Jürgen Nuss

Dr Martin Jansen
email: m.jansen@fkf.mpg.de

Synthesis via Disproportionation of Gold upon Interaction with Pnictides as Bases

Keywords: Gold, Ternary pnictides, Alkaline earth metals, Conducting materials, Crystal structures Gold, Ternary pnictides, Alkaline earth metals, Conducting materials, Crystal structures

Gold disproportinates in the presence of the Zintl phases Ba 3 P 2 or Ba 3 As 2 forming BaAuP and BaAuAs, respectively, and BaAu 2 . The air and moisture sensitive ternary compounds crystallize in the ZrBeSi type of structure, an ordered variant of AlB 2 : BaAuP (P6 3 /mmc; a = 440.68(6) pm; c = 899.8(2) pm; R 1 = 0.0237; 183 independent reflections); BaAuAs (P6 3 /mmc; a = 453.53(5) pm; c = 902.7(1) pm; R 1 = 0.0255; 168 independent reflections).

Both compounds, which are accessible as pure phases by reacting BaP, or BaAs, with Au, can be classified as poor metallic conductors.

Introduction

Ascribing to gold certain halogen-like behaviours has become a well established view in solid state chemistry [1]. Impressive factual evidence is provided by the isostructural couples of Ba 8 As 5 Au [2] and Ba 8 P 5 Cl [3], [NMe 4 ]Au and [NMe 4 ]Br [4], or Rb 5 [AuO 2 ]Au 2 [5] and K 5 [AuO 2 ]I 2 [6]. The auride anion acting as an acceptor in a N-H ... Au - hydrogen bond is displaying further halide-like behaviour [7]. Finally, gold and halogens, both can disproportionate in the presence of bases [5,8].

We have tried to use gold as an oxidizing reagent for Zintl phases, in analogy to e.g. iodine which oxidizes phosphorous in Ba 3 P 2 forming BaI 2 and Ba 3 P 3 I 2 , or Ba 5 P 5 I 3 , respectively [9]. However, when reacting gold with Ba 3 P 2 , aiming at 'Ba 3 P 3 Au 2 ' or 'Ba 5 P 5 Au 3 ', instead gold disproportionated, yielding a two-phase product consisting of BaAu 2 and BaAuP.

Results and Discussion

The Zintl phases Ba 3 X 2 (X = P, As), containing isolated X 3-ions, react with gold, and a two-phase product consisting of BaAuX and BaAu 2 in an approximate molar ratio of 2:1 is formed quantitatively. Figure 1 shows the powder pattern of the product mixture. The power of reduction of X 3-is not sufficient to achieve the formation of Au -anions together with X 2-or X 1-species. Such a redox process would have been necessary for obtaining a hypothetic compound 'Ba 3 P 3 Au 2 '. Instead elemental gold disproportionates under these conditions, with X 3-acting as a base, according to equations ( 1) and (2).

Ba 3 As 2 + 4 Au ±0 → 2 BaAu +1 As + BaAu -1 2 (2)
The formation of BaAu 2 as a by-product can be avoided by using BaX as starting material. In this case pure samples of BaAuP and BaAuAs were obtained, which have been used for conductivity measurements. The specific electrical resistivity of BaAuP (BaAuAs) increases from 0.076 (0.041) Ωcm at 5 K to 0.137 (0.052) Ωcm at 300K, thus displaying a typical behaviour of a poor metallic conductor (Fig. 2). Because of the metallic conductivity a description Ba +2 Au +1 X -3 would not be valid in a strict sense; this should result in semiconducting behaviour. In equations ( 1) and ( 2) conventional oxidation states are given with the signs indicating the correct polarisation: gold has a positive charge (Au +δ ) in BaAuX and a negative one (Au -δ ) in BaAu 2 . Both, BaAuP and BaAuAs, crystallize in the space group P6 3 /mmc (no. 194, Pearson code hP6) as was found by determination of the lattice constants from powder diffraction data, previously [10]. The structure represents an ordered variant of the AlB 2 type of structure, similar to ZrBeSi. Complex anions 2 ∞ [AuX] 2-are forming sheets corresponding to hexagonal boron nitride, with the barium cations in-between (Fig. 3), resulting in a 6+6 coordination for Ba, and a mutual trigonal planar coordination for the pnictides and gold atoms (Table 1).

Conclusions

BaAuP and BaAuAs have been synthesized by solid state reactions from Ba 3 X 2 (BaX) and gold. The reaction products have been characterized by powder diffraction, single crystal diffraction and resistivity measurements. It has been demonstrated that elemental gold disproportionates in the presence of pnictides Ba 3 X 2 , whereby the pnictides are acting as bases.

Experimental Section

Synthesis

The BaAuX (X = P, As) compounds were prepared by the reaction of Ba 3 X 2 and Au (1:4), with BaAu 2 as a by-product in stoichiometric amounts. An optimized synthesis, realised by the reaction of BaX with elemental gold, leads to pure samples of BaAuP and BaAuAs. The barium pnictides were prepared from the elements in corundum crucibles. The crucibles were sealed in quartz ampoules, and heated at 1020 K for 48 h.

Stoichiometric amounts of BaX and gold powder (fine Au powders can be obtained as described previously [11]) were mixed in a dry-box (MBraun, Garching, Germany) in an argon atmosphere (< 0.1ppm O 2 , H 2 O) and sealed in tantalum ampoules. The reaction was carried out with the following temperature profile: 298 → 1220 K (50 K h -1 , subsequent annealing for 48 h); 1220 → 298 K (50 K h -1 ).

The solidified black reguli are sensitive to humid air and must be handled in an inert atmosphere. For single crystal X-ray diffraction experiments, black single crystals were selected in a dry-box and mounted in sealed glass capillaries. Data collection, at 296(2) K, was carried out on a STADI4 four circle diffractometer (Stoe & Cie GmbH, Darmstadt, Germany) and a Smart 1000K three circle diffractometer (Bruker AXS, Karlsruhe, Germany), respectively. The structures were solved by direct methods and refined by full-matrix least squares using the SHELXTL program package [12]. Experimental details are given in Table 2, atomic coordinates and anisotropic displacement parameters are given in Table 3 and4

X-ray Analysis

Electrical Resistivity

Temperature dependent resistivity has been obtained for pressed pellets using van der Pauw method [START_REF] Van Der Pauw | [END_REF]. The ground powders were pressed into 6 mm diameter by 1 mm thick pellets, and annealed for 24h at 723 K under argon atmosphere. The pellets were then connected to four probes of the resistivity measurement apparatus. Data were recorded in the temperature range 5-300 K at 5 K intervals.

Figure 1 .

 1 Figure 1. Powder diffraction pattern of the reaction product between Ba 3 P 2 and Au (middle), with calculated intensity plots of BaAu 2 (top) and BaAuP (bottom). For clarity, the intensities (arbitrary units) are drawn in positive direction for BaAuP and the product, and in negative direction for BaAu 2 .

Figure 2 .

 2 Figure 2. Electric resistivity vs. temperature for BaAuP and BaAuAs.

Figure 3 .

 3 Figure 3. Perspective representation of the crystal structure of BaAuP, with the margins of the unit cell (grey).

  All samples were examined by X-ray powder diffraction. Powder patterns were collected with a linear position-sensitive detector on a STADI P diffractometer in Debye-Scherrer geometry, with Gemonochromated MoKα 1 radiation, λ = 1.540598 Å, 5 < 2θ < 100° (Stoe & Cie GmbH, Darmstadt, Germany). Samples were sealed in glass capillaries of 0.3 mm diameter.

  of the reaction product between Ba3P2 and Au (middle), with calculated intensity plots of BaAu2 (top) and BaAuP (bottom). For clarity, the intensities (arbitrary units) are drawn in positive direction for BaAuP and the product, and in negative direction for BaAu2. 120x69mm (600 x 600 DPI)

Table 1 .

 1 Interatomic distances /pm for BaAuX, X = P, As (standard deviations).

		Atomic contact	BaAuP	BaAuAs	multiplicity
	Au	-X	254.42(3)	261.84(3)	3
		-Ba	339.61(4)	345.68(3)	6
	Ba	-X	339.61(4)	345.68(3)	6
		-Au	339.61(4)	345.68(3)	6
	X	-Au	254.42(3)	261.84(3)	3
		-Ba	339.61(4)	345.68(3)	6

Table 2 .

 2 . Crystal data and structure refinement data of BaAuP and BaAuAs.

