Z 0,n = •) .
This is the density of particles at time n. The density d n can also be viewed as an evaluation of the distance between Z n and δ • Z . Indeed, for any finite subset E of Z, consider the Hamming distance on {•, •} E , and denote by W H the corresponding Wasserstein distance on

M({•, •} E ). Setting Z E,n = (Z k,n ) k∈E , we have: W H (Z E,n , δ • E ) = |E| d n .
Theorem 5.2. Let T be the time that a simple symmetric random walk on Z needs to reach 2, starting from 0. We have

d n = P (T > 2n) (10) = 4 -n 2n + 1 n . ( 11
)
In particular, d n ∼ 2/ √ π n, hence converges to 0 as n grows.

In continuous time, when the particles perform a simple symmetric random walk, Bramson & Griffeath [START_REF] Bramson | Clustering and dispersion rates for some interacting particle systems on Z[END_REF] obtain the same asymptotic behavior for d n , up to a scaling factor, as expected.

Corollary 5.3. Model C is ergodic with unique invariant measure δ • Z .
Proof. We first prove [START_REF] Toom | Cellular automata with errors: problems for students of probability[END_REF], assuming [START_REF] Mountford | A coupling of infinite particle systems[END_REF]. Let S = (S k ) k∈N be a realization of the simple symmetric random walk on Z, starting from 0. Define M k = max{S i , 0 ≤ i ≤ k}, the maximum of the random walk at time k. Recall that T = inf{i ≥ 0 | S i = 2}. We have

P (T > 2n) = P (M 2n ≤ 1) = 1 -P (M 2n ≥ 2) = 1 - ℓ∈Z P (M 2n ≥ 2, S 2n = ℓ) = 1 -P (S 2n ≥ 2) - ℓ≤1 P (M 2n ≥ 2, S 2n = ℓ)
According to the reflection principle, for ℓ ≤ 1, P (M 2n ≥ 2, S 2n = ℓ) = P (S 2n = 4 -ℓ). Therefore,

P (T > 2n) = 1 -P (S 2n ≥ 2) -P (S 2n ≥ 3) = 1 -P (S 2n ≤ -2) -P (S 2n ≥ 3) = P (S 2n ∈ {0, 2}) = 4 -n 2n n + 2n n + 1 = 4 -n 2n + 1 n .
Using Stirling's formula, we get

4 -n 2n + 1 n ∼ 2 √ πn .
Now let us prove [START_REF] Mountford | A coupling of infinite particle systems[END_REF]. Recall that Z = (Z n ) n∈N is a realization of model C with Z 0 = • Z . One can extend the definition of Z via coupling from the past. Consider the i.i.d. r.v.'s (U k,n ) (k,n)∈Z×Z with P (U i,j =↑) = P (U i,j =→) = 1/2. For each s ∈ Z, define

Z (s) = (Z (s) n ) n≥s by Z (s) s = • Z , ∀n ≥ s, Z (s)
n+1 = C(Z (s) n , (U k,n ) k∈Z )
. The starting time of Z (s) is s, but, besides that, the dynamic is the same as that of process Z. Observe that Z (0) = Z. More generally, Z (s) has the same distribution as (Z -s+n ) n≥s . Thus, we have In Figure 4, we have represented space-time diagrams for the model. The point of coordinate (k, n) corresponds to site k at step n. We have also represented the updating variables with the following convention: at the point (k, n), there is an arrow pointing north if U k,n =↑ and an arrow pointing north-east if U k,n =→. This allows to visualize the evolution of particles in the processes Z (s) . In Figures 4.b and 4.c, the processes Z (-3) and Z (-5) are represented; the grey nodes are the ones whose color depend on updating variables outside of the represented window. In We have a 0 = b 0 = 0, and

d n = P Z (-n) 0,0 = • .
|I 0 | = 1. Define ρ = 1 4 δ -1 + 1 2 δ 0 + 1 4 δ 1 .
We check the following:

