G Turchetti
email: giorgio.turchetti@bo.infn.it

S Sinigardi

G Servizi

F Panichi

S Vaienti

Errors, correlations and fidelity for noisy Hamilton flows. Theory and numerical examples

Keywords: numbers: 05.40.Ca, 05.45.Pq, 02.50.Ey Noisy Hamiltonian flow, Fidelity, Reversibility error, Observational noise

We analyse the asymptotic growth of the error for Hamiltonian flows due to small random perturbations. We compare the forward error with the reversibility error, showing their equivalence for linear flows on a compact phase space. The forward error, given by the root mean square deviation σ(t) of the noisy flow, grows according to a power law if the system is integrable and according to an exponential law if it is chaotic. The autocorrelation and the fidelity, defined as the correlation of the perturbed flow with respect to the unperturbed one, exhibit an exponential decay as exp -σ 2 (t) . Some numerical examples such as the anharmonic oscillator and the Hénon Heiles model confirm these results. We finally consider the effect of the observational noise on an integrable system, and show that the decay of correlations can only be observed after a sequence of measurements and that the multiplicative noise is more effective if the delay between two measurements is large.

Introduction

The Hamiltonian description of geophysical fluids has been extensively developed [START_REF] Morrison | Hamiltonian fluid mechanics Encyclopedia of mathematical physics[END_REF][START_REF] Morrison | [END_REF]3,[START_REF] Swaters | Introduction to Hamiltonian Fluid Dynamics and Stability Theory Monographs and Surveys in Pure and Applied Mathematics[END_REF] and is suited to analyse the conservation laws and to investigate approximation schemes which preserve the geometric invariants. These methods are also applicable to the Poisson-Vlasov equations for a collisionless plasma [START_REF] Morrison | [END_REF]6]. Examples of Hamiltonian models of geophysical fluids are the canonical formulation of Rossby waves in a rotating sphere and the system of N vortices.

Finite dimensional Hamiltonian systems are well understood in the integrable and uniformly hyperbolic limits, which correspond to an ordered quasiperiodic and a chaotic evolution [START_REF] Arnold | Mathematical Aspects of Classical and Celestial Mechanics Encyclopaedia of Mathematical Sciences[END_REF]. The effect of small perturbations is amenable to analytical treatment in both situations, but the case of large perturbations and the transition from regular to chaotic regimes can only be explored numerically. In addition the effect of a weak noise and of round off errors in numerical computations deserves a special attention. The divergence of orbits and the memory loss rate of a given system are intimately related and reflect its dynamic behaviour. The Lyapunov exponents, and related indicators, specify the asymptotic separation rate of two initially close orbits [START_REF] Skokos | The lyapunov characteristic exponents and their computation Dynamics of Small Solar System Bodies and Exoplanets[END_REF], whereas the autocorrelation decay measures how fast the evolution loses its memory of a given initial condition. The asymptotic divergence of nearby orbits, the decay of correlations and the spectrum of Poincaré recurrences are intimately related. In particular the correlation decay rate is related to the Lyapunov spectrum [9].

When a small deterministic or stochastic perturbation is introduced into the system, the perturbed orbit diverges from the unperturbed one (with the same initial condition) and we call forward error (FE) their distance at time t or the root mean square deviation at time t, if the perturbation is stochastic. The reversibility error (RE) is the distance, or the root mean square deviation, from the initial point, of its image after the evolution forward and backward for the same time interval t. The loss of memory with respect to the reference orbit is given, for instance, by the classical fidelity [10], first introduced in quantum systems [11].

For linear symplectic maps the asymptotic equivalence or RE and FE was proved in [12] and a relation with the fidelity decay was established. The fidelity behaviour for maps on the torus and the cylinder with random perturbations was first analysed in [13] and compared with the perturbation induced by round off errors [14,15]. The RE analysis was first introduced to investigate the global effects of round off errors in symplectic maps in [16].

In this note we examine the asymptotic behaviour of FE and RE for Hamiltonian flows with a small random perturbation. In the case of linear flows we recover, with simpler proofs, the results previously obtained for linear symplectic maps. If the phase space is a compact set such as the torus T 2d then the autocorrelation and fidelity can be computed and their decay rate is related to the growth of the forward and reversibility errors.

Denoting with σ(t) the FE and RE for a random perturbation, the asymptotic decay of fidelity is given by F (t) ∼ e -2π 2 σ 2 (t) . For an integrable system defined on the cylinder we prove that the error growth follows a power law σ(t) ∼ t β where β = 3/2 if the system is anisochronous, β = 1/2 if it is isochronous. The exponent 3/2 is due to the conditional mixing, known as filamentation in the case of a rotating fluid. This kind of mixing is also responsible for the algebraic decay of the spectrum of Poincaré recurrences [17]. These results easily extend to integrable systems of higher dimension. If the unperturbed system is not skew the result very likely still holds, as suggested by numerical computations, though the proof is no longer elementary. The proposed method could be applied to any finite dimensional dynamical system with a small additive noise because it is based on the linear equation satisfied by the Gaussian stochastic process which defines the forward and reversibility errors. Infinite dimensional extensions might be considered since the stochastic process satisfies a linear partial differential equation.

We have analysed numerically the behaviour of the anharmonic oscillator, the Hénon Heiles model, and a system of point vortices [START_REF] Newton | The N-Vortex Problem: Analytical Techniques Applied Mathematical Sciences[END_REF][START_REF] Aref | [END_REF][START_REF] Saffman | Vortex Dynamics[END_REF][START_REF] Luo | Hamiltonian Chaos Beyond the KAM Theory: Dedicated to[END_REF]. The results obtained for these models agree with the ones previously obtained for the standard map and the three body system [12,15]. Except for the vortex model the Hamiltonian reads H = T (p) + V (q, t) -x ξ(t) where ξ(t) is a white noise and a symplectic integrator was used, 4-th order accurate for the deterministic component, just as for the three body problem. For a discussion on numerical stochastic integration see [START_REF] Mannella | A gentle introduction to the integration of stochastic differential equations Stochastic Processes in Physics[END_REF]. In the integrable and chaotic regions the results proved for linear systems are confirmed. In the transition regions the results might be interpreted following the model proposed for Poincaré recurrences [17].

Random perturbations are introduced by the environment and by observations: the effect of observational noise in dynamical systems is actively investigated, see [23,24] and references therein. When the system is integrable the effect of a single observation is small. However a sequence of observations may cause an exponential decrease of the correlations or the fidelity. The multiplicative noise changes the signal in a way dependent on the signal itself and may determine a faster decay of fidelity when the change of phase between two observations is large.

Dynamical systems with additive noise

In a previous paper we have considered prototype model maps defined on the torus or the cylinder, whose invariant measure is the normalized Lebesgue measure. We consider first a 1D dynamical system with a stochastic perturbation defined by the Langevin equation

dx dt = Φ(x) + ξ(t) (1)
where x ∈ R and ξ(t) is a white noise. We denote by average with respect to all the realizations of the stochastic process

ξ(t) = 0 ξ(t)ξ(t ) = δ(t -t ) (2)
We denote the deterministic solution with initial point x 0 with x = S t (x 0 ) and with f (x, t) the distribution function at time t corresponding to an initial value f (x 0 , 0) = f (x 0 ). The distribution function f for the deterministic system satisfies the continuity equation

∂ ∂t f (x, t) + ∂ ∂x (Φ(x)f (x, t)) = 0 (3)
The fundamental solution of the continuity equation is given by

G(x, x 0 ; t) = δ(x -S t (x 0 )) = δ(x 0 -S -t (x)) η(S -t (x), t) η(x 0 , t) = ∂S t (x 0 ) ∂x 0 (4)
since the solution x = S t (x 0 ) is assumed to have a unique inverse x 0 = S -t (x). The solution for the distribution function f (x, t) corresponding to an initial distribution

f (x 0 , 0) = f (x 0 ) reads f (x, t) = +∞ -∞ G(x, x 0 ; t) f (x 0 ) dx 0 = f (S -t (x)) η(S -t (x), t) (5)
When the noise is present we can still write a stochastic continuity equation given by (3) where Φ is replaced by Φ(x) + ξ(t). In this case after averaging over the noise the distribution function satisfies the Fokker-Planck equation (see [10]).

∂ ∂t f (x, t) + ∂ ∂x (Φ(x)f (x, t)) = 2 2 ∂ 2 ∂x 2 f (x, t) (6)
whose solution is given by

f (x, t) = +∞ -∞ G (x, x 0 ; t) f (x 0 ) dx 0 (7)
We denote with G (x, x 0 ; t) the fundamental solution of the Fokker-Planck equation, namely its solution with initial condition G (x, x 0 ; 0) = δ(x -x 0 ). In the limit → 0 we recover the fundamental solution of equation (3). We write the solution of the Langevin equation as x(t) = S , t (x 0 ) where

S , t (x 0 ) = S t (x 0 ) + χ(t) = S t (x 0 ) + Ξ(t) + O( 2 ) (8)
The stochastic process Ξ(t) is the limit of χ(t) for → 0 and proves to be Gaussian since it satisfies the linearised equation

d Ξ dt + M(t) Ξ = ξ(t) M(t) = Φ (S t (x 0 ))
Indeed Ξ(t) can be written as a convolution of the white noise Ξ(t) =

t 0 K(t, s) ξ(s) ds. If Φ(x) is linear Φ(x) = ω + βx (9)
then χ(t) and Ξ(t) are equal. In this case the deterministic component is

S t (x 0 ) = e βt x 0 + φ(t) φ(t) = ω e βt -1 β (10)
and the fluctuating component is defined by

χ(t) = t 0 e β(t-s) ξ(s)ds σ 2 (t) = 2 e 2βt -1 2β (11)
As a consequence the fundamental solution of the Fokker-Planck equation (6) reads

G (x, t) = exp - (x -x ) 2 2σ 2 (t) 2πσ 2 (t) x(t) = S t (x 0 ) = e βt x 0 + φ(t) (12)
We notice that for = 0 the fundamental solution of the Fokker-Planck reduces to δ(x -S t (x 0 )), namely the fundamental solution of the deterministic continuity equation (3). We remark also that the root mean square deviation for β > 0 has an exponential growth σ(t) ∼ e βt . If β = 0 the motion becomes a translation and in this case σ(t) = t 1/2 .