		BaAuP	BaAuAs
	Formula weight	365.28	409.23
	Space group (no.), Z	P6 3 /mmc (194), 2	P6 3 /mmc (194), 2
	Lattice constants /pm a = 440.68(6)	453.53(5)
		c = 899.8(2)	902.7(1)
		c/a = 2.042	1.990
	V /Å 3 , ρ xray /g cm -3	151.33(4), 8.016	160.80(3), 8.452
	Diffractometer	STADI4 (Stoe &	Smart 1000K
		Cie), scintillations	(Bruker AXS),
		counter	CCD-detector
	Radiation	MoKα (λ = 71.073 pm), graphite
		monochromatated
	Absorption correction X-Shape [13]	SADABS [14]
	2θ range /°	9.06 to 74.82	9.04 to 70.08
	Index range	-7 ≤ h ≤ 7,	-7 ≤ h ≤ 7,
		-7 ≤ k ≤ 7,	-7 ≤ k ≤ 7,
		-15 ≤ l ≤ 15	-14 ≤ l ≤ 14
	Rfelection collected	2976	2637
	Data, R int	183, 0.0196	168, 0.0535
	No. of parameters	8	8
	R 1 [F 2 > 2σ(F 2 )]	0.0237	0.0255
	wR(F 2 )	0.0631	0.0585
	Extinction coefficient 0.023(2)	0.013(2)
	∆ρ max , ∆ρ min /e Å -3	2.56, -3.31	1.41, -2.02
	Deposition no. [15]	CSD-420340	CSD-420341

Table 3 .

 3 Atomic coordinates and equivalent isotropic displacement parameters U eq /pm2 for BaAuP and BaAuAs.

	Compound Atom Site x	y	z	U eq
	BaAuP	Au	2d	1/3	2/3	3/4	150(2)
		Ba	2a	0	0	0	133(3)
		P	2c	1/3	2/3	1/4	147(7)
	BaAuAs	Au	2d	1/3	2/3	3/4	114(2)
		Ba	2a	0	0	0	102(2)
		As	2c	1/3	2/3	1/4	95(3)

Table 4 .

 4 Anisotropic displacement parameters U ij /pm 2 for BaAuP and BaAuAs (U 13 = U 23 = 0).

	Compound Atom	U 11	U 22	U 33	U 12
	BaAuP	Au	149(2)	U 11	151(3)	75(1)
		Ba	126(3)	U 11	146(4)	64(1)
		P	115(9)	U 11	210(19) 58(5)
	BaAuAs	Au	106(2)	U 11	132(3)	53(1)
		Ba	96(3)	U 11	114(4)	48(2)
		As	63(4)	U 11	161(6)	32(2)

Wiley-VCH ZAAC

Acknowledgments

The authors greatfully acknowledge the help of Mrs. S. Prill-Diemer for carrying out the synthesis, and Mrs. G. Siegle for resistivity measurements. ____________


T Lelièvre

K Minoukadeh

Long-time convergence of an Adaptive Biasing Force method: the bi-channel case

, where the efficiency of the ABF approach is demonstrated for such a multiple-channel situation.

Introduction

We consider a system of N particles with positions q ∈ D ⊂ R 3N . In statistical physics, one is interested in calculating averages with respect to the Boltzmann-Gibbs measure dν(q) = Z -1 exp(-βV (q))dq, [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] with V : D → R the potential energy function, Z = D exp(-βV (q)) dq the partition function, D = {q, V (q) < ∞} the configuration space, and β = 1/(k B T ), where k B is the Boltzmann constant and T is the temperature. The function V is the energy associated with the positions of the particles and is assumed to be such that Z < ∞. The probability measure ν represents the equilibrium measure sampled by the particles in the canonical ensemble. A typical dynamics that can be used to sample this measure through trajectorial averages is the overdamped Langevin dynamics

dQ t = -∇V (Q t )dt + 2β -1 dB t , (2) 1
where (B t ) t≥0 is a 3N -dimensional standard Brownian motion. Indeed, under loose assumptions on V , one has the ergodic property: for any smooth test function ϕ,

lim T →∞ 1 T T 0 ϕ(Q t ) dt = D ϕ dν.
The efficiency of this sampling procedure, which can be shown to be related to the convergence rate to equilibrium of the above dynamics is often hindered by metastabilities in the potential function V , namely regions of low energy are separated by high energy barriers. Equivalently, in terms of the probability measure, ν is typically a multimodal measure, with regions of high probability separated by regions of low probability. To circumvent this issue, a one-dimensional collective variable (or reaction coordinate) ξ : D → M is introduced, which will be used to define a biasing potential for [START_REF] Arnold | On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF]. In the following, we will assume that |∇ξ| > 0 on D, and that M = T, where T = R/Z is the one-dimensional torus (which typically corresponds to the case where the reaction coordinate represents a dihedral angle, for example to characterize the conformation of a molecule). Before defining more precisely the biased dynamics in the next section, we need to introduce a few notation.

The image of the measure ν in ξ is given by

dν ξ (z) = Z -1 exp(-βA(z)) dz,
where A is the so-called free energy, defined by

A(z) = -β -1 ln(Z z ) (3)
where

Z z = Σz exp(-βV (q)) δ ξ(q)-z (dq) (4)
is the partition function on the submanifold Σ z = {q ∈ D | ξ(q) = z}. The measure δ ξ(q)-z (dq) is defined through the conditioning formula: for any smooth test function ϕ : D → R, D ϕ(q) dq = M Σz ϕ(q)δ ξ(q)-z (dq) dz.

Using the co-area formula (see [14, Appendix A]), one can also identify this measure as δ ξ(q)-z (dq) = |∇ξ| -1 dσ Σz , where σ Σz is the Lebesgue measure on Σ z . We assume in the following that ξ and V are such that Z z < ∞, for all z ∈ M.

Practitioners are typically interested in free energy differences A(z) -A(z 0 ), which can be obtained by computing (and integrating) the derivative A ′ (z), the so-called mean force

A ′ (z) = Z -1
z Σz f (q) exp(-βV (q)) δ ξ(q)-z (dq) [START_REF] Darve | Calculating free energies using average forces[END_REF] where f is the local mean force defined by

f = ∇V • ∇ξ |∇ξ| 2 -β -1 div ∇ξ |∇ξ| 2 . ( 6
)
The function f can be understood as the negative force projected onto the reaction coordinate, plus some correction term related to the curvature of the submanifolds Σ z . Notice that the mean force [START_REF] Darve | Calculating free energies using average forces[END_REF] is in fact a conditional expectation

A ′ (z) = E ν [f (Q)|ξ(Q) = z] = Σz f dν |z , (7)
where dν |z = exp(-βV (q))δ ξ(q)-z (dq) exp(-β(A(z)))

denotes the probability measure ν conditioned to a fixed value z of ξ(q). This measure is supported on the submanifold Σ z . For the derivation of ( 5)-( 6)- [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] which is again based on the co-area formula, the reader is referred to [START_REF] Ciccotti | Projection of diffusions on submanifolds: Application to mean force computation[END_REF][START_REF] Sprik | Free energy from constrained molecular dynamics[END_REF][START_REF] Otter | The calculation of free energy differences by constrained molecular dynamics simulations[END_REF].

The adaptive biasing force method

The adaptive biasing force (ABF) method [START_REF] Darve | Calculating free energies using average forces[END_REF][START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF] uses an estimate of the mean force A ′ to bias the standard overdamped Langevin dynamics [START_REF] Arnold | On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] in order to overcome metastabilities in ξ. The bottom line of the approach is that it should be easier to sample the probability measure with density proportional to exp(-β(V -A • ξ)) than the original Boltzmann-Gibbs measure ν, since the marginal probability of the former in ξ is a uniform probability measure on T, while the marginal of the latter (namely exp(-βA(z)) dz) is typically multimodal.

The ABF dynamics is given by

dX t = -∇(V -A t • ξ)(X t )dt + 2β -1 dB t , A ′ t (z) = E [f (X t )|ξ(X t ) = z] , (8)
where A ′ t is an on-the-fly estimate of the mean force, which, in view of the definition [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF], is expected to be a good estimate of A ′ . The law of X t has density ψ(t, •), which satisfies the non-linear Fokker-Planck equation:

             ∂ t ψ = div(∇(V -A t • ξ)ψ) + β -1 ∆ψ, A ′ t (z) = Σz f ψ δ ξ(q)-z (dq) Σz ψ δ ξ(q)-z (dq) . (9)
Roughly speaking, the biasing force ∇(A t • ξ) "flattens the free-energy barriers in ξ". To support this claim, let us simply indicate that if |∇ξ| is constant, the marginal density in ξ satisfies a simple heat equation, with zero bias, see [START_REF] Lelièvre | Computation of free energy profiles with parallel adaptive dynamics[END_REF][START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] and also Equation ( 16) below. Existence and uniqueness of solutions to [START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF] are studied in [START_REF] Jourdain | Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process[END_REF] and a study of the longtime convergence of (9) can be found in [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF], the results of which are briefly discussed below.