• Assume that a n < b n . Then, We define a new PCA, called model D, which is a coupling of models B and C. The alphabet is {•, b, g} and the set of sites is Z. Given a configuration u ∈ {•, b, g} Z , the interpretation is as follows: if u i = • then site i is empty; if u i = b then site i contains a blue particle; if u i = g then site i contains a green particle. Particles move as in models B and C. When two particles collide, they get merged into one particle as in model C. In absence of collision, particles keep their color. In case of a collision, the merged particle is colored according to the rules:

a n+1 = a n -1 if U an-1,-n-1 =→ a n if U an-1,-n-1 =↑ , b n+1 = b n -1 if U bn,-n-1 =→ b n if U bn,-n-1 =↑ . Thus, |I n+1 | -|I n | ∈ {0, ±1},
(13) b + b → g, g + g → g, b + g → b, g + b → b .
We have represented a realization of model D on Figure 5. The "question mark" nodes are the ones whose color depend on updating variables outside of the represented window. Define

π B :{•, b, g} -→ {•, •}, π B (•) = •, π B (b) = •, π B (g) = • , π C :{•, b, g} -→ {•, •}, π C (•) = •, π C (b) = •, π C (g) = • .
We keep the same notations for the product applications: andd A n ≤ d n . Let (D n ) n be a realization of model D with D 0 being defined as follows: the r.v.'s D i,0 are i.i.d. with P (D 0,0 = b) = P (D 0,0 = g) = 1/2. At step n, the colors of the remaining particles are still i.i.d. and uniform : whatever the shape of the binary tree of coalescences leading to the presence of a particle at a given position at time n (see Figure 5 for an example), if the colors of the initial particles are independent and if one of these particle's color is uniform, then, due to [START_REF] Toom | Discrete local Markov systems[END_REF], the resulting color will still be uniformly distributed. Therefore we have ( 14 This completes the proof of Prop. 6.1.

π B : {•, b, g} Z → {•, •} Z , (u i ) i → (π B (u i )) i , and π C : {•, b, g} Z → {•, •} Z , (u i ) i → (π C (u i )) i . Lemma 6.2. If (D n ) n is a realization of model D, then (π B (D n )) n is a realization of model B, and
(π C (D n )) n is a realization of model C. As consequences, model D is ergodic with unique invariant measure δ • Z ,

Conclusion.

The following question remains: does there exist a positive-rates PCA which is non-ergodic with a unique invariant measure?

Let us provide some context. By definition, a PCA has positive-rates if all its probability transitions are different from 0 and 1 (more formally, if f : Σ V → M(Σ) is the transition function, then ∀u ∈ Σ V , ∀v ∈ Σ, f (u)(v) ∈ (0, 1)). It had been a long standing conjecture that all 1-dimensional positive-rates PCA are ergodic. In [START_REF] Gács | Reliable cellular automata with self-organization[END_REF], Gács disproved the conjecture by exhibiting a complex counter-example with several invariant measures. The existence of the intermediate case (unique but non-attractive invariant measure) remains open. A priori, it is not possible to perturbate model A to get a positive rates example.

3 .

 3 Statement of the main result 3.1. Model A. Consider the PCA F A on the alphabet Σ = {0, 1}, with neighborhood V = {-1, 0}, and transition function a defined by:

Figure 1 .

 1 Figure 1. The transition function of the PCA F A .

4. 1 .

 1 Model B. It is the Markov chain on {•, •} Z defined as follows. Consider the function

Figure 2 .

 2 Figure 2. The transition function of model B.

  a realization of a PCA on the alphabet {•, •} × U , with neighborhood V = {-1, 0}. The same remark holds for model C below.The continuous time version of model B, with exponential holding times, is called an annihilating random walk (cf.[START_REF] Griffeath | Additive and cancellative interacting particle systems[END_REF] Ch. 3, Sec. 5]).

Figure 3 .

 3 Figure 3. The transition function of model C.

4. 2 .

 2 Model C. It is the Markov chain on {•, •} Z defined as follows. Consider the function

Figure 4 .