Observables on the torus and correlations

In order to have a chaotic behaviour an exponential divergence of the flow and the phase space compactness are required. This can be achieved by considering a dynamical system on the torus T. The torus can be seen as the interval [0, 1], where the endpoints are identified. Any periodic function of period 1 defined on R is a dynamic variable on T 1 . If f (x) is any function defined on R we may construct a dynamic variable f T (x) in the following way

f T (x) = +∞ j=-∞ f (x + j) (13)
provided that the decrease of f (x) at infinity is sufficiently fast to insure the convergence of the series. The function f T (x) is periodic of period 1 by construction and therefore can be expanded in a Fourier series

f T (x) = +∞ k=-∞ fk e 2π i k x (14)
whose coefficients are given by fk =

1 0 e -2π i k x f T (x) dx = +∞ j=-∞ 1 0 e -2π i k x f (x + j) dx = +∞ j=-∞ j+1 j e -2π i k (x-j) f (x) dx = +∞ -∞ e -2π i k x f (x) dx (15) If f (x) ∈ L 1 (R) its Fourier transform f (k) for k ∈ R is defined by f (k) = +∞ -∞ e -2π i k x f (x) dx (16)
and consequently fk = f (k) for k ∈ Z.

If f (x, t) is the solution of the continuity equation on R the corresponding solution on T is given by

f T (x, t) = +∞ j=-∞ f (x + j, t) = +∞ j=-∞ +∞ -∞ δ(x + j -S t (x 0 )) f (x 0 ) dx 0 = = +∞ -∞ G T (x, x 0 , t) f (x 0 ) dx 0 (17)
By G T (x, x 0 , t) we denote the fundamental solution on T 1 whose Fourier expansion is given by

G T (x, x 0 ; t) = +∞ j=-∞ δ(x + j -S t (x 0 )) = +∞ k=-∞ e 2π i kx e -2π i k St(x 0 ) (18)
By replacing ( 18) into ( 17) we obtain the Fourier expansion of f T (x, t) according to

f T (x, t) = +∞ k=-∞ fk (t) e 2π i k x fk (t) = +∞ -∞ dx 0 e -2π i k St(x 0 ) f (x 0 ) (19)
We introduce also the following coefficients

f k (t) = 1 0 dx 0 e -2π i k St(x 0 ) f T (x 0 ) (20)
Notice that for t = 0 we have fk (0) = fk . The new time dependent coefficients differ from the Fourier components unless the evolution is just a translation S t (x) = x 0 + ωt or f (x) has support on [0, 1] vanishing elsewhere. In this case

f k (t) = fk (t)
and the following equality holds

1 0 f T (x) f T (x, t) dx = 1 0 f T (x 0 ) f T (S t (x 0 )) dx 0 (21)

Correlations

We introduce two distinct definitions of the correlation on the torus according to

Ĉ(t) = 1 0 f T (x) f T (x, t)) dx -f 2 0 = k =0 f-k fk (t) (22)
see equation ( 19)

C(t) = 1 0 f T (x 0 ) f T (S t (x 0 )) dx 0 -f 2 0 = k =0 f-k f k (t) (23)
see equation [START_REF] Saffman | Vortex Dynamics[END_REF]. These definitions are equivalent when S t is a translation or f (x) has support on [0, 1]. In general we can write C(t) according to

C(t) = (k ,k) =(0,0) fk A k ,-k f-k A k ,-k = 1 0 dx 0 e 2π i(k x 0 -k St(x 0 ) ) (24)
When the vector field Φ(x) is linear, see equation ( 9), and the flow is linear, see equation (10), then C(t) = Ĉ(t) for a sequence of times t = t m where t m = β -1 log m for m ∈ Z. Indeed from equations ( 19), ( 10) and ( 16) we have

fk (t) = e -2π i k φ(t) f (ke βt ) (25)
The coefficients f k (t), according to equations ( 23) and ( 24), are given by

f k (t) = +∞ k =-∞ fk A k ,-k (26)
where A k ,-k is given by

A k ,-k = e -2π i kφ(t)      e πi(k -k e βt ) sin(π(k -ke βt )/(π(k -k e βt )) if e βt = m δ k ,mk if e βt = m (27)
As a consequence, for t = t m , from ( 26) and ( 27) we have

f k (t m ) = fk (t m ) = e -2π i kφ(tm) fkm = e -2π i kφ(tm)
fke βtm (28)

Noisy systems on the torus: correlations and fidelity

For a noisy system the evolution of a given initial distribution is given by equation ( 7) and if the system is defined on the torus T the evolution of the corresponding distribution f T (x) defined by ( 13) is a periodic function in x which can be expanded in a Fourier series according to

f , T (x, t) = +∞ -∞ G T (x, x 0 ; t) f 0 (x 0 ) dx 0 = +∞ k=-∞ f , k (t) e 2π i kx (29)
where G T (x, x 0 ; t) denotes the fundamental solution of the Fokker-Planck equation on the torus and f T (x, t) the distribution function on the torus at time t

G T (x, x 0 ; t) = +∞ j=-∞ G (x + j, x 0 ; t) f , T (x, t) = +∞ j=-∞ f (x + j, t) (30)
Even in this case we introduce two distinct definitions of correlations as an obvious extension of the noiseless case

Ĉ (t) = 1 0 f T (x) f , T (x, t) dx -f 2 0 = k =0 f-k f , k (t) (31)
and

C (t) = 1 0 f T (x 0 ) f T ( S , t (x 0 ) ) dx 0 -f 2 0 = k =0 f-k f , k (t) (32)
where

f ,k (t) = 1 0 e -2π i k S ,t(x0) f T (x 0 ) dx 0
The fidelity gives the correlation between a dynamic variable f (x) evaluated along the perturbed orbit at time t and f (x) evaluated along the unperturbed orbit at the same time. We have a unique definition of fidelity given by

F (t) = 1 0 f T (S t (x 0 )) f T ( S , t (x 0 ) ) dx 0 -f 2 0 = = (k,k ) =(0,0) f-k fk 1 0 dx 0 e 2π i (k S , t(x0)-kSt(x0)
(33)

Linear systems

For a linear Langevin equation, namely for Φ(x) = βx + ω, explicit expressions can be obtained. Indeed, to evaluate f , k (t), we expand G T (x, x 0 ; t) in Fourier series, taking ( 8) and ( 11) into account. The result is

G , k (x 0 , t) = 1 0 dx e -2π i kx G T (x, x 0 ; t) = +∞ -∞ e -2π i kx exp - (x -S t (x 0 )) 2 2σ 2 (t) 2πσ 2 (t) = e -2πik St(x 0 ) e -2π 2 σ 2 (t) (34) The Fourier components of f T (x, t) are given by f , k (t) = e -2π 2 σ 2 (t) +∞ -∞ e -2πik St(x 0 ) f (x 0 ) dx 0 = e -2π 2 σ 2 (t) fk (t) (35)
In order to evaluate f , k (t) we recall that for a linear Langevin equation (1) where Φ(x) is given by ( 9) the solution S , t (x 0 ) is given by ( 8) and ( 11) so that

e -2π i kS , t(x0) = e -2π i k St(x 0 ) e -2π i k χ(t) = e -2π i k St(x 0 ) e -2π 2 k 2 σ 2 (36)
As a consequence taking equation ( 16) into account we can write

f , k (t) = e -2π 2 k 2 σ 2 f k (t) (37)
The explicit expression for the correlation is

Ĉ (t) = k =0 f-k f k (t) e -2π 2 k 2 σ 2 (t) (38)
and

C (t) = (k,k ) =(0,0) fk A k ,-k f-k , e -2π 2 k 2 σ 2 (t) (39)
In this case the fidelity has the following expression

F (t) = (k,k ) =(0,0) f-k fk 1 0 dx 0 e 2π i (k -k) Stx 0 e -2π 2 σ 2 (t) (40)

Model systems

We consider first three different prototype models i) Translations on the torus T 1 .

The field is Φ(x) = ω and S t (x 0 ) = x 0 + ωt and the Fourier coefficients are

f k (t) = fk (t) = e -2π i k ωt fk (41)
The correlations do not decay. When the noise is introduced the root mean square deviation with respect to the unperturbed trajectory is σ = t 1/2 and the Fourier coefficients are given by

f k (t) = f k (t) = e -2π i k ωt fk e -2π 2 2 t (42)
The decay of correlations and fidelity is exponential with decay time τ = (2π 2 2 ) -1 . The difference is that the correlations oscillate and decay, while the fidelity goes to zero without oscillations, due to the absence of the phase factor. ii) Anisochronous translations on the cylinder This is a skew system. Letting x = (x, y) ∈ C = T × I be the phase space coordinates, Φ = (y, 0) the vector field and S t (x 0 ) = (x 0 + y 0 t, y 0 ) the flow, for an observable f (x) to which corresponds f T (x) on the torus we have

fk (t) = f k (t) = e -i k y 0 t fk (43)
Even though f T does not depend on the torus label y 0 the coefficient f k (t) depend on it according to the previous equation. The correlation is computed by averaging on the interval I = [0, a] to which y 0 belongs. Taking account of the normalization factor a -1 the correlation reads

C(t) = Ĉ(t) = k =0 |f k | 2 1 a a 0 e -2π i k y 0 t dy 0 = k =0 |f k | 2 e -π i k a t sin(π k a t) π k a t (44)
as a consequence supposing that |f k | < A/|k| we have

|C(t)| < 3A 2 πa 1 t (45)
and the decay follows a power law. This slow decay is due to the conditional mixing, known in physics as the filamentation phenomenon, which occurs because the frequency on the tori has a continuous and monotonic variation. For an observable f (x, y) such that the average on x is constant, the result still holds provided that the Fourier coefficients fk (y) are C 1 and decay faster than 1/y at infinity. Any autonomous 1D system in the region around a stable fixed point delimited by the separatrices is defined, using action angle variables (φ, ), by the vector field Φ = (ω(j), 0). If ω(j) is monotonic, introducing the new variables x = φ/(2π) and y = ω() we are back to the case considered above.