Existing convergence results, and the multiple channel scenario

It has been shown in [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] that, under appropriate assumptions, the biasing force A ′ t in (8) (actually for a slightly different dynamics which reduces to [START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF] if |∇ξ| is constant for example) converges to the mean force A ′ exponentially fast in the longtime limit. The rate of convergence was estimated as the minimum of (i) the rate at which the law of ξ(X t ) converges to equilibrium, and (ii) the smallest logarithmic Sobolev inequality constant (LSI constant, discussed in Section 2.3) of the conditional probability measures ν |z , for z ∈ M. Thanks to the bias in the direction of the reaction coordinate, it can be shown that the rate of convergence of the marginal in ξ is actually not the limiting rate in practice since it satisfies a simple diffusion equation. The real limitation is thus the metastable features (i.e. the multimodality) of the family of laws ν |z , which is quantified through the LSI constants associated to these measures: roughly speaking, the smaller the constant the more multimodal the probability measure. These constants may be in some cases extremely small, at least for some values of z ∈ M. The question we address in this paper is the optimality of this theoretical rate of convergence for the ABF method.

The generic situation is indeed that the LSI constants for the measures ν |z are not small uniformly in z. Typically, there exists some values of z for which these measures are multimodal. This situation is often encountered in practice due to two reasons. First, finding a suitable reaction coordinate, namely in our context one that ensures that there are no metastabilities associated to the equilibrium measures ν |z , is not trivial for a large-dimensional system. Secondly, a lowdimensional reaction coordinate may simply not be sufficient to describe all metastabilities of the system. In such cases, the results of [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] predict a very small rate of convergence for the ABF dynamics [START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF], and thus, the inefficiency of this biasing procedure.

As a typical case for which such difficulties appear, we will consider in the following the socalled 'multiple channel situation' (see Figure 1 on the left, for a bi-channel case). In such a situation, starting from a metastable state and as the system evolves in the direction of increasing values of the reaction coordinate ξ, it can follow different channels, which are separated (in the 'orthogonal direction to ξ') by arbitrary high energy barriers. In other words, the energy landscape features parallel valleys which are orthogonal to the isocontours of the reaction coordinate. In such a prototypical situation, the conditional probability measures ν |z are indeed multimodal, for the values of z corresponding to the system being in one of these channels.

However, recent numerical experiments [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF] (based on a discretization of the ABF dynamics (8) by multiple walkers simulated in parallel) suggest that in fact, high energy barriers in Σ z do not always hinder the convergence of the ABF method. The multiple walkers are made to follow similar dynamics [START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF], but driven by independent Brownian motions. The chemical system considered in [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF] is the compact states of the deca-alanine peptide (the reaction coordinate ξ being thus the endto-end distance of the peptide). Due to some 'buckling effects', this is a typical multiple channel situation, since the molecule can shrink to various compact states (see [START_REF] Hénin | Exploring multidimensional free energy landscapes using time-dependent biases on collective variables[END_REF]). In [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF], numerical results show that the ABF approach indeed yields reliable results in such a situation. We interpret this as follows. When encountered with a fork in the channel, each walker is likely to travel down a different channel. Thus, it is indeed almost impossible for a given walker to switch from one channel to another, once it has entered one of them, but this appears not to be necessary to obtain reliable results. It suggests that the theoretical rate of convergence obtained in [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] is actually not optimal.

Inspired by these numerical results, we present herein an improved theoretical rate of convergence of the ABF method. The rate will be shown to depend on the LSI constants of the family of equilibrium measures conditioned to being in Σ z and a channel. By doing so, we show that high energy barriers separating the channels do not in fact affect the rate of convergence of the method. The crucial assumptions needed to show our result are: (i) exchange between the two channels is possible for some values of ξ (see [H1] below) and (ii) the free energy is a 'good bias' in each channel (see [H4] below). This is formalized in the main result of this paper, namely Theorem 1 below.

For some technical reasons, we were actually unable to prove this result on the original ABF dynamics [START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF]. We will thus consider a slightly different system (that we call the bi-channel model) which retains the most important features of the dynamics (8) when applied to a potential exhibiting two parallel channels in the direction of ξ, separated by a high energy barrier.

The paper is organized as follows. In Section 2 we give details of the bi-channel model, define some probability measures and recall some entropy definitions before presenting the main result. Finally, the proof of the main result is given in Section 3.

The bi-channel model and statement of the main results

In this section, we present a model to describe the bi-channel scenario. In the following, we treat the case d = 2 (so that the position is q = (x, y)), D = T × R, M = T and ξ : D → T, where ξ(x, y) = x. We further assume without loss of generality that β = 1, which can be done by a change of variables t = β -1 t, ψ( t, x) = ψ(t, x), Ṽ (x, y) = βV (x, y). With these assumptions, some notation may be simplified: |∇ξ| = 1, Σ z = R and δ ξ(q)-z (dq) = dy. Furthermore, the definition of the local mean force in [START_REF] Otter | The calculation of free energy differences by constrained molecular dynamics simulations[END_REF] simplifies to

f = ∂ x V
and the free energy and its derivative are given by

A(x) = -ln R exp(-V (x, y)) dy and A ′ (x) = R ∂ x V (x, y) exp(-V (x, y)) dy R exp(-V (x, y)) dy . ( 10
)
We would like to emphasize that the choice of the domain M = T and reaction coordinate ξ(x, y) = x is merely to reduce technicalities, see [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] for appropriate tools to treat general ξ.

In particular, the results can be straightforwardly generalized to the case D = T × R d-1 and ξ(q 1 , . . . , q d ) = q 1 . Likewise, the generalization to a situation with multiple channels (more than two) is straightforward (see Remark 1 below for another generalization).

The bi-channel model

The bi-channel situation is characterized by the existence of two channels joining an initial and final state on a potential energy surface V , separated from each other by a region of high energy, as depicted in the left of Figure 1. As explained above, we were not able to analyze the original ABF dynamics [START_REF] Hénin | Overcoming free energy barriers using unconstrained molecular dynamics simulations[END_REF] in this situation, because of some technical difficulties in expressing the probability density flux from one channel to the other.

We therefore analyze the convergence of the ABF method for a slightly different model, which is schematically represented in right of Figure 1. Each channel is described by a potential energy function V i : D → R, where i ∈ {0, 1} denotes the channel index. The stochastic process we consider now is actually a couple (X t , I t ), where the position vector X t lives at time t on the potential V It , I t ∈ {0, 1} being the index of the visited channel at time t. The channel index I t is allowed to switch to 1 -I t only if ξ(X t ) lies in some designated regions (typically at the two ends of the two channels).

The dynamics for the ABF dynamics in the bi-channel model is then

         dX t = -∇(V It -A t • ξ)(X t )dt + √ 2 dB t , A ′ t (x) = E [∂ x V It (X t )|ξ(X t ) = x] , I t ∈ {0, 1} is a jump process with generator Lϕ(x, y, i) = -λ(x)(ϕ(x, y, i) -ϕ(x, y, 1 -i)). (11)
In terms of the stochastic process I t , switching between the two potentials (namely change of I t to 1 -I t ) occurs at times

τ n+1 = inf t > τ n t τn λ(ξ(X s ))ds > T n ,
where τ 0 = 0 and T n are i.i.d. random variables drawn from the exponential distribution with parameter 1. In this way λ(x) denotes the rate at which the trajectories jump from potential V i to potential V 1-i . Note that this rate depends only on the position, x, in the reaction coordinate. The bi-channel feature of the model is related to the fact that we assume that the rate λ is zero (there is no switching) in some region of the reaction coordinate. Outside of this region, the rate is supposed to be constant, and the potential functions are assumed to be identical (the particles live in the same potential). Let us state this as a formal assumption, The region E ⊂ T in the above hypothesis represents the region where the two channels are separated by high energy barriers, see Figure 1. It is assumed to have a Lebesgue measure different from 0 and 1.