 4 Figure 4. a. The updating (U k,n ) (k,n)∈[[-9,0]]×[[-6,0]] . b. The process Z (-3) during the span [[-3, 0]]. c. The process Z (-5) during the span [[-5, 0]]. d. The merging of particles.

Figure 4 .

 4 d, the particles painted in orange (gray) are those that merged into the particle present at time 0 and site 0. Let I n , n ≥ 0, be the set of indices of particles present at time -n in Z (-n) that merge into particle 0 at time 0. Either I n = [[a n , b n ]], a n ≤ b n , in which case Z (-n) 0,0 = •, or I n = ∅, in which case Z (-n) 0,0 = •. We focus on |I n |. For instance, in Figure 4.d, we have (|I n |) n∈[[0,6]] = (1, 2, 1, 2, 3, 4, 3). Observe that (12) d n = P Z (-n) 0,0 = • = P (|I n | ≥ 1) .

  and the conditional law of |I n+1 | -|I n | is ρ.

Figure 5 .

 5 Figure 5. A realization of process D (gray for g(reen), black for b(lue)).

2 P 2 P

 22 )P (π B (D 0,n ) = •) = P (D 0,n = b) = 1 (D 0,n ∈ {b, g}) = 1 (π C (D 0,n ) = •) = 1 2 d n ,where the last equality follows from Theorem 5.2.Now let ( D n ) n be a realization of model D with D 0 = b Z . Define E = (E i ) i by E i = b if D i,1 = b g if D i,1 = • or g . The r.v.'s (E i ) i are i.i.d. with P (E 0 = b) = P (E 0 = g) = 1/2.Let us justify this point. The state at time 1 of a realization of model A that starts from 0 Z is uniformly distributed by definition. Hence, the state at time 1 of a realization of model B that starts from • Z is uniformly distributed. And E has the same law as the latter up to the transformation b ↔ •, g ↔ •.So we have E ∼ D 0 . Observe also that π B ( D 1 ) = π B (E). We deduce that, for all n ≥ 1, we have π B ( D n ) ∼ π B (D n-1 ). In particular, using (14), P π B ( D 0,n ) = • = P D 0,n = b = P (D 0,n-1 = b) = d n-1 /2 .

• Assume that a n = b n . Then

For instance, the third case appears between lines -3 and -2 in Figure 4.d. Here again, the conditional distribution of

Consequently (|I n |) n∈N is a random walk with step ρ, starting from 1, and killed when it reaches 0. Using [START_REF] Toom | Contours, convex sets, and cellular automata[END_REF], we obtain (10).

Speed of convergence for models A and B

Let (A n ) n∈N be a realization of model A, with A 0 ∼ µ, µ ∈ M({0, 1} Z ). The possible limits for weakly-converging subsequences of (A n ) n∈N are of the form pδ (01

]. An evaluation of the distance to the limits is given by

Since model A is not monotone, we do not know for which initial measure µ this distance will be maximized. Hence we evaluate the "speed of convergence" for model A by the quantity:

The quantity d A n is also the speed of convergence to δ • Z for model B. Indeed we have P (A 0,n A 1,n ∈ {00, 11}) = P (ϕ(A n ) 0 = •), which implies that

where (B n ) n denotes a realization of model B and ν denotes its initial distribution (B 0 ∼ ν).

Recall that d n = 4 -n 2n+1 n is the speed of convergence for model C, see (9) and Theorem 5.2. Proposition 6.1. We have

Let (A n ) n be a realization of model A, with A 0 ∼ µ. If µ is the uniform distribution on {0, 1} Z , i.e. the r.v.'s A i,0 are i.i.d. with P (A 0,0 = 0) = P (A 0,0 = 1) = 1/2, then we shall see that The end of the section is devoted to the proof of Prop. 6.1, through the study of model A with the two initial distribution mentioned previously.