When a stochastic perturbation is introduced the vector field becomes Φ(x, y) = ( y + ξ x (t), ξ y (t) ) where ξ x (y) and ξ y (t) are two independent white noises. In this case the stochastic flow S , t (x 0 , y 0 ) is given by x = x 0 + y 0 t + w (1) y (t) + w x (t)

y = y 0 + w y (t) (46)
where w(t) = t 0 ξ(s) ds denotes a Wiener noise and w (1) (t) = t 0 w(s) ds. Recalling that w 2

x (t) = t, (w

y (t)) 2 = t 3 /3, w x (t)w (1)
y (t) = 0, the root mean square deviations for x and y are given by

σ x = t + t 3 3 1/2 σ y = t 1/2 (47)
The correlation on the cylinder for the observable f T (x) is given by

C(t) = k =0 |f k | 2 e -π i k a t sin(π k a t) π k a t e -2π 2 k 2 (t+t 3 /3) (48)
The fidelity has the same expression where the term within the square parenthesis is replaced by 1. If the frequency is the same on each torus, or if the y variable is not affected by noise, the decay of correlations or fidelity is much slower since the cubic term in the exponential is absent. iii) Expanding flow Given the vector field Φ(x) = βx, the flow is S t (x 0 ) = e βt x 0 and the Fourier coefficients are given by fk

(t) = +∞ -∞ f (x) e 2π i k x 0 e βt = f (k e βt ) ( 49
) If f (x) ∈ L 2 then f (k) ∈ L 2 and consequently | f (k)| decreases to 0 for |k| → ∞. If | fk | ≤ A |k| -1
then f (ke βt ) decreases exponentially fast to zero with t and the following estimate holds

| Ĉ(t)| ≤ 2A 2 e -βt ∞ k=1 1 k 3 ≤ 3A 2 e -βt (50)
The Fourier coefficients entering the other definition of correlation are, for e βt ∈ N,

f k (t) = 1 0 f T (x 0 ) e -2π i k x 0 e 2βt dx 0 = +∞ k =-∞ f k e +π i (k -ke βt ) sin π(k -ke βt ) k -ke βt (51)
If e βt = m ∈ N then letting t m = β -1 log m we have fk (t m ) = f k (t m ) = fmk where | fmk | ≤ e -βtm A/|k|. If f (x) has a compact support on [0, 1] the two definitions agree for any t ∈ R.

When the noise is present the root mean square deviation is given by

σ = e 2βt -1 2β 1/2 (52)
The correlations consequently are given by ( 38) and (39) , the fidelities by (40) where fk (t) and f k (t) are given by equations ( 49) and ( 51) respectively.

The explicit expressions of correlations for e βt ∈ N is

Ĉ (t) = k =0 f-k f (ke βt ) e -2π 2 k 2 σ 2 (t) (53)
and

C (t) = (k,k ) =(0,0) f -k f k e π i (k -ke βt ) sin π((k -ke βt ) π(k -ke βt ) e -2π 2 k 2 σ 2 (t) (54)
The explicit expression for the fidelity reads

F (t) = (k,k ) =(0,0) f -k f k e π i (k -k)e βt sin π((k -k)e βt π i(k -k)e βt e -2π 2 k 2 σ 2 (t) (55)
For e βt = m ∈ N we have

Ĉ (t m ) = C (t m ) = k =0 f-k fmk e -2π 2 k 2 σ 2 (tm) F (t m ) = k =0
|f k | 2 e -2π 2 k 2 σ 2 (tm) (56)

Reversibility error and related fidelity

The reversibility error is the distance from the initial point of its forward evolution up to time t and backward for the same time interval. Any autonomous flow is reversible since the flow S t has the group property S -t • S t = I. This property is lost when the system is stochastically perturbed. In this case the local errors ξ(±t) at times ±t are independent and we have S ,-t • S ,t = I.

We define a fidelity associated to the reversibility error according to

F R (t) = 1 0 dx 0 f T (x 0 ) f T (S ,-t • S ,t (x 0 )) -f 2 0 ( 57
)
The FE and the RE are related, and for linear flows we prove they are equal up to a factor √ 2. We consider first the translation perturbed with a white noise whose flow is S , t x 0 = x 0 + ωt + w(t) so that S , -t • S , t x 0 = x 0 + (w(t) + w(-t)). The reversibility error is given by the root mean square deviation of this process σ R (t) = √ 2 t 1/2 , since w(t)w(-t) = 0. If the flow is expanding, namely Φ = βx + ξ(t) then we have

S , -t • S , t x 0 = x 0 + χ R (t) χ R (t) = e -βt χ(t) + χ(-t) (58)
As a consequence from χ R (t) = t 0 e -βs (ξ(s) -ξ(s -t))ds we obtain the reversibility error

σ R (t) = χ 2 R (t) 1/2 = √ 2 1 -e -2βt 2β (59)
The reversibility error σ R (t) is equal to the forward error σ(t) (times √ 2) but for the system whose vector field is Φ(x) = -βx rather than Φ(x) = βx. The forward flow expanding, the reverse flow is contracting, but RE for the expanding flow is equal to the FE for the contracting flow. To recover the equivalence it is sufficient to consider the flow on the torus T 2 defined by the following noisy vector field Φ = (-βx+ ξ x (t), βy+ ξ y (t)) where ξ t (t) and ξ y (t) are independent white noises. A simple computation shows that in this case σ R (t) = √ 2 σ(t). The deterministic flow in this case is area preserving. More generally linear Hamiltonian flows exhibit the same equivalence asymptotically.

Quadratic Hamiltonians

We consider the general case of a quadratic Hamiltonian H(x) = 1 2 x • Bx where x ∈ R 2d for an arbitrary number of degrees of freedom d and B is a symmetric matrix. The equations of motion and the flow are

dx dt = A ≡ JB J = 0 I -I 0 S t x 0 = e At x 0 60)
where the matrix e At is symplectic namely e tA J e tA T = J. When the system is stochastically perturbed the equations of motion and the stochastic flow become

dx dt = Ax+ ξ(t) S , t x 0 = e At x 0 + χ(t) χ(t) = t 0 e A(t-s) ξ(s) ds (61)
The RE is computed starting from the reversed flow

S , -t S , t x 0 = x 0 +χ R (t) χ R (t) = [e -At χ(t)+χ(-t)] = t 0 e -sA ( ξ(s)-ξ(s-t)) ds
(62) and explicitly reads

σ 2 R (t) = 2 χ R (t) 2 = 2 2 t 0
ds Tr e -s A e -s A T (63)

The forward error σ(t) is defined by

σ 2 (t) = 2 χ(t) 2 = 2 t 0 ds Tr e s A e s A T (64)
It is not hard to prove that σ(t) and σ R (t)/ √ 2 are asymptotically equal taking into account the property of symplectic matrices. Indeed e At and e -At have the same eigenvalues. The fidelity F R (t) for a noisy reversed flow on the torus T 2d is defined by

F R (t) = 1 0 dx 0 f T 2d (x 0 ) f T 2d (S , -t S , t x 0 ) - 1 0 dx 0 f T 2d (x 0 2 (65)
The reversed flow is given by equation ( 62) where the global error χ R (t) is defined. Expanding f T 2d (x) in a Fourier series we find

F R (t) = (k,k ) =(0,0) f -k f k 1 0 dx 0 1 • • • 1 0 dx 0 2d e -2π i (k -k)•x 0 e 2π i k χ R (t) = = (k,k ) =(0,0) f -k f k e -4π 2 k•Σ 2 (t) k (66) where Σ 2 (t) = 2 t 0 ds e -As e -A T s (67)
We recall that the square of the reversibility error is expressed by

σ 2 R (t) = 2 Tr (Σ 2 (t)) ( 68
)
If all the eigenvalues are complex of unit modulus then e sA is conjugated to an orthogonal matrix. If e sA is orthogonal σ 2 (t) = 2d 2 t supposing the phase space is R 2d . If A has real eigenvalues ±λ i ordered in an increasing sequence -λ, -λ 2 , . . . , λ 2 , λ then e tA is conjugated to the diagonal matrix e tΛ = diag (e λt , e λ 2 t , . . . , e -λ 2 t , e -λt ). Supposing for simplicity that A = Λ is diagonal (in the general case only trivial constant factors appear) we obtain that asymptotically

σ(t) e λt σ R (t) √ 2 e λt k • Σ 2 R (t)k = k 2 1 2 e 2λt (69)
Also in this more general case the asymptotic decay rate of fidelity is related to the asymptotic growth of the error.