[H1] ∃ E ⊂ T, λ(x) = λ1 T\E (x) and ∀x ∈ T\E, V 0 (x, •) = V 1 (x, •).
The main qualitative difference between the bi-channel model we study, and the original ABF dynamics ( 8) is that the switching only depends on the x-position and not on the y-position. However, the ABF dynamics [START_REF] Jourdain | Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process[END_REF] conserves the main difficulty of the original one, namely the metastability of the dynamics in terms of visited channels for some values of ξ. At times t such that ξ(X t ) ∈ E, I t cannot switch to 1 -I t . In particular, it can be checked that the proof of [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] applied to [START_REF] Jourdain | Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process[END_REF] in the case E = ∅ leads to an estimated rate of convergence limited by λ and is thus eventually zero if λ goes to zero (see Remark 3 below). The aim of this work is to study the case E = ∅ and to obtain an exponential rate of convergence even if λ = 0 in some region (see Theorem 1 below).

Remark 1 On Figure 1, we represent the bi-channel case with two metastable states linked by two different channels. The region where the stochastic process can jump from one channel to another has two connected components. We would like to emphasize that our result also applies to the case where this region is simply connected, namely a situation where two channels start from a metastable state (along the reaction coordinate value) but do not end in another metastable state. This situation is of course less favorable in terms of speed of convergence to equilibrium, which would be reflected in our theoretical result through the parameter θ (see [H4] below). This is actually typically the situation of the numerical experiments in [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF] mentioned above since the various compact states obtained do not belong to the same metastable basin.

A partial differential equation formulation

Let us introduce the time marginal of the process (X t , I t ) in [START_REF] Jourdain | Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process[END_REF]:

dµ t = 1 i=0 ψ(t, x, y, i)δ i dx dy,
where δ i is the Dirac measure on the singleton {i}. When necessary, we shall denote the idependency of the density by a subscript ψ i , so that ψ 1-i denotes ψ(t, x, y, 1i) for example. The evolution of the densities are described by a system of non-linear partial differential equations: ∀i ∈ {0, 1},

∂ t ψ = div(∇(V i -A t • ξ)ψ) + ∆ψ -λ • ξ [ψ -ψ 1-i ] on D, (12)
where, we recall, the last term is zero for ξ(x, y) ∈ E. The non-linearity is due to the definition of the mean force estimate, given by

A ′ t (x) = 1 i=0 R ∂ x V i (x, y) ψ(t, x, y, i) dy 1 i=0 R ψ(t, x, y, i) dy . ( 13
)
Using hypothesis [H1], it can be checked that if ψ ∞ is defined as a probability density proportional to e -(Vi-A∞•ξ) where A ∞ is a given long-time limit for A t , then ψ ∞ is a stationary solution to [START_REF] Lelièvre | A general two-scale criteria for logarithmic Sobolev inequalities[END_REF]. Then, by replacing ψ by ψ ∞ in ( 13) and comparing with the definition of the mean force [START_REF] Holley | Logarithmic Sobolev inequalities and stochastic Ising models[END_REF], it is clear that by choosing A ′ ∞ = A ′ , one obtains a stationary solution of the system ( 12)-( 13) written as:

ψ ∞ (x, y, i) = e -(Vi(x,y)-A(x)) 1 i=0 D e -(Vi-A•ξ) dx dy . ( 14
)
The associated equilibrium measure for the process (X t , I t ) writes:

dµ ∞ = 1 i=0 ψ ∞ (x, y, i)δ i dx dy.
We will need further notation for marginal and conditional laws associated to µ t and µ ∞ . The next two sections give precise definitions for these measures, and Table 1 summarizes the notation.

Marginal laws

We are now in the position to define marginal probability densities. The image of the probability measure µ t in ξ and I is denoted by

dµ ξ,I t = 1 i=0 ψ ξ,I (t, x, i)δ i dx where ψ ξ,I (t, x, i) = R ψ(t, x, y, i) dy.
The evolution of the density ψ ξ,I is described by the (non-closed) partial differential equation

∂ t ψ ξ,I = R ∂ x (∂ x (V i -A t • ξ)ψ) dy + ∂ xx ψ ξ,I -λ(x)(ψ ξ,I -ψ ξ,I 1-i ), (15)
obtained by integrating [START_REF] Lelièvre | A general two-scale criteria for logarithmic Sobolev inequalities[END_REF] in y. The associated equilibrium measure is

dµ ξ,I ∞ = 1 i=0 ψ ξ,I ∞ (x, i)δ i dx where ψ ξ,I ∞ (x, i) = R ψ ∞ (x, y, i) dy.
The marginal measure in ξ only is denoted by By summing (15) over i and using the definition (13) of A ′ t , it is easy to check that ψ ξ satisfies a closed, very simple partial differential equation (this is similar to the original ABF dynamics, see [START_REF] Lelièvre | Computation of free energy profiles with parallel adaptive dynamics[END_REF][START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF]):

dµ ξ t = ψ ξ (t, x) dx where ψ ξ (t, x) = 1 i=0 ψ ξ,I (t, x, i) = 1 i=0 R ψ(t,
∂ t ψ ξ = ∂ xx ψ ξ on T. (16)
Thanks to the adaptive bias, along ξ, the barriers have been flattened. The long-time limit of ψ ξ is given by

ψ ξ ∞ (x) = 1 i=0 R ψ ∞ (x, y, i) dy = 1,
which corresponds to the uniform probability measure on the torus T.

Conditional laws

Let us introduce the measure µ t|x of (X t , I t ) conditioned to being at a specified point x in the reaction coordinate:

dµ t|x = 1 i=0 ψ(t, x, y, i)δ i dy ψ ξ (t, x) . ( 17
)
Its long-time limit is

dµ ∞|x = 1 i=0 ψ ∞ (x, y, i)δ i dy ψ ξ ∞ (x) . ( 18
)
To study the bi-channel model, we will also need to introduce the measures conditioned to being at fixed ξ and a particular channel i ∈ {0, 1}. This measure and its long-time limit are defined respectively by

dµ t|x,i = ψ(t, x, y, i) dy ψ ξ,I (t, x, i) and dµ ∞|x,i = ψ ∞ (x, y, i) dy ψ ξ,I ∞ (x, i) . (19)
Finally, we will also need the marginals in I of the probability measures µ t|x and µ ∞|x . These Bernoulli probability measures µ I t|x and µ I ∞|x represent the proportion of the marginal distribution ψ ξ (t, •) and ψ ξ in each channel. They are formally defined as

dµ I t|x = 1 i=0 ψ ξ,I (t, x, i)δ i ψ ξ (t, x) and dµ I ∞|x = 1 i=0 ψ ξ,I ∞ (x, i)δ i ψ ξ ∞ (x)
.

(20)

Entropy and Fisher information

In this section we recall some well-known results for defining relative entropy between two probability measures, which can be seen as a measure of the 'distance' between those. A general introduction to this topic can be found in [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF][START_REF] Villani | Topics in Optimal Transportation[END_REF] and applications to study the longtime behavior of Fokker-Planck type equations are presented in [START_REF] Arnold | On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF].

Definition 1 (Entropy). For any two probability measures µ and ν such that µ is absolutely continuous with respect to ν (denoted as µ ≪ ν), the relative entropy is defined as

H(µ|ν) = ln dµ dν dµ.
The positivity of the relative entropy can be shown using the inequality ln(1/x) ≥ 1x. Furthermore, H(µ|ν) = 0 if and only if µ = ν.

Definition 2 (Csiszar-Kullback inequality).

For measures µ and ν which have densities with respect to the Lebesgue measure, the following holds

µ -ν L 1 ≤ 2H(µ|ν).
This allows us to control the L 1 -norm of the difference of two probability measures by their relative entropy.

Definition 3 (Fisher information). For any probability measure µ absolutely continuous with respect to ν, the Fisher information is given by

F (µ|ν) = ∇ln dµ dν 2 dµ.
Definition 4 (Logarithmic Sobolev inequality). The probability measure ν is said to satisfy a logarithmic Sobolev inequality with constant ρ > 0 (in short: LSI(ρ)) if for all probability measures µ such that µ ≪ ν,

H(µ|ν) ≤ 1 2ρ F (µ|ν).
Such an inequality holds for Gaussian measures [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] for example, and more generally [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF] for any measure ν with density proportional to e -V , where V is α-convex (in which case the LSI constant ρ is equal to α). Besides, there exists a perturbation result [START_REF] Holley | Logarithmic Sobolev inequalities and stochastic Ising models[END_REF]: if ν is a probability measure such that dν/dν = e U , where ν satisfies a LSI(ρ) and U is a bounded function, then ν satisfies a LSI with constant ρ = ρ e -osc(U) , where osc(U ) = sup(U )inf(U ). Thus, a very large class of probability measures satisfy a LSI. An important feature of the LSI constant is that it degenerates to zero in the case where the probability measure is multimodal. For example, if dν = Z -1 exp(-βW (x)) dx and W (x) = x 4 /4x 2 /2 is the double-well potential in dimension 1, then the LSI constant scales as exp(-β∆W ) where ∆W = W (0) -W (1) > 0 is the height of the barrier. Such inequalities thus hold under rather loose assumptions, but the constant ρ is very small for a multimodal measure. For example, in typical situations encountered in molecular dynamics, the LSI constant for the measure ν defined in (1) is extremely small, which is related to the fact that it is difficult to sample directly this measure.