T Geisler

A.-M Seydoux-Guillaume

P Poeml

U Golla-Schindler

J Berndt

R Wirth

K Pollok

A Janssen

A Putnis

Experimental hydrothermal alteration of crystalline and radiation-damaged pyrochlore

We have performed hydrothermal experiments with a crystalline microlite and a heavily self-irradiation-damaged (i.e., X-ray amorphous) betafite in a solution containing 1 mol/l HCl and 1 mol/l CaCl 2 at 175°C. The well-crystalline microlite grains were partly (~ 5 -10 µm rim) replaced by a Ca and Na-poorer, defect pyrochlore phase with a larger unit-cell and a sharp chemical gradient at the interface (on a nm scale) to the unreacted core. The amorphous betafite grains (up to ~ 2 mm in diameter), on the other hand, were completely transformed into an intergrowth of different crystalline phases (polycrystalline anatase and rutile, a yet unidentified Nb-Ta oxide, and a Y-REE phase), showing complex non-equilibrium structures. Our experimental observations resemble those made on natural samples. They indicate that the processes of the fluid-pyrochlore interaction are influenced by self-irradiation structural damage and that thermodynamic equilibrium models can hardly be applied to adequately describe such systems.

Introduction

Considering the enormous amounts of high-level nuclear waste accumulated worldwide in the past decades, solutions for its safe immobilisation are urgently needed. Pyrochlore is a component of some Synroc versions (i.e., titanate-based ceramics [1]) and has been proposed as immobilization matrix for excess weapons-grade plutonium. Although considerable effort has been put to understanding radiation damage effects in pyrochlore [2], almost nothing is known about the effect of radiation damage on fundamental atomic mechanisms causing the aqueous alteration features seen in natural samples [3,4].

When a solid phase interacts with a fluid, several reaction mechanisms are possible, depending on the physicochemical conditions of the fluid, but also on the structural and thermodynamic properties of the solid phase (e.g., "porosity" of the structure, defect concentration, degree of amorphization, elastic strain). The simplest process that can occur is stoichiometric dissolution, following the breaking of cation-oxygen bonds controlled by protonation or de-protonation of the surface. Stoichiometric dissolution can be spatially coupled with precipitation of a thermodynamically more stable phase at a moving front. In other cases, the solid-fluid interaction is controlled by complex diffusion-reaction processes (often referred to as "leaching" reactions), which occur while the parent phase is in the solid state. The importance of each of these mechanisms during the interaction of crystalline and radiation-damaged pyrochlore with a hydrothermal fluid has not yet been fully understood.

In this contribution, we present data from crystalline and radiation-damaged pyrochlore treated at 175° in an acidic solution. The aim of the study was to investigate the influence of self-irradiation damage on the mechanisms and kinetics of the pyrochlore-fluid interaction.

Experimental details

Analytical methods

The starting material and the run products were studies by powder X-ray diffraction (XRD) analyses, electron microprobe (EMP), Raman spectroscopy, and transmission electron microscopy (TEM). The focussed ion beam (FIB) technique was used to prepare electron transparent foils from the reaction products [5]. The U, Ta and Ti content in the experimental fluid was analysed by inductive-coupled plasma optical emission spectroscopy (ICP-AES). Details of the analytical procedures are given in Geisler et al. [6]. All errors are given at the 2σ level.

Starting material

We have performed hydrothermal experiments with an X-ray amorphous betafite-(Y) from a rare earth pegmatite from Lindvikskollen near Kragerø, South Norway and a crystalline microlite from an evolved granite near Lueshe, Republic of Congo.

The crystalline microlite was virtually unaltered. The empirical structural formula obtained from 32 EMP analyses is (Ca 1.23 Na 0.75 A 0.02 )(Ta 1a). More details about this sample are given in Geisler et al. [6].

In contrast, the XRD spectrum of the betafite-(Y) shows no Bragg diffraction peaks, even after counting 90 s per 0.02° 2θ step. Instead, two very broad diffraction humps occur at about 30° and 50° 2θ (Fig. 1b), which are typical for heavily damaged, i.e. metamict pyrochlore [7]. Surprisingly, electron diffraction patterns obtained from powder drilled from unaltered areas exhibit in some cases minor reflections (not shown here), which could unambiguously be assigned to pyrochlore.