Models

We give some examples of application of the previous results. To this end we consider as integrable system the anharmonic oscillator and as non integrable system the Hénon Heiles Hamiltonian. The Hamilton's equations are integrated by using symplectic 4th order integrators, based on the splitting method. Finally we consider the system of vortices for which explicit symplectic integrators are not available: we use then a Runge-Kutta integrator. The restricted 3 body problem has been discussed elsewhere [12] just as native discrete systems expressed by symplectic maps [14]. Denoting the time step by ∆t = T /n s , where T is a characteristic time of the system, and using fourth order methods, we insure that the local integration error is close to the round off choosing n s = 10 4 . In addition symplectic integrators insure that the error on the invariants has a null average growth. We consider also the Lyapunov error defined as the distance at time t from the reference orbit of an orbit whose initial point has an error of amplitude . We have checked that for integrable systems the forward error due to a stochastic perturbation of amplitude grows as σ(t) ∼ t 3/2 just as the reversibility error up to a factor √ 2. However when the system is isochronous the growth is σ(t) ∼ t 1/2 . We have checked that decay of correlations and fidelity follows an exponential law according to C(t) ∼ F (t) ∼ e -2π 2 σ 2 (t) . For non integrable systems in the regions of regular motion, the error growth follows a power law with exponent 3/2, whereas in the chaotic regions the growth is exponential σ(t) ∼ e λt , where λ is the maximum Lyapunov exponent.

The anharmonic oscillator

The Hamiltonian is given by

H = ω 2 (x 2 + p 2 ) + η 4 x 4 -pξ(t) = ω + η  2 cos 4 θ + (2) 1/2 sin θ ξ(t) (70)
The transformation to new action angle variables (Θ, J) allows to render H integrable up to a remainder of order λ 2 and neglecting this remainder as well as terms of order λ the Hamiltonian reads

H = ωJ + η 3 8 J 2 ω 2 + (2J) 1/2 sin Θ ξ(t) (71)
In these new coordinates the noise become multiplicative. A much simpler model to solve analytically is the one in which the noise is additive

H = H 0 (J) -Θ e J ξ J (t) + J e Θ ξ Θ (t) H 0 (J) = ωJ + η 3 8 J 2 ω 2 (72)
where e = (e Θ , e J ) is a unit vector and ξ Θ , ξ J (t) are independent white noises.

In this case the equations of motion can be solved explicitly by retaining only the first order in and for initial conditions

J(0) = J 0 , Θ(0) = Θ 0 read Θ = Θ 0 + Ω(J 0 )t + Ω (J 0 ) e J w (1)
J (t) + e θ w θ (t) J(t) = J 0 + e J w J (t) (73)
where Ω = dH 0 /dJ and Ω = dΩ/dJ . As long as the distance on the cylinder agrees with the Euclidean distance we can write for the forward error The figure shows the result of a fit to the exponent α for power law growth of the forward error σ(t) = C t α as a function of the anisochronicity strength η. The function α(η) varies between 1/2 and 3/2 with a rather sharp transition in agreement with theoretical estimate (74).

σ(t) = t + e 2 J Ω 2 (J 0 ) t 3 /3 1/2 (74)

The Hénon Heiles Hamiltonian

This model describes the motion of an elliptical galaxy and the Hamiltonian reads

H ± = 1 2 (p 2 x + p 2 y ) + 1 2 (x 2 + y 2 ) + x 2 y ± 1 3 y 3 (75)
The Hamiltonian H -is not integrable, whereas the model H + is integrable. For the first one we have considered the reversibility error due to the round off. In Figure 2 we compare the colour plot for the reversibility error with the phase portrait. Both refer to the Poincaré section y = 0, ẏ > 0

The N vortex model

For inviscid plane fluid the dynamics of point vortices provides a complete description of the dynamic evolution. The Hamiltonian for N point vortices located at The coordinates x k and y k are canonically conjugated. There are two independent first integrals in involution. As a consequence the system of 3 vortices is integrable, while a system of 4 is not . If the vorticity of one of them is very low, a reduced Hamiltonian can be written which looks similar to the restricted planar three body problem. Introducing a canonical change of coordinates the Hamiltonian reads

z k = x k + iy k with a strength Γ k is H(x) = - 1 4π j =k Γ j Γ k log |z j -z k | (76)
H = - 1 2 (x 2 + p 2 ) + (1 -λ) log[(x + λ) 2 + p 2 ] 1/2 + λ log[(x + λ -1) 2 + p 2 ] 1/2 (77)
We remark that the splitting cannot be used to construct an explicit symplectic integrator. Only symplectic implicit schemes are available. However one can use a 4-th order Runge-Kutta integrator choosing n s = 10 4 such that the Hamiltonian is conserved within an error very small with respect to the stochastic perturbation. In Figure 3 we show level curves of the Hamiltonian (77). The behaviour of the forward and reversibility error follows the power law with exponent 3/2 in agreement with the theory.

The observational noise

To complete the present analysis on the effect of noise on a dynamical system we consider now the effect of observations. We suppose they are made regularly with a time interval T . We assume our system to be regular and to be given by the translation on the torus T and that each observation, having a small duration τ T , behaves as a white noise of amplitude . The correlation and the fidelity for an observable f T (x) during the first measurement T < t < T + τ decay. After this time interval the correlation oscillates, whereas the fidelity remains constant, namely for t > T + τ is

C (t) = k =0 |f k | 2 e 2π i ω t e -2π 2 2 τ F (t) = k =0 |f k | 2 e -2π 2 2 τ (78)
If we repeat measurements of the same duration τ and same time separation T , after n of them, namely for t ≥ n(T + τ ), the decrease factor e -2π 2 2 nτ becomes significant if n 2 τ > 1. The threshold t * to observe the exponential decrease is

t * ∼ T 2 τ (79)

Map formulation

If the disturbance introduced by the measurement is proportional to the signal itself, then we must treat it as a multiplicative noise. In the continuous time case analytical estimates are difficult to obtain also for translations on the torus. To overcome the difficulty we consider the case of a discrete time system. When the noise is additive the results are similar to the continuous time case. We consider a map M which gives the evolution on the same time interval T which separates two subsequent observations. Letting x n+1 = M (x n ) we have

x n = x n-1 + ωT + τ 1/2 ξ n mod 1 ( 80
)
where ξ n are random variables uniformly distributed in [-1, 1]. The correlation after n = 1 iteration (observation) is

C (1) = 1 0 dx 0 f T (x 0 ) f T (x 0 + ωT + τ 1/2 ξ) = = k,k =(0,0) f k f -k e 2π i k ωT 1 2 1 -1 dξ e 2π i k τ 1/2 ξ × 1 0 dx 0 e 2π(k -k)x 0 = = k = |f k | 2 e 2π i k ωT S( τ 1/2 k) S(x) = sin(2πx) 2πx (81)
After n iterations (observations) the correlation becomes

C (n) = k = |f k | 2 e 2π i knωT S n ( τ 1/2 k) (82)
The same result holds for the fidelity where the phase factor is absent. We recall that |S(x)| ≤ e -x 2 log(2π) for |x| ≤ 1 and |S(x)| ≤ (2π|x|) -1 for |x| > 1, see reference [13], and follow the same procedure outlined there. Assuming τ 1/2 1 and that the Fourier coefficients are bounded by |f k | ≤ A/|k|, we have

|C(n)| ≤ 2A 2 e -n τ 2 log(2π) (83)

Multiplicative noise

If the random perturbation introduced by an observation is proportional to the signal effect and we make a single observation after N iterations the map (82) is replaced by

x n = (x n-1 + ω T )(1 + τ 1/2 ξ n ) mod 1 (84)
After the first iteration the correlation reads

C (1) = k,k =(0,0) f k f -k e 2π i kωT 1 2 1 -1 dξ e 2π i k ωT τ 1/2 ξ 1 0 dx 0 e 2π i (k(1+ τ 1/2 ξ)-k )x 0 = = k,k =(0,0) f k f -k (-1) k-k e 2π i kωT 1 2 1 -1 dξ S k -k + k τ 1/2 ξ 2 • • exp 2π i k τ 1/2 ξ ωT + 1 2 (85) If f T (x) is a trigonometric polynomial, namely f k = 0 for |k| ≥ K and τ 1/2 K 1 then S((k -k -k τ 1/2 ξ)/2) ∼ δ k,k so that C (1) = k =0 |f k | 2 e 2π i knωT S k τ 1/2 ωT + 1 2 (86)
Supposing that τ 1/2 ωT 1 a single observation produces a significant decrease in the correlation roughly proportional to (τ 1/2 ωT ) -1 . On the contrary if τ 1/2 ωT 1 the decrease is negligible and we need to make several observations. After n observations we have

C (n) = k =0
|f k | 2 (-1) k-k e 2π i knωT S n k τ 1/2 ωT + 1 2 (87)

Supposing that τ 1/2 1 and ωT 1 we have the following estimates

C (n) ≤ 3A 2      exp -log(2π) nτ 2 (ωT ) 2 if τ 1/2 ωT 1 (2π τ 1/2 ωT ) -n if τ 1/2 ωT 1 (88)
To conclude, if the stochastic perturbation, introduced by an observation, is multiplicative rather than additive, a phase factor ωT is introduced in the correlation decay. Since the perturbation amplitude and duration are assumed to be small (τ 1/2 1), if the phase factor is large (ωT 1), the decay is much faster. If τ 1/2 ωT ∼ 1, a few observations determine a significant decay of correlations in the multiplicative case.