To analyze the convergence of the ABF dynamics for the bi-channel model, we make use of LSIs for the conditional measures µ ∞|x,i (see assumption [H3] below), which are the equilibrium canonical measures, conditioned to being at a fixed value of ξ and in a channel. In [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF], log-Sobolev inequalities for the equilibrium canonical measures conditioned only to being at a fixed value of ξ were considered, but in the bi-channel case, those are typically very small due to the presence of high energy barriers 'orthogonal' to the isocontours of the reaction coordinate (see again Figure 1).

Let us now define the Wasserstein distance between two probability measures.

Definition 5 (Wasserstein distance). The Wasserstein distance with linear cost between probability measures µ and ν is defined as

W (µ, ν) = inf π∈Π(µ,ν) D×D |y -y ′ | π(dy, dy ′ ),
where Π(µ, ν) denotes the set of coupling probability measures on D × D, with marginals µ and ν.

Definition 6 (Talagrand inequality). The probability measure ν is said to satisfy a Talagrand inequality with constant ρ > 0 (or T (ρ)) if for all probability measures µ such that µ ≪ ν,

W (µ, ν) ≤ 2 ρ H(µ|ν). (21)
Logarithmic Sobolev inequalities and Talagrand inequalities are related (see [START_REF] Otto | Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality[END_REF]):

Lemma 1 If ν satisfies LSI(ρ), then ν satisfies T (ρ).
Below, we present entropies that prove useful in obtaining convergence results of the bi-channel ABF model. In this paper, we are primarily interested in the convergence to a stationary state of the Fokker-Planck equation ( 12)-( 13), and thus of the associated partial differential equations [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF] and [START_REF] Otto | Generalization of an inequality by Talagrand, viewed as a consequence of the logarithmic Sobolev inequality[END_REF]. Relative entropies will therefore be defined for some probability measures with respect to their long-time limits.

The total entropy will be denoted by

E(t) = H(µ t |µ ∞ ) = 1 i=0 D ln ψ ψ ∞ ψ dx dy.
The so-called macroscopic entropy is defined as

E M (t) = H(µ ξ t |µ ξ ∞ ) = T ln ψ ξ ψ ξ ∞ ψ ξ dx,
the local entropy at a fixed value x in the reaction coordinate by

e m (t, x) = H(µ t|x |µ ∞|x ) = 1 i=0 R ln ψ ψ ξ ψ ∞ ψ ξ ∞ ψ ψ ξ dy,
and the microscopic entropy by

E m (t) = T e m (t, x) ψ ξ (t, x) dx.
With the above definitions, it is easy to show that

E(t) = E M (t) + E m (t). (22)
In order to treat the bi-channel case, we define a channel-local entropy, defined by

e cl (t, x, i) = H(µ t|x,i |µ ∞|x,i ) = R ln ψ ψ ξ,I ψ ∞ ψ ξ,I ∞ ψ ψ ξ,I dy.
Two hypothesis that will be essential to the results presented below are: a so-called 'bounded coupling' assumption on the cross derivative ∂ x,y V i (see [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF][START_REF] Lelièvre | A general two-scale criteria for logarithmic Sobolev inequalities[END_REF]), and an assumption on the logarithmic Sobolev inequality constants for the probability measures µ ∞|x,i . The hypotheses read

[H2]      ∀i ∈ {0, 1}, V i and ξ are sufficiently differentiable functions such that ∃ C, M > 0 ∂ x,y V i L ∞ (T×R) ≤ M < ∞ and R ∂ x V i e -Vi dy R e -Vi dy L ∞ (T) ≤ C < ∞,
[H3] ∀i ∈ {0, 1}, V i and ξ are such that ∃ρ > 0, ∀x ∈ T, ∀i ∈ {0, 1}, the conditional measure µ ∞|x,i satisfies LSI(ρ).

The hypothesis [H3] gives us that ∀x ∈ T, ∀i ∈ {0, 1}

H(µ t|x,i |µ ∞|x,i ) ≤ 1 2ρ F (µ t|x,i |µ ∞|x,i ), or equivalently R ln ψ ψ ξ,I ψ ∞ ψ ξ,I ∞ ψ ψ ξ,I dy ≤ 1 2ρ R ∂ y ln ψ ψ ∞ 2 ψ ψ ξ,I dy. (23)
Finally, an entropy that appears in later calculations, is that of the Bernoulli measure µ I t|x with respect to its long-time limit

e c (t, x) = H(µ I t|x |µ I ∞|x ) = 1 i=0 ln ψ ξ,I ψ ξ ψ ξ,I ∞ ψ ξ ∞ ψ ξ,I ψ ξ .
The so-called channel entropy is then given by

E c (t) = T e c (t, x)ψ ξ dx = 1 i=0 T ln ψ ξ,I ψ ξ ψ ξ,I ∞ ψ ξ ∞ ψ ξ,I . (24)

The free energy as a bias in each channel

As well as the hypothesis [H3] on the conditional measures µ ∞|x,i , an assumption will also be necessary to ensure that the free energy is a 'good bias' in each channel. More precisely, once the bias has converged, the marginals along ξ in each channel must converge sufficiently quickly to their long-time limit. Roughly speaking, the channels should not be too 'asymmetrical'. The aim of this section is to state this more formally, see assumption [H4] below. Consider that the system is already nearly at equilibrium, in the sense that i)

A ′ t = A ′ ∞ = A ′ , ii) ∀i ∈ {0, 1}, R ∂ x V i dµ t|x,i = R ∂ x V i dµ ∞|x,i .
Then the marginal density ψ ξ,I can be shown to satisfy (see [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF] and (40) below)

∂ t ψ ξ,I = -L i ψ ξ,I = ∂ x ψ ξ,I ∞ ∂ x (ψ ξ,I /ψ ξ,I ∞ ) -λ(x)(ψ ξ,I -ψ ξ,I 1-i ). ( 25
)
It can be shown that the operator L = (L 0 , L 1 ) is symmetric and positive definite with respect to the inner product f, g =

1 i=0 T f i (x)g i (x) 1 ψ ξ,I ∞ (x, i)
dx and has a spectral gap θ > 0 (see Lemma 12 below). In other words, L is such that for all functions f : T × {0, 1} → R, with

f i ∈ H 1 1 ψ ξ,I ∞ (x,i) dx , f = 0 and 1 i=0 T f i (x) dx = 0 , we have f, f ≤ 1 θ f, Lf . ( 26
)
As will be apparent in the proof, it will be necessary for the spectral gap θ to be sufficiently large.

[H4]

     θ > θ min with θ min = 8(C + M ρ -1/2 ) 2 M c where 0 < c, M < ∞ are such that inf x,i ψ ξ,I ∞ = c and sup x ψ ξ (0, x) = M .
We recall that the constants C, M and ρ have been introduced in assumptions [H2] and [H3] above. The fact that c > 0 is not restrictive since ∀i ∈ {0, 1}, V i is a continuous function and ψ ξ,I ∞ ∝ R e -Vi dy > 0 is continuous and defined on the compact space T. Similarly, since ψ ξ satisfies the heat equation ( 16), the assumption M < ∞ is not restrictive. If, for example, the initial condition has a Dirac mass marginal in ξ, for any positive time t 0 > 0, ψ ξ (t 0 , •) is a bounded function, and one has simply to consider the dynamics on [t 0 , ∞).

Remark 2 In the hypothesis [H2],

the assumption on the cross derivative ∂ x,y V i could in fact be replaced by ∂ x V i L ∞ (T×R) ≤ M < ∞, in which case, in [H4], the minimum value for θ would be changed to θ min = 8M 2 M c and the Talagrand inequalities in Lemmas 6 and 11 would be replaced by Csiszar-Kullback inequalities.

Main result

We are now in position to present the main result of the paper.