In contrast to the microlite, the betafite-(Y) exhibits already severe alteration features. The altered areas occur along cracks and as large patchy zones (Fig. 2a). They are characterized by lower backscattered electron (BSE) intensity. Chemically, the alteration zones can be distinguished from unaltered areas by significantly higher Ca contents, which is typical feature for primary alteration of pyrochlore and microlite at temperatures between 350 and 650°C [1]. We have estimated a U-Th-total Pb model age of 1020 ± 10 Ma from the U, Th, and Pb content of 134 analyses of the unaltered areas. This age is in good agreement with U/Pb ages obtained from minerals of other REE-rich granite pegmatites in South Norway and Southwest-Sweden [8]. From the age and the average U and Th concentration we have estimated that the sample has experienced an average cumulated α-decay dose of (38.7 ± 4.2) × 10 19 α-decays/g. We are aware that the betafite-(Y) is not an ideal sample since the experimental alteration is superimposed on the natural alteration. However, we will show that this will not affect the main conclusions drawn in this paper.

Experimental methods

The experiments were carried out with powder and two larger grains (1 -4 mm) and a solution containing 1 mol/l HCl and 1 mol/l CaCl 2 in Teflon reactors at 175 ± 3°C under autogeneous pressures (< 50 bar). Such a solution can be considered as a simplified natural fluid composition. The fluid-powder weight ratio was ~20. A second run at the same conditions yielded identical results, confirming the reproducibility of the experiments.

Results and Discussion

Fig. 1 shows the powder XRD pattern from the experimental run products compared with the pattern from the untreated material. In the XRD spectrum of the hydrothermally treated microlite new diffraction peaks occur at the low 2θ side of each peak observed in the untreated microlite, indicating that a new pyrochlore phase with a larger unit-cell volume crystallized (Fig. 1a). BSE images of individual grains reveal that this new pyrochlore phase partly replaces the microlite grains (Fig. 3). The thickness of the replacement varies between about 5 to 10 µm. This new pyrochlore phase was found to be characterized by significantly lower Ca (Fig. 3b), Na, and F concentrations and a high amount of A-site vacancies and molecular water [6] Geisler et al. [6] have demonstrated that the experimental alteration resembles low-temperature alteration features of natural pyrochlore-group minerals from world-wide localities. The chemical alteration of natural samples has been explained by selective "leaching" of A site cations associated with hydration as a consequence of a remarkable stability of the B 2 X 6 framework of the pyrochlore structure [3,4]. Geisler et al. [5] further discussed the experimental results within the framework of (i) the proposed leaching mechanism and (ii) a coupled dissolution-reprecipitation (D-R) process that operates at a moving interface [9]. The latter process is often associated with the development of porosity, irrespectively of the relative molar volumes of parent and product solid phases. It was shown by Putnis [9] that the development of porosity is often controlled by the relative solubility of the parent and product phase in the fluid. However, it is evident from the BSE images (Fig. 2a; see also [6]) that the new defect pyrochlore does not exhibit porosity on the micrometer scale. Porosity could also be excluded from mass balance calculations based on the measured concentration of Ta in the fluid [6]. However, in order to investigate the microstructure of the defect pyrochlore, i.e. to look for nano-sized pores by TEM, an electron-transparent foil across the interface between the parent and the product pyrochlore phase was prepared by the FIB technique. Selected area electron diffraction (SAED) patterns reveal that the replacement is topotactic (Figs. 4a,b). The new defect pyrochlore is further characterized by a high density of dislocations (Fig. 4d). The most important observations, however, are (i) that defect pyrochlore does not exhibit porosity and (ii) that the Ca and Na concentration gradient at the interface is sharp on the nanometer scale (Fig. 4c).