Conclusions

We have examined the asymptotic behaviour of the forward and reversibility errors (FE, RE) introduced by stochastic perturbations in a dynamical system defined on a compact phase space and we have compared these errors with the decay rates of correlation and fidelity. We have first considered the translations on the torus T and the cylinder, showing that both errors follow the same power law with an exponent equal to 1/2 and 3/2 respectively. For expanding flows on T the FE grows exponentially, but the equivalence of FE and RE occurs only if we have a linear flow on the torus T 2 which preserves the area. This is a special case of linear Hamiltonian flows for which the asymptotic equivalence of FE and RE is proved. We have analysed the relation between the memory loss and the error growth. For linear Hamiltonian systems we have shown that the correlation and fidelity decay as e -2π 2 σ 2 (t) where σ(t) is the FE and that a similar relation holds between the fidelity and the RE. The RE method can be used to investigate the transition regions from regular to chaotic motions and the effect of round off errors in numerical simulations. The proposed framework is general and an extension to classical fields, by using the theory of linear stochastic partial differential equations to define the FE and RE, could be considered. Finally we have examined the effect of the observational noise showing that for an integrable system only a sequence of observations can lead to the decay of the correlation and that the multiplicative noise can increase the decay rate if the phase advance between two observations is large. and the following equality can be proved

1 0 f T (x, n) f T (x) dx = q n -1 0 1 0 1 q n f T x + j q n f T (x) = = q n -1 j=0
(j+1)/q n j/q n dx 0 f T (x 0 ) f T (q n x 0 -j) = 1 0 dx 0 f T (x 0 ) f T (q n x 0 ) (A5)

where we have made the coordinates change x + j = q n x 0 . Letting f k be the Fourier coefficients of f T (x) the order k Fourier coefficients of f T (x, n) are f k (n) = f q n k . The equality (A4) is a consequence of continuity of the map M on T. For a linear flow S t (x 0 ) = e βt x 0 we have seen that a similar equality, given by equation ( 22), holds only for the sequence of values of t m such that e βtm = m is an integer, since the map S tm (x 0 ) = m x 0 mod 1 is continuous on T.

Figure 1 .

 1 Figure 1. Plot of the exponent α for the fit σ(t) = C n α to the forward error of the anharmonic oscillator with a stochastic perturbation of amplitude as a function of the nonlinearity strength η. The noise is additive in Cartesian coordinates, according to equation (70). The red line is drawn just to guide the eye. When η 1 the system is almost isochronous and α = 1/2, when η 1 the system is strongly anisochronous and α = 3/2, but in the intermediate region the exponent varies continuously between these two values.

Figure 2 .Figure 3 .

 23 Figure 2. Hénon-Heiles non integrable Hamiltonian H -given by equation (75). Left side: colour plot of the reversibility error due to round off for H -= 1/8 and initial conditions on a grid of points in the (x, p x ), with y = 0, p y > 0 and t = 100 dynamic periods T = 2π. Right figure: phase space plot for the projection into the (x, p x ) phase plane of the orbits in the Poincaré section y = 0, p y > 0

Acknowledgements

GT and SV would like to thank the Newton Institute of Cambridge for the kind hospitality during the semester Mathematics for Planet Earth, during which this paper was initiated. SV was supported by the Leverhulme Trust thorough the Network Grant IN-2014-021 and by the MATH AM-Sud Project "Physeco". SV also thanks the Department of Physics of the University of Bologna for the kind hospitality during the completion of this work.

Appendix A. Correlation on the torus and Perron Frobenius operator

Given a map M (x) continuous on the torus there is a unique definition of correlation. Indeed the evolution of a given initial distribution f T (x) is governed by the Perron Frobenius equation

where P is the Perron-Frobenius operator. Supposing M (x 0 ) has q preimages, with x j = (M -n ) j (x) for 1 ≤ j ≤ q n we have indicated all the preimages of M n (x 0 ). For any A ⊂ T 1 the invariance of the probability measure µ is expressed by µ(M -1 (A)) = µ(A).

For example the Bernoulli map M (x) = qx mod 1 where q ∈ N has q inverses and M -1 ≡ ∪ k M -1 k (A) where µ(M -1 k (A)) = q -1 µ(A). Since the density of the absolutely continuous invariant measure is 1, we define the correlation according to

where the result has been obtained by interchanging the integration order. As a specific example we consider the linear map M (x) = qx mod 1 continuous on T for integer q. In this case the preimages x j of M n are

where j i = 0, 1, . . . , q -1 and j = j 1 + qj 2 + . . . + q n-2 j n-1 + q n-1 j n has a range from 0 to q n -1. As a consequence, from the Perron-Frobenius equation we have

q -n f T (q -n (x + j) (A4)


J Pety

J R Goicoechea

M Gerin

P Hily-Blant

D Teyssier

E Roueff

E Habart

A Abergel

THE HORSEHEAD MANE: TOWARDS AN OBSERVATIONAL BENCHMARK FOR CHEMICAL MODELS

1 The intrinsic complexity of chemical models Photodissociation region (PDR) models are used to understand the evolution of the FUV illuminated matter both in our Galaxy and in external galaxies. To prepare for the unprecedented spatial and spectroscopic capabilities of ALMA and Herschel, two different kinds of progresses are currently taking place in the field. First, numerical models describing the chemistry of a molecular cloud are being benchmarked against each other to ensure that all models agree not only qualitatively but also quantitatively on at least simple cases1 (Röllig et al. 2007). Second, new or improved chemical rates are being calculated/measured by several theoretical and experimental groups. However, the difficulty of this last effort implies that only a few reactions (among the thousand ones used in chemical networks) can be thoroughly studied. New numerical tools are thus being developed for taking into account the impact of the uncertainties of the chemical rates on the chemical model predictions, and their comparison with observed abundances (e.g. Wakelam et al. 2005Wakelam et al. , 2006)). In view of the intrinsic complexity of building reliable chemical networks and models, there is an obvious need of well-defined observations that can serve as basic references. PDRs are particularly well suited to serve as references because they make the link between diffuse and dark clouds, thus enabling to probe a large variety of physical and chemical processes.

The Horsehead edge as a chemical laboratory

The illuminated edge (PDR) of the western condensation presents one of the sharpest infrared filament (width: 10 ′′ or 0.02 pc) detected in our Galaxy by ISOCAM. The most straightforward explanation given by Abergel et al. (2003) is that most of the dense material is within a flat structure viewed edge-on and illuminated in the plane of the sky by σOri. The H 2 fluorescent emission observed by Habart et al. (2005) is even sharper (width: 5 ′′ ), implying the inclination of the PDR on the plane-of-sky to be Fig. 1. Emission maps obtained with the IRAM Plateau de Bure Interferometer or 30m single-dish, except for the H2 v=1-0 S(1) emission observed with the NTT/SOFI and the PAH mid-IR emission observed with ISO-LW2. The maps have been rotated by 14 • counterclockwise around the image center to bring the exciting star direction in the horizontal direction as this eases the comparison of the PDR tracer stratifications. Maps have also been horizontally shifted by 20 ′′ (compared to the projection center: RA = 05h40m54.27s, Dec = -02 • 28'00 ′′ , J2000) to set the horizontal zero at the PDR edge delineated as the vertical red line. Either the synthesized beam or the single dish beam is plotted in the bottom left corner. The emission of all the lines is integrated between 10.1 and 11.1 km s -1 . Values of contour level are shown on each image lookup table (contours of the H2 image have been computed on an image smoothed to 5 ′′ resolution). The green crosses display the positions where we derived column densities and abundances (see Table 1).

less than 5 • . The Horsehead ridge thus offers the opportunity to study at small linear scales (1 ′′ corresponds to 0.002 pc at 400 pc) the physics and chemistry of a PDR with a simple geometry, very close to the prototypical kind of source needed to serve as a reference to chemical models. Since 2001, we started to study the Horsehead PDR mainly with the IRAM Plateau de Bure interferometer at 3 mm and the IRAM-30m at 1 mm achieving spatial resolutions from 3 to 11 ′′ (except the H 2 fluorescent emission observed at 1 ′′ -resolution with the NTT/SOFI instrument). Figure 1 displays all the observed, high resolution maps already acquired. Those maps trace the different layered structures predicted by photochemical models according to chemical reaction networks, excitation conditions and radiative transfer. Abergel et al. (2003) deduced from the distance between σOri and the PDR that the intensity of the incident far UV radiation field is χ ∼ 60 relative to the interstellar radiation field in Draine's units. Through the modelling of the H 2 and CO emission, Habart et al. (2005) showed that the PDR has a very steep density gradient, rising to n H ∼ 10 5 cm -3 in less than 10 ′′ (i.e. 0.02 pc), at a roughly constant pressure of (2.0 ± 1.0) 10 13 3.7 10 -10 CS (+4, 0) (+23.9, -1.0) 10 (8.1 ± 1.0) 10 13 2.5 10 -9 C 34 S 16 (3.7 ± 0.5) 10 12 1.2 10 -10 HCS + 29 (6.8 ± 0.5) 10 11 2.0 10 -11 H 13 CO + (+19.7, +20.4) (+44.0, +15.0) 6.7 × 4.4 (0.5 -1) 10 13 (0.5 -1) 10 -10 DCO + 12 (0.5 -1) 10 13 (0.5 -1) 10 -10 CS (+21, +15) (+44.0, +9.5) 10 (1.2 ± 1.0) 10 14 3.9 10 -9 C 34 S 16 (5.3 ± 0.5) 10 12 1.8 10 -10 HCS + 29 (6.8 ± 0.5) 10 11 2.3 10 -11 CS (+36, +25) (+61.0, +15.5) 10 (1.7 ± 1.0) 10 14 6.0 10 -9 C 34 S 16 (7.9 ± 0.5) 10 12 2.5 10 -10 HCS + 29 (9.0 ± 0.5) 10 11 3.8 10 -11

Table 1. Column densities and abundances of several chemical species from the UVilluminated PDR (upper part) to the shielded region (lower part) of the Horsehead edge.