Theorem 1 Assume hypotheses [H1]-[H4]

. There exists a smooth function Λ : (θ min , ∞) → (0, ρ) which is increasing and such that:

Λ(ρ + 2θ min ) = ρ 2 and Λ(θ) → 0 as θ → θ min ρ as θ → ∞
for which we can prove the following convergence result: for any ε ∈ (0, Λ(θ)), there exists a constant K > 0 such that, ∀t > 0,

E m (t) ≤ Kexp -2 min{(Λ(θ) -ε), 4π 2 } t . ( 27
)
This implies that the total entropy E and thus ψ(t, •)ψ ∞ 2 L 1 converge exponentially fast to zero with the same rate. Furthermore, the biasing force A ′ t converges to the mean force A ′ in the following sense: ∀t ≥ 0,

T |A ′ t (x) -A ′ (x)| 2 ψ ξ (t, x) dx ≤ 2(C + M ρ -1/2 ) 2 E m (t).
As a consequence, for any positive time t 0 > 0 and ε ∈ (0, Λ(θ)), there exists a constant K such that ∀t ≥ t 0 ,

T |A ′ t (x) -A ′ (x)| 2 dx ≤ Kexp -2 min{(Λ(θ) -ε), 4π 2 } t .
The term 4π 2 corresponds to the exponential rate of convergence of ψ ξ to ψ ξ ∞ (see ( 16)), and is clearly not the bottleneck. There are actually various ways to make this rate as small as needed (see Remark 11 in [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF]).

Thus, this result essentially shows that the ABF method converges at a rate which is limited by the multimodality of the equilibrium canonical measures conditioned to being at fixed value of ξ and in a channel (quantified by the constant ρ), if the free energy is a bias which enables a fast exploration of each channel (this is quantified by the constant θ, which should be sufficiently large for Λ(θ) to be indeed close to ρ). Thus, the convergence may be fast even if switching between the two channels is impossible for some values of ξ. If the spectral gap θ is sufficiently large, we thus recover a similar expression for the rate of convergence of the ABF method as the one derived in [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF], with ρ being now the LSI constant of the canonical measures µ |x,i .

We would like to emphasize that our arguments hold under the following assumption of existence of regular solutions: We assume that we are given a process (X t , I t ) and a function A ′ t which satisfy [START_REF] Jourdain | Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process[END_REF], and such that X t , conditionally on I t = i has a smooth density ψ(t, x, y, i). We suppose that this density is sufficiently regular so that the entropy estimates below are valid. We refer for example to [START_REF] Arnold | On logarithmic Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF] for an appropriate functional framework in which such entropy estimates hold.

Proof of main result

In order to prove the exponential decay of the microscopic entropy in Theorem 1, we use the fact that the time evolution of the microscopic entropy can be expressed as a combination of the evolution of the total and the marginal entropies, from ( 22)

dE m dt = dE dt - dE M dt .
In order to obtain results for the microscopic entropy, we begin by treating the time evolutions of the total entropy and the macroscopic entropy separately.

Preliminary computations on the total entropy E

In order to study the evolution of the total entropy, some auxiliary results will be needed and are given in the lemmas below. First, it will be useful to write the Fokker-Planck equation associated to ψ in a different form.

Lemma 2

The Fokker-Planck equation [START_REF] Lelièvre | A general two-scale criteria for logarithmic Sobolev inequalities[END_REF] for ψ can be rewritten as

∂ t ψ = div(ψ ∞ ∇(ψ/ψ ∞ )) + ∂ x ((A ′ -A ′ t )ψ) -λ(x)(ψ -ψ 1-i ).
Proof : By developing the right hand side, we obtain

∂ t ψ = div ∇ψ - ψ ψ ∞ ∇ψ ∞ + ∂ x ((A ′ -A ′ t )ψ) -λ(x)(ψ -ψ 1-i ) = div (∇ψ + ∇(V i -A • ξ)ψ) + ∂ x (A ′ ψ) -∂ x (A ′ t ψ) -λ(x)(ψ -ψ 1-i ) = div (∇(V i -A t • ξ)ψ) + ∆ψ -λ(x)(ψ -ψ 1-i ),
which is indeed the Fokker-Planck equation [START_REF] Lelièvre | A general two-scale criteria for logarithmic Sobolev inequalities[END_REF]. ♦

The above may now be used to estimate the time evolution of the total entropy.

Lemma 3 The total entropy E satisfies

dE dt ≤ - 1 i=0 T×R ∇ln ψ ψ ∞ 2 ψ - 1 i=0 T×R (A ′ -A ′ t )∂ x ln ψ ψ ∞ ψ. ( 28
)
Proof : First, by definition of the total entropy, we have

dE dt = d dt 1 i=0 T×R ln ψ ψ ∞ ψ = 1 i=0 T×R ln ψ ψ ∞ ∂ t ψ, (29)
using the fact that ψ is a probability density. Next, we use Lemma 2 to obtain

dE dt = 1 i=0 T×R div ψ ∞ ∇ ψ ψ ∞ ln ψ ψ ∞ + 1 i=0 T×R ∂ x ((A ′ -A ′ t )ψ)ln ψ ψ ∞ - 1 i=0 T×R λ(x) ln ψ ψ ∞ (ψ -ψ 1-i ) (30)
≤ -

1 i=0 T×R ∇ln ψ ψ ∞ 2 ψ - 1 i=0 T×R (A ′ -A ′ t )∂ x ln ψ ψ ∞ ψ,
where the last line is a result of integration by parts and the fact that (30) is non-positive, which is proved below in Lemma 4. ♦ Lemma 4 For ψ satisfying (12) and ψ ∞ its long-time limit, the following holds

- 1 i=0 T×R λ(x) ln ψ ψ ∞ (ψ -ψ 1-i ) ≤ 0.
Proof : First, recall that λ(x) = 0, ∀x ∈ E. We consider therefore the left hand side of the inequality at fixed (x, y) ∈ (T\E) × R. At fixed (x, y), we consider renormalized (Bernoulli) probabilities denoted by ψi = ψ i /(ψ 0 + ψ 1 ), so that ψ0 + ψ1 = 1.

- From hypothesis [H1], ∀(x, y) ∈ (T\E) × R, ψ∞,1 = ψ∞,0 = 1 2 , which allows the above to be written as

1 i=0 ln ψ ψ ∞ (ψ -ψ 1-i ) = (-ψ 0 + ψ 1 ) ln ψ 0 ψ ∞,0 -ln ψ 1 ψ ∞,1 = (ψ 0 + ψ 1 )(-ψ0 + ψ1 ) ln ψ0 ψ∞,0 -ln ψ1 ψ∞,1 = (ψ 0 + ψ 1 ) -ln ψ0
- 1 i=0 ln ψ ψ ∞ (ψ -ψ 1-i ) = (ψ 0 + ψ 1 ) -2 1 i=0 ln ψi ψ∞ ψi + 2 1 i=0 ln ψi ψ∞ ψ∞ = -2(ψ 0 + ψ 1 ) 1 i=0 ln ψi ψ∞ ψi + 1 i=0 ln ψ∞ ψi ψ∞ (31) ≤ 0.
The last line is due to the fact that the two terms between brackets are non-negative, since they are relative entropies. ♦ Remark 3 Let us consider the case where E = ∅ and thus ∀x ∈ T, λ(x) = λ > 0, which implies (see [H1])

V 0 = V 1 and therefore ψ ∞,0 = ψ ∞,1 everywhere on T × R.
In this case, it follows from (31) that

- 1 i=0 T×R λ(x)ln ψ ψ ∞ (ψ -ψ 1-i ) ≤ -2λ T×R (ψ 0 + ψ 1 ) 1 i=0 ln ψi ψ∞ ψi ≤ -2λ T×R 1 i=0 ln ψ ψ ∞ ψ + 2λ T×R ln ψ 0 + ψ 1 ψ ∞,0 + ψ ∞,1 (ψ 0 + ψ 1 ).
Furthermore, since the marginal ψ x,y := ψ 0 + ψ 1 satisfies in this specific case

∂ t ψ x,y = div(ψ x,y ∞ ∇(ψ x,y /ψ x,y ∞ )) + ∂ x ((A ′ -A ′ t )ψ x,y ), ( 32
)
one can show, using the results of [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF], that ∃C > 0, ∀t ≥ 0,

- 1 i=0 T×R λ(x)ln ψ ψ ∞ (ψ -ψ 1-i ) ≤ -2λE + 2λCe -2 min{ρ,4π 2 } t .
Using this, and the fact that -E ≤ -E m , one can show that E m converges to zero exponentially fast with rate 2 min{ρ, 4π 2 , λ}.

The convergence rate thus depends on λ > 0, the rate at which switching occurs between the two channels. This is comparable to the original result obtained for the ABF algorithm, see [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF]. The aim of what follows is to obtain a result in the case where E = ∅, namely when λ = 0 in some region.