Whereas the topotactic relationship between the parent and product pyrochlore has frequently been considered as a characteristic of a diffusion-controlled ion exchange process, it is also typical for a coupled D-R process, where dissolution occurs simultaneously with precipitation at a moving interface [9]. The latter process would be more compatible with the sharp chemical gradient seen in Fig. 4c than a diffusion-based process. Furthermore, a high number of extended defects such as those seen in Fig. 4d are typical for crystals that grow from highly supersaturated solutions. A high number of dislocations were also observed in hydroxyapatite that has replaced a chlorapatite by a coupled D-R process [10]. As discussed by Geisler et al. [6], one may consider (H 2 O) x H 2 Ta 2 O 6 , described in [11], as the thermodynamically stable end member in the simplified solid solution/aqueous solution system (H 2 O) x H 2 Ta 2 O 6 -(Ca,Na) 2 Ta 2 O 6 F -HCl -H 2 O. However, as discussed by Geisler et al. [6], the circumferential cracking seen in Fig. 3a, which occurs some distance behind the interface, indicates the occurrence of stress as a result of a 3.4% larger volume of the defect pyrochlore. It is difficult to explain the formation of stress that caused cracking by a coupled D-R process. Nevertheless, whatever the mechanism at the interface (i.e., stoichiometric dissolution versus selective bond breaking), the fact that porosity has not been detected by TEM verifies previous conclusion that the reaction rate is controlled by solid state diffusion through the defect pyrochlore phase. The elucidation of the process, however, warrants further research. In contrast to the microlite, the amorphous betafite-(Y) completely recrystallized to a new phase assemblage as seen in the XRD spectrum (Fig. 1b). Most of the peaks in the XRD pattern could be assigned to rutile and anatase structure but with slightly higher d values compared to the pure TiO 2 phases (anatase: pdf 01-071-1169; rutile: pdf 01-073-2224). In order to study the phases and their relationship by BSE imaging and EMP, we have polished one large grain to about half of its thickness. It is noteworthy that the shape of this grain is almost the same as it was before the experiment. The BSE images reveal a complex intergrowth of at least two phases and a large porosity (Fig. 2). The most frequent phase shows complex, non-equilibrium banding structures (Figs. 2bd).

Compared with the untreated betafite this phase is depleted in Y, REEs, and U and relative enriched in Nb, Ti, and Ta (Fig. 2d, Fig. 5). The underlying phase can actually be identified as a mixture of anatase and rutile from electron diffraction pattern (Fig. 6b) and µ-Raman measurements. From the intensity of the strongest (101) anatase and the (110) rutile reflection of the XRD spectrum and the empirical relationship given in Despero et al. [12], we can estimate that the anatase to rutile ratio is about 0.4:0.6. The banding is based on large variations of the Nb-Ti ratio (Fig. 2d, Fig. 5a). It is noteworthy that two trends can be distinguished in the diagram Nb 2 O 5 versus UO 2 (Fig. 5b). Whereas anatase was detected as the major phase in the Nb-rich areas by µ-Raman spectroscopy, rutile dominates in areas with no more than about 30 wt.% Nb 2 O 5 (see also Fig. 5b). We note that the Nb content is anomalous high for the anatase or rutile structure. Another phase appears bright in the BSE images of Figs. 2b,c. This phase is a yet unidentified Nb-Ta oxide. Two XRD reflections can be identified that do not belong to rutile or anatase, which thus may be assigned to this phase (d ≈ 0.33 nm and d ≈ 0.27 nm). It contains almost no U, REEs, and Y. Furthermore, a minor third, yet unidentified phase could be detected, which has captured Y and REEs.

All new phases contain less U than the untreated betafite-(Y) (Fig. 5b). This indicates that significant U was lost to the fluid phase. Indeed, we found (11.88 ± 0.40) × 10 -3 g U in the fluid phase by ICP-AES. We also detected minor amounts Ti (= (0.075 ± 0.0010) × 10 -3 g) in the fluid. From the average U content in the betafite-(Y) (= 9.8 ± 2.2 wt. %) and the amount of betafite used for the experiment (= 0.17520 g) we have estimated that ~ 30 wt.% of the U of the betafite was retained in the new solid phase assemblage, i.e. that ~ 70% was lost to the fluid phase. This estimate is in good agreement with mass balance calculations based on the phase proportions estimated from BSE images.