Coordinates offsets are given both in the Equatorial system from RA = 05h40m54.27s, Dec = -02 • 28'00 ′′ (J2000) and in the coordinate system adapted to the source geometry and used in Figure 1. Abundances are computed with respect to the density of protons (nH).

P ∼ 4 × 10 6 K cm -3 . These observations were followed by a chemical study of small hydrocarbons (CCH, c-C 3 H 2 , C 4 H). Pety et al. (2005) showed that the abundances of the hydrocarbons are higher than the predictions based on pure gas phase chemical models (Le Petit et al. 2002 and reference therein). These results could be explained either by the photoerosion of the large aromatic molecules and/or small carbon grains (C. Joblin, 2007, this volume) or by a turbulent mixing which would transport in the illuminated part of the PDR molecules after their productions in the dark part (M. Gerin, 2007, this volume). Goicoechea et al. (2006) linked the PDR model predictions with detailed non-LTE, nonlocal excitation and radiative transfer models adapted to the Horsehead geometry. They showed that the gas sulfur depletion invoked to account for CS and HCS + abundances is orders of magnitude lower than in previous studies of the sulfur chemistry. Finally, Pety et al. (2007) studied the deuterium fractionation in the Horsehead edge from observations of several H 13 CO + and DCO + lines. A large [DCO + ]/[HCO + ] abundance ratio (≥ 0.02) is inferred at the DCO + emission peak, a condensation shielded from the illuminating far-UV radiation field where the gas must be cold (10-20 K) and dense (n H ≥ 4 × 10 5 cm -3 ). DCO + is not detected in the warmer photodissociation front, implying a lower [DCO + ]/[HCO + ] ratio (< 10 -3 ). To our knowledge, this is the brightest DCO + emission (4 K) detected in an interstellar cloud close (angular distance < 40 ′′ ) to a bright H 2 /PAH emitting region. This opens the interesting possibility to probe at high resolution the chemical transition from far-UV photodominated gas to "dark cloud" shielded gas in a small field of view.

3 Towards an observational benchmark An ideal observational benchmark would deliver to chemists a set of abundances (with the associated uncertainties) as a function of the distance (or extinction) to the illuminating star. This goal is difficult to achieve for several reasons: 1) The geometry of the source is never as simple as wished when it is known at all; 2) The spectra produced by the instruments must be inverted to obtain abundances; 3) The spectra are often measured at very different angular resolutions, implying beam dilution and/or mixing of different gas components. For several years now, we have started to systematically study the western edge of the Horsehead nebula because its geometry is not only well understood but also quite simple (almost 1D and viewed edge-on). The density profile across the PDR is well constrained and there are several current efforts to constrain the thermal profile. The combination of low distance to Earth (400 pc), low illumination (χ ∼ 60) and high density (n ∼ 10 5 cm -3 ) implies that all the interesting physical and chemical processes can be probed in a field-of-view of less than 50 ′′ with typical spatial scales ranging between 1 and 10 ′′ .

All those observations are done by the same team, using the same instruments (mainly IRAM Plateau de Bure and 30m) and the same methods both of data reduction and data analysis. For each species, we are trying to observe several transitions at similar angular resolutions (from 5 to 15 ′′ ) to constrain properly the excitation conditions and derive accurate column densities and abundances. Obtaining emission maps when possible has proved essential to understand the spatial distributions of the species. We are preparing the public release of cuts of spectra so that the radiative transfer analysis can be refined as knowledge of collisional rates progresses. In the meantime, Table 1 is our first attempt at quantitatively summarizing the results obtained up to now. In the future, the zoom capacity and resolving power of ALMA will enable to measure many different specie transitions at a resolution of 1 ′′ , enabling to resolve all the physical and chemical gradients.

IRAM & Observatoire de Paris, France

LERMA-LRA, France

IRAM & LAOG, France

European Space Astronomy Centre, Spain

LUTH, France

IAS, France 1 See also http://www.ph1.uni-koeln.de/pdr-comparison/intro1.htm c Observatoire de Paris & Université de Cergy-Pontoise


Alexandre Govin

Wolfram Schmidt

Marie-Claude Bartholin

Philippe Grosseau

Combination of guar gum derivatives

Keywords: Admixtures, rheology, polysaccharides, guar derivatives, superplasticizers. 1

come

Introduction

In order to obtain specific properties from the fresh state to the hardened material, the formulations of cementitious materials become more and more complex due to combining several groups of admixtures. For many years, cementitious materials with high fluidity have been developed, thanks to their economic and technical advantages. For this purpose, high-range-water-reducing (HRWR) admixtures, also known as superplasticizers, are commonly introduced in the formulations of cementitious materials. These admixtures can dramatically reduce the amount of mixing water, while maintaining or increasing workability at identical or similar performance (1), [START_REF] Yamada | Effects of the chemical structure on the properties of polycarboxylate-type superplasticizer[END_REF], [START_REF] Burgos-Montes | Compatibility between superplasticizer admixtures and cements with mineral additions[END_REF]. Today, polycarboxylate ethers (PCE) are commonly used as superplasticizers to disperse cement particles in concretes and mortars [START_REF] Ramachandran | Concrete admixtures handbook[END_REF]. Through the anionic charges of the backbone chain, the PCEs can be adsorbed on surfaces of cement phases and hydrates. Through the side chains and the main chain they induce steric repulsion forces between the particles, leading to improved dispersion (2), [START_REF] Plank | Impact of zeta potential of early cement hydration phases on superplasticizer adsorption[END_REF]. However, inhomogeneities such as segregation or bleeding may occur for highly flowable mixtures with negative effects on the durability. To overcome this adverse effect, viscosity-enhancing admixtures (VEA) or stabilizing agents (STA) can be introduced into the formulation ( 6), [START_REF] Khayat | Effect of Welan Gum-High-Range Water Reducer Combinations on Rheology of Cement Grout[END_REF], [START_REF] Lachemi | Self-consolidating concrete incorporating new viscosity modifying admixtures[END_REF], [START_REF] Sonebi | Rheological properties of grouts with viscosity modifying agents as diutan gum and welan gum incorporating pulverised fly ash[END_REF]. Natural polysaccharides or their derivatives, such as sphingan gums, starch derivatives, cellulose ethers or guar gum derivatives, are frequently used as STA. These admixtures induce modifications of the rheological behavior of the cementitious materials by usually increasing the yield stress and the apparent viscosity and by supplying shear thinning behavior [START_REF] Paiva | Effects of a water-retaining agent on the rheological behaviour of a single-coat render mortar[END_REF] [START_REF] Khayat | Viscosity-enhancing admixtures for cement-based materials -An overview[END_REF].

It is worth noting that the yield stress and the viscosity are very important properties for the flow of concrete, since the both affect its workability and casting properties. The viscosity is an important parameter during casting and application of concrete since it influences the filling velocity, the resistance to flow and the segregation stability during the flow process. The yield stress, which is responsible of the stability at rest (no segregation) and the maintaining of the shape at rest, is more linked to the properties of flowable concrete after casting. The rheological parameters of concrete are generally dependent of the solid volume fraction and the rheological properties of the fluid surrounding the aggregates, i.e. the cement paste.

The consideration of environmental concerns, the reduction of greenhouse gas emissions and energy consumption lead the cement and concrete industry to develop more and more complex formulations. Consequently, a greater control of the formulation and a better understanding of the effects of all admixtures in a formulation are necessary. Indeed, an understanding of the mechanism of action of the admixtures is a way to improve the selection of admixtures in relationship with the practical applications. However, most of the academic studies focused on the effect of individual admixture, despite competitive adsorption which can occur inhibiting the intended effect [START_REF] Plank | Competitive adsorption between superplasticizer and retarder molecules on mineral binder surface[END_REF] or positive interaction which can promote the efficiency of the polymers [START_REF] Lummer | Combination of lignosulfonate and AMPS®-co-NNDMA water retention agent-An example for dual synergistic interaction between admixtures in cement[END_REF], [START_REF] Bessaies-Bey | Organic admixtures and cement particles: Competitive adsorption and its macroscopic rheological consequences[END_REF].

The objectives of this paper are to propose a method allowing the quantification of the specific adsorption of hydroxypropyl guar and polycarboxylate-based superplasticizers when they are introduced together in the mix, and to improve the understanding of synergetic or competitive effects between these two admixtures on the rheological aspect.

Materials and methods

Mineral products

The cement used for this study was an ordinary Portland cement, referenced as CEM I 42.5 R (CEMEX) according to the European standard EN 197-1. The chemical composition was determined by means of Xray fluorescence spectroscopy and the resulting clinker composition according to the modified Bogue calculation according to Taylor [START_REF] Taylor | Modification of the Bogue calculation[END_REF] is given in Table 1.

Organic admixtures

Hydroxypropyl guars (HPGs) are obtained from the original guar gum via a nucleophilic substitution, using propylene oxide in the presence of an alkaline catalyst (Figure 1). Guar gum is a natural polysaccharide extracted from the endospermic seed of Cyamopsis tetragonolobus. The guar gum is considered as a non-ionic polymer in water. The guar gum exhibits a high molecular weight and hydrophilic properties. The structure of the guar consists of a D-mannopyranose backbone with random branchpoints of galactose. This branched-chain structure and the high amount of hydroxyl groups lead to the guar gum a high chemical reactivity and an easy dissolution in cold water. The HPGs manufacturing hardly impacts the environment, since the extraction of the guar gum requires only a simple thermo-mechanical process and the chemical modification of the original guar gum requires normal reaction conditions of temperature and pressure, which does not generate large quantity of by-products, and requires minimal purification procedure [START_REF] Biasotti | Hydraulic Binder Composition[END_REF]. In this paper, two different HPG, provided by Lamberti S.p.A., were studied. They are hereafter referred to HPG1 and HPG3. The only difference between the both HPGs is the molar substitution ratio, which increases from HPG 1 to HPG 3. The molar substitution ratio (MSHP) represents the number of hydroxypropyl units per hexose unit (mannose or galactose), which is, according to the manufacturer, less than 3 for the investigated HPGs. Since the two different HPGs are originated from the same original guar gum, they exhibits roughly the same molecular weight, around 2.10 6 g.mol -1 [START_REF] Poinot | Use of ultrasonic degradation to study the molecular weight influence of polymeric admixtures for mortars[END_REF]. These HPGs were chosen because they are able to adsorb on the cement particles [START_REF] Govin | Modification of water retention and rheological properties of fresh state cement-based mortars by guar gum derivatives[END_REF].