In order to estimate the last term on the right hand side of (28), it will be helpful to express the difference of the biasing force and the mean force in terms of densities Lemma 5 The difference between the biasing force A ′ t and the mean force A ′ can be expressed in the following way

A ′ t -A ′ = 1 i=0 R ∂ x ln ψ ψ ∞ ψ ψ ξ dy -∂ x ln ψ ξ ψ ξ ∞ .
Proof : We develop the expression on the right hand side and use the fact that

ψ ξ ∞ ≡ 1 1 i=0 R ∂ x ln ψ ψ ∞ ψ ψ ξ dy -∂ x ln ψ ξ ψ ξ ∞ = 1 i=0 R ∂ x ln(ψ) ψ ψ ξ dy - 1 i=0 R ∂ x ln(ψ ∞ ) ψ ψ ξ dy -∂ x ln ψ ξ = 1 i=0 R ∂ x ψ ψ ξ dy + 1 i=0 R ∂ x (V i -A • ξ) ψ ψ ξ dy - ∂ x ψ ξ ψ ξ = 1 i=0 R ∂ x V i ψ ψ ξ dy - 1 i=0 R A ′ ψ ψ ξ dy = A ′ t -A ′ ,
The last line is a result of the definition of A ′ t in ( 13) and the fact that A is a function of x only. ♦ Another useful estimate for the difference between A ′ t and A ′ is given in the following lemma.

Lemma 6

The difference of the biasing force and the mean force can be bounded by the microscopic entropy as:

T |A ′ t -A ′ | 2 ψ ξ dx ≤ 2R 2 E m (t), where R = C + M ρ -1/2 .
Proof : We begin by showing that

|A ′ t (x) -A ′ (x)| ≤ M 2e m (t, x)/ρ.
By definition, we have

A ′ t (x) -A ′ (x) = 1 i=0 R ∂ x V i ψ dy 1 i=0 R ψ dy - 1 i=0 R ∂ x V i ψ ∞ dy 1 i=0 R ψ ∞ dy = 1 i=0 R ∂ x V i ψ ξ,I ψ ξ ψ ψ ξ,I -∂ x V i ψ ξ,I ∞ ψ ξ ∞ ψ ∞ ψ ξ,I ∞ dy = 1 i=0 R×R ∂ x V i (x, y) ψ ξ,I ψ ξ -∂ x V i (x, y ′ ) ψ ξ,I ∞ ψ ξ ∞ π(dy, dy ′ )
where π(dy, dy ′ ) is any coupling measure on R × R with marginals µ t|x,i and µ ∞|x,i . Next, using Taylor's expansion on ∂ x V i (x, y), we have

A ′ t (x) -A ′ (x) = 1 i=0 R×R (∂ x V i (x, y ′ ) + ∂ x,y V i (x, η(y, y ′ ))(y -y ′ )) ψ ξ,I ψ ξ -∂ x V i (x, y ′ ) ψ ξ,I ∞ ψ ξ ∞ π(dy, dy ′ ) = 1 i=0 R×R ∂ x V i (x, y ′ ) ψ ξ,I ψ ξ - ψ ξ,I ∞ ψ ξ ∞ + ∂ x,y V i (x, η(y, y ′ ))(y -y ′ ) ψ ξ,I ψ ξ π(dy, dy ′ ) = 1 i=0 ψ ξ,I ψ ξ - ψ ξ,I ∞ ψ ξ ∞ R ∂ x V i (x, y ′ ) ψ ∞ ψ ξ,I ∞ dy ′ + 1 i=0 ψ ξ,I ψ ξ R×R ∂ x,y V i (x, η(y, y ′ ))(y -y ′ )π(dy, dy ′ ) where η(y, y ′ ) ∈ [y, y ′ ]. Recall from [H2] that ∃C, M > 0 such that R ∂ x V i dµ ∞|x,i L ∞ (T) ≤ C and ∂ x,y V i L ∞ ≤ M .
Furthermore, with the use of the Csiszar-Kullback inequality, we have

|A ′ t (x) -A ′ (x)| ≤ C 1 i=0 ψ ξ,I ψ ξ - ψ ξ,I ∞ ψ ξ ∞ + M 1 i=0 ψ ξ,I ψ ξ R×R |y -y ′ |π(dy, dy ′ ) ≤ C 2H(µ I t|x , µ I ∞|x ) + M 1 i=0 ψ ξ,I ψ ξ R×R |y -y ′ |π(dy, dy ′ ).
Next, by the Talagrand inequality (21) and the concavity of the function x → √ x,

|A ′ t (x) -A ′ (x)| ≤ C 2H(µ I t|x , µ I ∞|x ) + M 1 i=0 ψ ξ,I ψ ξ 2 ρ H(µ t|x,i , µ ∞|x,i ) ≤ C 2H(µ t|x , µ ∞|x ) + M 2 ρ 1 i=0 H(µ t|x,i , µ ∞|x,i ) ψ ξ,I ψ ξ ≤ C 2H(µ t|x , µ ∞|x ) + M 2 ρ H(µ t|x , µ ∞|x ) Therefore |A ′ t (x) -A ′ (x)| 2 ≤ 2 C + M ρ -1/2 2 e m (t, x).
The result follows immediately, since E m (t) = T e m (t, x) ψ ξ dx. ♦

We are now equipped with the right tools to control dE/dt, and are left to handle the evolution of the macroscopic entropy.

Controlling E M

Due to the free diffusion equation ( 16) satisfied by ψ ξ , the macroscopic entropy E M is easily controlled.

Lemma 7

The macroscopic entropy satisfies

dE M dt = - T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ = -F (ψ ξ |ψ ξ ∞ ). (33)
Proof : Using ( 16) and integration by parts

dE M dt = d dt T ln ψ ξ ψ ξ ∞ ψ ξ = T ln ψ ξ ψ ξ ∞ ∂ xx ψ ξ = - T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ .
Lemma 8 The Fisher information for the marginal density ψ ξ decays exponentially fast with rate

r = 8π 2 F (ψ ξ (t, •)|ψ ξ ∞ ) ≤ F 0 exp(-8π 2 t), where F 0 = F (ψ ξ (0, •)|ψ ξ ∞ ).
Proof : See Lemma 12 of reference [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF].

♦

In light of the lemmas presented above, hypotheses [H1]-[H4] may now be used to control the evolution of the microscopic entropy.

Controlling E m

The aim of this section is to obtain an estimation on the evolution of th microscopic entropy E m , see (36) below. We first begin by using the fact that the channel-local conditional measures µ ∞|x,i satisfy LSI(ρ) to bound the microscopic entropy.

Lemma 9 Under the hypothesis [H3], the microscopic entropy E m satisfies

E m ≤ 1 2ρ 1 i=0 T×R ∂ y ln ψ ψ ∞ 2 ψ dx dy + E c ,
where E c is defined in (24). Proof : We fist consider the local entropy e m (t, x), which can be decomposed into the sum of the channel-local entropy and the entropy of the Bernoulli measures.

e m (t, x) = H(µ t|x |µ ∞|x ) = 1 i=0 R ln ψ ψ ξ ψ ∞ ψ ξ ∞ ψ ψ ξ dy = 1 i=0 R ln ψ ψ ξ,I ψ ∞ ψ ξ,I ∞ ψ ψ ξ dy + R ln ψ ξ,I ψ ξ ψ ξ,I ∞ ψ ξ ∞ ψ ψ ξ dy ≤ 1 i=0 1 2ρ R ∂ y ln ψ ψ ∞ 2 ψ ψ ξ dy + 1 i=0 ln ψ ξ,I ψ ξ ψ ξ,I ∞ ψ ξ ∞ ψ ξ,I ψ ξ = 1 2ρ 1 i=0 R ∂ y ln ψ ψ ∞ 2 ψ ψ ξ dy + e c (t, x),
where the inequality is a direct result of [H3]. The microscopic entropy is then

E m (t) = T e m (t, x)ψ ξ dx ≤ 1 2ρ 1 i=0 T×R ∂ y ln ψ ψ ∞ 2 ψ dx dy + E c , as required. ♦
The time evolution of E m may now be expressed using results of Lemmas 3 and 7.

dE m dt = dE dt - dE M dt ≤ - 1 i=0 T×R ∇ln ψ ψ ∞ 2 ψ - 1 i=0 T×R (A ′ -A ′ t )∂ x ln ψ ψ ∞ ψ + T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ .
We may now apply Lemma 5 and use integration by parts

dE m dt = - 1 i=0 T×R ∂ y ln ψ ψ ∞ 2 ψ - 1 i=0 T×R ∂ x ln ψ ψ ∞ 2 ψ + 1 i=0 T×R 1 i=0 R ∂ x ln ψ ψ ∞ ψ ψ ξ dy -∂ x ln ψ ξ ψ ξ ∞ ∂ x ln ψ ψ ∞ ψ + T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ = - 1 i=0 T×R ∂ y ln ψ ψ ∞ 2 ψ - 1 i=0 T×R ∂ x ln ψ ψ ∞ 2 ψ + T 1 i=0 R ∂ x ln ψ ψ ∞ ψ dy 2 1 ψ ξ (34)
-