In order to further study the reaction mechanism, we investigate the microstructure of the banded anataserutile phase assemblage by TEM. For this, we cut out another foil perpendicular to the banding by the FIB technique (Fig. 2c). Bright field images and SAED patterns reveal that the anatase/rutile phase assemblage is polycrystalline with a random grain orientation (Fig. 6). Based on textural considerations, it seems very likely that anatase and rutile were formed within the unaltered areas of the betafite-(Y), whereas the Nb-Ta phase (bright in the BSE images of Figs. 2d,e) crystallized within the previously altered area, i.e. along pre-existing cracks. This interpretation is consistent with the orientation of the banding, which runs away from the zones where the Nb-Ta phase is located. Such a complex intergrowth and the polycrystalline structure of the anatase -rutile assemblage suggest that anatase and rutile were formed by a diffusion- reaction process while the betafite phase was in the solid state. During the alteration process the REEs, Y, and most of the U were selectively removed from the amorphous betafite, leaving behind the B-site cations, allowing the nucleation and growth of anatase and the anatase-to-rutile transformation to occur. We suggest that the diffusion of hydrogen plays a key role during such a process. It is also conceivable that the feedback, necessary for the development of patterns from an unpatterned state, is based on the well-known inhibition of grain growth of TiO 2 when Nb is incorporated [13,14]. However, the atomistic mechanism that has generated the banding structures is not yet understood.

The occurrence of the Nb-Ta and Y-REE phases also indicates that dissolution-reprecipitation reactions are involved, suggesting a complex interplay of various processes. We note here that rutile and anatase were also found as alteration products in natural betafite [3,4] and that complex, non-equilibrium structures similar to those produced experimentally have also been observed in a natural betafite crystal from Betafo, Madagascar [15]. This indicates that our experimental results are relevant for understanding natural alteration of pyrochlore and for predicting the long-term behaviour of a synthetic actinidebearing pyrochlore in the geological environment of a nuclear waste repository. However, in natural systems rutile and anatase are accompanied by liandratite (ideally UNb 2 O 8 ) and uranopyrochlore as alteration products of betafite, which apparently captured most of the U released from the altered betafite [3,4]. Under the experimental conditions neither liandratite nor uranopyrochlore was stable. Instead, the unidentified Nb-Ta and Y-REE oxides are stable phases, which did not incorporate U in significant amounts, resulting in the high loss of U to the fluid phase.

Concluding remarks

Our results on the aqueous alteration of crystalline and metamict pyrochlore in an acidic solution at 175°C indicate that the alteration proceeds via different reaction mechanisms, which are based on a complex interplay of diffusion, reaction, dissolution and reprecipitation of more stable phases. It is clear that the different chemistry of both pyrochlore samples can have an effect on their aqueous stability and also on the phase relationships. However, we emphasize that the altered volume was significantly larger in the heavily self-irradiation damaged betafite than it was in the microlite sample, indicating a strong impact of self-irradiation structural damage on the alteration kinetics.

Whatever the atomistic details of the reactions, our preliminary experimental results point to the importance of gaining a more detailed understanding of the aqueous alteration mechanisms and the impact of the radiation damage before reliable predictions about the long-term stability of an actinide-bearing titanate pyrochlore waste form in a geological environment can be made. Our observations indicate that the aqueous stability of pyrochlore-group minerals and the fluid-pyrochlore reaction mechanisms are influenced by self-irradiation structural damage and that thermodynamic equilibrium models cannot be applied to adequately describe such systems.

Fig. 1 .

 1 Fig. 1. Powder X-ray diffraction patterns of the untreated and hydrothermally treated (a) microlite (data from [5]) and (b) betafite-(Y) sample. The HasterisksH in (a) mark new peaks belonging to a new pyrochlore phase with a larger unit-cell. The HasterisksH and circles in (b) mark diffraction peaks, which belong to anatase and rutile, respectively.