Two polycarboxylate-based superplasticizers (PCE) were studied (Glenium Master Sky 591 called PCELC and Glenium Master Sky 595 called PCEHC). Since these two polymers are commercial products, no details of the chemical composition were given by the manufacturer (BASF). A qualitative description is given in Table 2. These superplasticizers were chosen because of their difference in the charge density, since it is one of the major influencing parameters for the performance of superplasticizers [START_REF] Plank | Competitive adsorption between superplasticizer and retarder molecules on mineral binder surface[END_REF]. For the following, LC and HC mean low charge and high charge density, respectively.

Preparation of fresh pastes

The investigations were carried out on pastes from cement and water. The water-to-cement ratio (W/C) was kept constant at 0.4. Deionized water was used for all the experiments. The hydroxypropyl guar was dry-blended with the cement, hand-premixed during one minute, before adding liquid. When superplasticizers were used, they were added into the dry mix with the water. The amount of water contained into the PCE solution was deduced from the mixing water in order to keep constant the liquidto-solid ratio. The mixing was made in a blender in accordance to EN 196-1. All the dosages in admixture were expressed in percent by weight of cement (% bwoc). The studied dosages were 0.1, 0.2 and 0.3% bwoc and 0.1% bwoc for PCEs and HPGs, respectively.

All tests were carried out at a controlled temperature of 20 ± 2 °C since rheological behavior of the cement paste and adsorption measurements are temperature-dependent. A reference test was also performed with cement without admixture.

Rheological behavior

The rheological measurements were performed with a Rheometer Viskomat NT (Schleibinger Geräte), in a room thermostated at 20 °C. The rheological properties of fresh pastes were investigated with a double gap cell and a basket probe composed of a structured grid (Vogel cell). The gap thickness was about 6 mm.

The paste was introduced into the measurement system at the end of the mixing cycle and was then held at rest. At 9 min the experiment was started. The mortar was first pre-sheared for 1 min at 100 s -1 in order to re-homogenize the sample and to eliminate its shear history because of potential thixotropic character of cementitious materials [START_REF] Phan | Influence of organic admixtures on the rheological behaviour of cement pastes[END_REF], [START_REF] Roussel | The origins of thixotropy of fresh cement pastes[END_REF]. After a period of rest of 5 min, the rheological measurements were started. The imposed shear profile consisted of 14 decreasing steps from 100 s -1 to 0.06 s -1 . The measuring time was adjusted for each shear rate, in order to obtain a steady state.

The shear stress () was expressed as a function of the shear rate (  ) and the Herschel-Bulkley model was applied to fit the experimental data and used to describe cement rheological behavior [START_REF] Herschel | Measurements of consistency as applied to rubber-benzene solutions[END_REF]:

n K      0 (1)
where 0 corresponds to the yield stress, K to the consistency coefficient and n to the fluidity index which characterizes shear-thinning or shear thickening behavior of the paste. In addition, the residual viscosity, as defined by Hot [START_REF] Hot | Adsorbing polymers and viscosity of cement pastes[END_REF], was determined.

      0 ) ( cos   app ity vis residual (2)
Where µapp is the apparent viscosity (=/), 0 is the yield stress.

The residual viscosity does not take into account the contribution of the energy needed to overcome the attractive interaction colloidal forces and to destroy the inter-particle network, characterized by the yield stress [START_REF] Hot | Adsorbing polymers and viscosity of cement pastes[END_REF].

Adsorption measurements

The adsorption isotherms were determined using the depletion method. The non-adsorbed polymers remaining within the pore solution was quantified. The pore solution was extracted from the cement pastes by means of centrifugation, 15 min after contact between water and cement. Around 150 g of cement paste were centrifuged at 5000 rpm for 5 min. Afterwards, the supernatant was centrifuged again at 14500 rpm for 10 min. It is worth noting that the amount of missing polymer into the pore solution was called "adsorbed" for the rest of the paper, despite that a small part of polymer could be intercalated into hydrates due to early chemical interactions [START_REF] Flatt | A simplified view on chemical effects perturbing the action of superplasticizers[END_REF]. Moreover, adsorption does not mean that the polymer becomes active in dispersing due to possible multilayer adsorption.

Total Organic Carbone (TOC) measurements:

The supernatant was diluted with hydrochloric acid solution at 0.1 mol.L -1 . The Total Organic Carbone (TOC) was determined by combustion at 850 °C with a Vario-TOC Cube (Elementar). The device was calibrated thanks to reference solutions of HPG and

Size Exclusion Chromatography (SEC) measurements:

Since the TOC measurements give the global amount of carbon in the solution, this method does not allow to quantify the amount of each polymer separately. An original method, involving the Total Organic Carbone (TOC) measurements and the Size Exclusion Chromatography analysis (SEC) was thus developed to achieve this objective. The SEC analysis was used to separate the two polymers (superplastisizer and polysaccharide) and to quantify the amount of PCE only, in the pore solution. After calibrating of the both analysis methods, the subtraction of the SEC results to the TOC results gives the amount of polysaccharide in the pore solution.

Size Exclusion Chromatography analyses were carried out on a Waters apparatus equipped with a pump Waters 916. The eluent was a 0.5 mol.L -1 NaCl solution, on-line degassed. The flow rate was set to 0.5 mL.min -1 . A column Shodex OHpak SB-806M-HQ was used and kept at 50 °C in an oven. Detection was achieved thanks to a refractometer-type detector Waters 2410.

Results and discussion

Adsorption measurements

It was first checked if the SEC method is a suitable method to quantify the adsorption of PCE. For this purpose, adsorption measurements were carried out without polysaccharide, and the concentration in PCE, remaining into the pore solution, was determined by the both analysis methods. The concentrations obtained by Size Exclusion Chromatography (SEC) analysis were compared to those obtained by Total Organic Carbone (TOC) measurements (Figure 2). The results show a good correlation between the both methods (slope very close to 1 and correlation coefficient higher than 0.99), validating the suitability of the Size Exclusion Chromatography method for quantifying the amount of PCE remaining in the pore solution. The adsorption isotherms obtained by Size Exclusion Chromatography analysis for the both PCEs (high charge and low charge) are shown in Figure 3. The adsorption measurements of PCEs are represented by Langmuir model (solid line). The results show that the adsorption of PCEHC is higher than that of PCELC, which is coherent with the charge density of the two PCEs ( 12), ( 24) and the greater size of PCELC molecules in the pore solution (result not shown here).

Figure 3. Adsorption isotherm of PCELC and PCEHC alone determined by SEC analysis.

The amounts of PCEs adsorbed in presence of a constant amount of HPG1 or HPG3 are shown in Figure 4 and Figure 5. The adsorption isotherms of PCEs, HPG1 and HPG3 alone are also represented for comparison. The adsorption of HPGs (0.1%) is of 0.74 mg/g and 0.53 mg/g for HPG1 and HPG3, respectively, whereas the adsorption of 0.3% of PCEs reaches 0.55 mg/g and 0.82 mg/g for PCELC and PECHC, respectively. However, due to the great difference in molecular weight between HPG and PCEs, more molecules of PCE are adsorbed than HPG. The increase in hydroxypropyl substitutions on the guar (increasing MSHP from HPG1 to HPG3) leads to a decrease in the adsorption of the admixture onto the solid surfaces. This result is consistent with previous studies conducted on HPGs and cellulose ethers (CE) in cementitious materials [START_REF] Brumaud | Cellulose ethers and yield stress of cement pastes[END_REF], [START_REF] Govin | Modification of water retention and rheological properties of fresh state cement-based mortars by guar gum derivatives[END_REF] and can be ascribed to the decrease in the polarity of the polysaccharide and to the increase in the size of the molecules.

The results highlight that both admixtures (PCE and HPG) are able to adsorb even when they are added together in the cement pastes. The combination of the PCE, exhibiting the lowest charge density (PCELC), with HPG1 leads to a modification of the adsorption of the two polymers (Figure 4). Indeed, for the both admixtures, their amounts adsorbed are lower when the two polymers are introduced together in the mix

PCE HC

than when they are introduced separately. By increasing the dosage of PCELC from 0.1% to 0.3% bwoc, the adsorption is reduced from 7% to 5% and 13% to 20% for PCELC and HPG1, respectively. When the charge density of PCE increases (PCEHC), the adsorption of the PCE seems to be unmodified by the presence of the HPG1 (Figure 5 (a)). It appears as if the adsorption of the HPG1 is strongly reduced by the presence of the PCEHC. The drop of adsorption ranges from 37% to 56% when the dosage of PCE increases from 0.1% to 0.3% bwoc. These values correspond approximately to three times those obtained with the low charge density PCE. The combination of HPG3 with PCEHC leads also to a low impact on the adsorption of PCEHC but to a great reduction of the HPG3 adsorption (Figure 5(b)). The adsorption of HPG3 is reduced by 0.36 and 0.46 mg/g, for PCE concentrations of 0.1 and 0.2%, respectively. In comparison, the losses in adsorption are equal to 0.27 and 0.29 mg/g for HPG1. Despite a lower adsorption of HPG3 alone than HPG1 alone, HPG3 appears to be more sensitive to the presence of the PCE. This result tends to indicate that the bonding energy, involved in the adsorption mechanism between the HPG and the surface of cement phases, decreases when the MSHP increases. However, the presence of 0.3% of PCEHC leads to a reduction of approximately 0.45 mg/g in adsorption for the both polysaccharides, seeming to indicate an insufficient available surface for these latter.

All the measurements were realized with a high dosage in PCE (>0.1% bwoc) leading to a high surface coverage of the cement particles. This surface coverage becomes more and more close to the surface saturation when the dosage in PCE increases (Figure 3). In these conditions, competitive adsorption between organic admixtures should occur [START_REF] Bessaies-Bey | Organic admixtures and cement particles: Competitive adsorption and its macroscopic rheological consequences[END_REF]. The results show clearly that the affinity of the polymer with the cement particles is a key factor in the competitive adsorption. The polymer with the highest affinity adsorbs first, leaving the remaining surface available for the second admixture. In the case of increasing dosage in PCEHC, the available surface for the polysaccharide decreases, leading to a decreasing adsorption of HPG. When the charge density of the PCE is lower, the gap of affinity between the PCE and the polysaccharide is reduced, and competitive adsorption occurs.

Impact of admixtures on the rheological properties of fresh pastes

Figure 6 and Figure 7 show the evolution of the relative yield stress, extracted from Herschel-Bulkley model, and the relative residual viscosity for all the studied pastes (with and without admixture). The relative value (yield stress or residual viscosity) corresponds to the ratio of the value obtained for the admixed pastes to the value determined for the non-admixed cement paste. The use of increasing dosage in PCEs leads to continuous decrease in the yield stress and residual viscosity. The charge density of PCE impacts the evolution of the yield stress and the residual viscosity. Indeed, pastes admixed with PCEHC exhibit a lower yield stress and a lower residual viscosity than pastes mixed with PCELC. This result is in agreement with the higher adsorption of PCEHC which improves the steric repulsions and limits the contacts between cement particles. In our case, cement pastes admixed with 0.1% and 0.2% of PCEHC give similar rheological behaviors than cement pastes mixed with 0.2% and 0.3% of PCELC, respectively. Consequently when the charge density of PCE decreases, higher dosage in PCE is required to obtain the same rheological behavior, and thus same slump flow [START_REF] Schmidt | Effects of the Characteristics of High Range Water Reducing Agents and the Water to Powder Ratio on Rheological and Setting Behavior of Self-Consolidating Concrete[END_REF].

The results show also that the presence of 0.1% of HPG (HPG1 or 3) leads to an increase in both yield stress and residual viscosity compared to the non-admixed paste. The results indicate that HPG is able to bridge two cement particles, which results in an increase of the yield stress [START_REF] Govin | Modification of water retention and rheological properties of fresh state cement-based mortars by guar gum derivatives[END_REF]. However, despite a strong drop of the adsorption with the increase in the MSHP (from HPG 1 to 3) the yield stress increases. This suggests that the non-adsorbed polymer may be responsible for the yield stress increase. The rise in the polymer concentration in the pore solution induces an increase in the pore solution viscosity and/or depletion flocculation, which could compensate the loss of bridging. When the PCEs are combined with HPGs, a general decrease in the yield stress and the residual viscosity is observed with increasing dosage in PCE. However, the yield stress is always higher than with PCE alone. In presence of HPG1, the impact of the dosage in PCE depends on the charge density of the PCE. For PCEHC, a continuous decrease in the yield stress is observed, whereas a plateau followed by a decrease is noticed for PCELC. These tendencies are consistent with the adsorption results. The significant reduction in HPG adsorption induced by PCEHC, leads to the progressive replacement of the bridging by the steric repulsions, and consequently, to the drop of the yield stress. In the case of the lowest dosage in PCELC, we showed that the reduction in adsorption of HPG and PCE was low. The reduction in HPG adsorption could be sufficiently low to maintain enough bridging and thus a similar yield stress than with HPG alone. At higher dosage in PCELC, the steric repulsion induced by the superplasticizer adsorbed could be responsible to the decrease in the yield stress.

The combination of HPG with PCELC induces a low modification of the residual viscosity compared to the use of PCELC alone. However, the viscosity of the pore solution is quite similar than that obtained with HPG alone (results not shown here). This result indicates that the modifications of the colloidal and electrostatic forces induced by the PCE are greater than the modification of the hydrodynamic forces induced by the polysaccharide. For highly charged PCE, the increase in the pore solution viscosity induced by the reduction in HPG adsorption can justify the greater residual viscosity observed when the pastes are admixed with the combination of PCE and HPG compared to the PCE alone. Figure 7 shows the impact of HPG3 on the rheological behavior of cement paste admixed with PCEHC.

The results obtained with the combination of HPG1 with PCEHC are also given for comparison. It appears clearly a strong decrease in the yield stress and the residual viscosity for increasing dosage in PCE.

Despite that HPG3 provides a greater initial yield stress than HPG1, the yield stress decreases faster when PCEHC is added. This signifies that the increase in the amount of HPG3 solubilized in the pore solution, resulting from the huge reduction in the adsorption of HPG3, is not sufficient to compensate the loss of bridging, and the steric repulsion induced by the PCE. However, this leads to an increase in the residual viscosity, in comparison with HPG1.

Conclusion

This paper deals with the effect of the combination of PCEs and HPGs on the adsorption and rheological behavior of cement pastes. An original method allowing the quantification of the adsorption of two admixtures was proposed. The results show that the combination of HPG with PCE superplasticizer strongly affects the adsorption of the polymers, more specially, leads to a decrease in the adsorption of the polysaccharide. Moreover, the combination of both admixtures modifies the rheological behavior of cement pastes. Despite the presence of HPG, the relative viscosity of the pastes decreased with increasing dosage in PCE. However, it allows maintaining a significant yield stress in the cement paste compared to PCE alone. The increase in the charge density of the PCE seemed to amplify the drop of the viscosity and to reduce the gain on the yield stress induced by HPG. The MSHP seems also to slightly affect the rheological behavior by leading to a supplementary decrease in the yield stress and a supplementary increase in the viscosity.

Figure 1 .

 1 Figure 1. Molecular structure of HydroxyPropyl Guar.

Figure 2 .

 2 Figure 2. Comparison of SEC and TOC methods for the quantification of PCE adsorption.

Figure 4 .

 4 Figure 4. Adsorption isotherm of HPG1 alone, PCELC alone, HPG1 and PCELC when they are combined.

  (a) Adsorption isotherm of HPG1 alone, PCEHC alone, HPG1and PCEHC when they are combined.(b) Adsorption isotherm of HPG3 alone, PCEHC alone, HPG3 and PCEHC when they are combined.

Figure 5 .

 5 Figure 5. Adsorption isotherm of HPGs alone, PCEHC alone, HPGs combined with PCEHC.

( a )

 a Yield stress as a function of solid content of PCEs, in presence of HPG1. (b) Relative residual viscosity as a function of solid content of PCEs, in presence of HPG1.

Figure 6 .

 6 Figure 6. Effect of PCEs in presence of HPG1 on the rheological behavior of cement pastes.

( a )

 a Yield stress as a function of solid content of PCEHC, in presence of HPG1 or HPG3. (b) Relative residual viscosity as a function of solid content of PCEHC, in presence of HPG1 or HPG3.

Figure 7 .

 7 Figure 7. Effect of HPG1 and HPG3 in presence of PCEHC on the rheological behavior of cement pastes.

Table 1 . Oxide composition (%, weight) of the investigated cement determined by XRF and clinker phase composition according to the modified Bogue calculation.

 1

	Chemical composition (% wt)	Phase composition (% wt)
	Oxides	XRF	Oxides	XRF	Phases	XRD (Rietveld)
	CaO	62,80	Na2O	0,28	C3S	61,8
	SiO2	20,56	K2O	0,95	C2S	20,5
	Al2O3	4,36	TiO2	0,2	C3A	6,2
	Fe2O3	2,27	SO3	3,45	C4AF	7,7
	MgO	2,14	LOI	2,40

Table 2 . Qualitative description of the PCEs used.

 2

		PCELC	PCEHC
	Backbone	Polycarboxylate ether (PCE)
	Side chains	Polyethylene oxide (PEO)
	Side chain length	Medium + low	Medium
	Grafting degree	High	Low
	Backbone charge density	Low	High
	Solid content (%)	30	20

.1% alone PCE LC alone HPG1-0.1% + PCE LC PCE HC alone HPG1-0.1% + PCE HC

	1000						1000
								(%)
	stress (%)	100						viscosity	100
	Relative Yield	1 10	HPG 1-0,1% PCE1 HPG 1-0,1% + PCE1 PCE2 HPG 1-0,1% + PCE2 HPG1-0.1% alone PCE LC alone HPG1-0.1% + PCE LC PCE HC alone HPG1-0.1% + PCE HC				Residual Relative	1 10	HPG 1-0,1% + PCE2 PCE1 HPG 1-0,1% + PCE1 PCE2 HPG 1-0,1% HPG1-0
		0.00%	0.05%	0.10%	0.15%	0.20%	0.25%	0.30%	0.00%	0.05%	0.10%	0.15%	0.20%	0.25%	0.30%
				Solid PCE content (%bwoc)					Solid PCE content (%bwoc)

Acknowledgement

The authors would like to acknowledge Lamberti S.p.A. and BASF for the admixtures they provided.

6.