Fig. 2 .

 2 Fig. 2. Backscattered electron (BSE) and elemental distribution images of untreated and hydrothermally treated grains of the radiationdamaged betafite-(Y) sample. (a) BSE image of a grain of the untreated betafite -(Y) sample. Note that this sample has already been altered during its geological history. The alteration zones give a lower BSE intensity and spread out along cracks, but also appear as lager patchy areas. (b) Hydrothermally treated betafite -(Y) grain. The hydrothermal treatment has caused the complete transformation into an intergrowth of Nb-rich anatase and rutile and unidentified Nb-Ta and Y-REE phases, which are reflected by various grey scales in the BSE image. The white square marks the magnified section of the BSE and the element distribution image shown in (c) and (d), respectively. (c) BSE image of the area marked in (b) by a rectangle. The location of the electron transparent foil prepared by the focussed ion beam (FIB) technique for transmission electron microscope (TEM) investigations is marked. Note the porosity in the banded areas. (d) Nb distribution image of the area shown in (c). The unidentified Nb-Ta oxide phase appears in the lower middle part of the image.

  Fig.1shows the powder XRD pattern from the experimental run products compared with the pattern from the untreated material. In the XRD spectrum of the hydrothermally treated microlite new diffraction peaks occur at the low 2θ side of each peak observed in the untreated microlite, indicating that a new pyrochlore phase with a larger unit-cell volume crystallized (Fig.1a). BSE images of individual grains reveal that this new pyrochlore phase partly replaces the microlite grains (Fig.3). The thickness of the replacement varies between about 5 to 10 µm. This new pyrochlore phase was found to be characterized by significantly lower Ca (Fig.3b), Na, and F concentrations and a high amount of A-site vacancies and molecular water[6]. The average structural formula of the new defect pyrochlore as determined by EMP analyses

Fig. 3 (

 3 Fig. 3 (a) BSE image of a hydrothermally treated grain of the crystalline microlite. The arrow points to the interface between a reaction rim and the parent microlite. (b) A Ca distribution map of the same grain shown in (a). The reaction rim is characterized by a low Ca content compared to the parent microlite (core) and represents a highly defective pyrochlore phase. See text for more details.

Fig. 4 .

 4 Fig. 4. TEM results from an electron transparent foil prepared by the FIB technique. The foil crosses the interface between the parent microlite and the reaction rim. (a) Selected area electron diffraction (SAED) pattern of the parent microlite (left hand side in (c)). (b) SAED pattern from the defect pyrochlore rim (right hand side in (c)). (c) Bright field (BF) image of the TEM foil along with energy-dispersive X-ray intensity profiles of Ca-Kα and Na-Kα radiation measured along the white line. Note the sharp chemical gradient at the interface. The contrast in the image is partly due to the underlying copper grid. (d) BF image of the defect pyrochlore rim, showing a high number of dislocations.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Chemical variation of the alteration phases of the hydrothermally treated betafite-(Y) in a plot of (a) Nb 2 O 5 versus TiO 2 and (b) Nb 2 O 5 versus UO 2 . The legend for both diagrams is shown in (b). Note that the anatase and rutile phases are unusually enriched in Nb. See text for further details-

  1.78 Nb 0.08 Ti 0.08 Si 0.06 ) Σ=2 O 6.28 Sr, Pb, Y and Fe. The structural formula was computed on the assumption that ΣB = 2. The sample further contains ~150 ppm U and ~ 700 ppm Th. The XRD pattern shows sharp diffraction signals (Fig.

	with A =
	X,Y
	0.15
	F 0.57

Acknowledgements

We would also like to thank G. Lumpkin for providing us with an Excel spreadsheet to calculate the chemical formula of pyrochlore. We are indebted to J. Schlüter from the Mineralogical Museum of the University of Hamburg for putting both samples used for the experiments at our disposal. We also acknowledge financial support by the Deutsche Forschungsgemeinschaft (grant GE 1094/5-1).