1 i=0 T×R ∂ x ln ψ ξ ψ ξ ∞ ∂ x ln ψ ψ ∞ ψ + T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ .
Notice that (34) is non-positive by the Cauchy-Schwartz inequality. We therefore have

dE m dt ≤ - 1 i=0 T×R ∂ y ln ψ ψ ∞ 2 ψ - 1 i=0 T×R ∂ x ln ψ ξ ψ ξ ∞ ∂ x ln ψ ψ ∞ ψ + T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ = - 1 i=0 T×R ∂ y ln ψ ψ ∞ 2 ψ - T ∂ x ln ψ ξ ψ ξ ∞ ψ ξ (A ′ t -A ′ ) (35)
≤ -2ρE m + 2ρE c + T |A ′ t -A ′ | 2 ψ ξ T ∂ x ln ψ ξ ψ ξ ∞ 2 ψ ξ .
Line (35) is a result of Lemma 5 and the last inequality is due to Lemma 9 and a further application of the Cauchy-Schwartz inequality. Now, using Lemmas 6 and 8, we obtain

dE m dt ≤ -2ρE m + 2ρE c + R 2E m F 0 e -8π 2 t , .
where we recall F 0 = F (ψ ξ (0, •)|ψ ξ ∞ ). Finally, using Young's inequality:

∀ε > 0, ∀a, b ∈ R, ab < εa 2 + 1 4ε b 2 , we obtain dE m dt ≤ -2 ρ -R 2 ε E m + 2ρE c + 1 4ε F 0 e -8π 2 t , (36)
where ε > 0 will be chosen optimally later in the proof. We are left to control the channel entropy term E c in order to conclude.

Controlling E c

The aim of this section is to obtain a control on the evolution of E c defined by (38), and more precisely an upper bound of E c , denoted P , which is the weighted χ 2 -distance between ψ ξ,I and ψ ξ,I ∞ , see (38) and (49) below.

Recall that E c = T H(µ I t|x |µ I ∞|x )ψ ξ dx, where the integrand is the relative entropy of a Bernoulli measure. Poincaré and logarithmic Sobolev inequalities have been studied for Bernoulli measures [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]. In order to obtain an exponentially decaying relative entropy, however, a suitable semi-group and its infinitesimal generator is needed for the measure. In the case of the bi-channel model (in particular [H1]), we face problems due to the region E ⊂ T, where no exchange is permitted between the two channels: in this region, the speed at which the measure µ I t|x reaches equilibrium cannot directly be controlled. To circumvent this issue, we consider the spectral gap of an adequate operator and resort to the Poincaré inequality.

By the definition of E c and using the inequality ∀x > 0, xln(x) ≤ x(x -1) and the fact that 1 i=0 T ψ ξ,I (t, x, i) dx = 1, we obtain

E c = 1 i=0 T ln ψ ξ,I ψ ξ ψ ξ,I ∞ ψ ξ ∞ ψ ξ,I = 1 i=0 T ln ψ ξ,I /ψ ξ,I ∞ ψ ξ,I -E M ≤ 1 i=0 T ψ ξ,I ψ ξ,I ∞ -1 2 ψ ξ,I ∞ dx -E M . (37)
We therefore have

E c ≤ P, (38)
where

P = 1 i=0 T ψ ξ,I ψ ξ,I ∞ -1 2 ψ ξ,I ∞ dx. (39)
In order to proceed and consider the time derivative of P , we will need some further results to express the evolution of the marginal density ψ ξ,I . The idea is to compare the evolution of ψ ξ,I with the dynamics of this density if A ′ t and T ∂ x V dµ t|x,i were already at equilibrium (see Section 2.4).

Lemma 10

The Fokker-Planck equation [START_REF] Minoukadeh | Potential of mean force calculations: A multiplewalker Adaptive Biasing Force approach[END_REF] for ψ ξ,I can be rewritten as

∂ t ψ ξ,I = ∂ x ψ ξ,I ∞ ∂ x ψ ξ,I ψ ξ,I ∞ + ∂ x R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ ψ ξ,I +∂ x ((A ′ -A ′ t )ψ ξ,I ) -λ(x)(ψ ξ,I -ψ ξ,I 1-i ) (40)
Proof : First we will show that

∂ t ψ ξ,I = ∂ x R ψ ∞ ∂ x ψ ψ ∞ dy + ∂ x ((A ′ -A ′ t )ψ ξ,I ) -λ(x)(ψ ξ,I -ψ ξ,I 1-i ). (41)
By developing the right hand side, we have

∂ t ψ ξ,I = ∂ x R ∂ x ψ - ψ ψ ∞ ∂ x ψ ∞ + ∂ x R (A ′ -A ′ t )ψ -λ(x)(ψ ξ,I -ψ ξ,I 1-i ) = ∂ x R ∂ x (V i -A • ξ)ψ + ∂ xx ψ ξ,I + ∂ x R (A ′ -A ′ t )ψ -λ(x)(ψ ξ,I -ψ ξ,I 1-i ) = R ∂ x (∂ x (V i -A t • ξ)ψ) + ∂ xx ψ ξ,I -λ(x)(ψ ξ,I -ψ ξ,I 1-i )
which is indeed the Fokker-Planck equation ( 15) associated to ψ ξ,I .

Next, we show that ∀x ∈ T, ∀i ∈ {0, 1}

R ψ ∞ ∂ x ψ ψ ∞ dy = ψ ξ,I ∞ ∂ x ψ ξ,I ψ ξ,I ∞ + R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ ψ ξ,I . (42)
To prove the above, notice that

R ψ ∞ ∂ x ψ ψ ∞ dy -ψ ξ,I ∞ ∂ x ψ ξ,I ψ ξ,I ∞ = R ∂ x ψ dy - R ψ ψ ∞ ∂ x ψ ∞ -∂ x ψ ξ,I + ψ ξ,I ψ ξ,I ∞ ∂ x ψ ξ,I ∞ = R (∂ x (V i -A • ξ))ψ dy -R ∂ x (V i -A • ξ)e -(Vi-A•ξ) dy R e -(Vi-A•ξ) dy ψ ξ,I = R (∂ x V i )ψ dy -A ′ (x)ψ ξ,I -R ∂ x V i e -(Vi-A•ξ) dy R e -(Vi-A•ξ) dy ψ ξ,I + A ′ (x)ψ ξ,I
Finally, by using the fact that the free energy A is independent of y, we obtain

R ψ ∞ ∂ x (ψ/ψ ∞ ) dy -ψ ξ,I ∞ ∂ x (ψ ξ,I /ψ ξ,I ∞ ) = R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ ψ ξ,I , (43)
as required. The final result (40) is obtained by substituting (42) into (41). ♦

Notice that (40) is comparable to (25), only with additional terms due to the fact that A t and

T ∂ x V dµ t|x,i
have not yet converged. The difference of the biasing force and mean force, A ′ t -A ′ , was already estimated in Lemmas 5 and 6. We are therefore left to control the remaining term.

Lemma 11 ∀x ∈ T, ∀i ∈ {0, 1}, R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ ≤ M 2 ρ H µ t|x,i µ ∞|x,i .
As a consequence,

1 i=0 T R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ 2 ψ ξ,I dx ≤ 2M 2 ρ E m . (44)
Proof : Let Π(µ t|x,i , µ ∞|x,i ) be the set of coupling measures on R × R with marginals µ t|x,i and µ ∞|x,i respectively and let π ∈ Π . Then

R ∂ x V i ψ ψ ξ,I -∂ x V i ψ ∞ ψ ξ,I ∞ = R×R (∂ x V i (x, y) -∂ x V i (x, y ′ )) π(dy, dy ′ ) ≤ ∂ x,y V i L ∞ R×R |y -y ′ | π(dy, dy ′ ) ≤ M 2 ρ H(µ t|x,i |µ ∞|x,i )
where we have used Lemma 1 since µ ∞|x,i satisfies LSI(ρ). Equation (44) follows immediately from the fact that 1 i=0 T H(µ t|x,i |µ ∞|x,i )ψ ξ,I dx ≤ E m . ♦ One result that is now needed to derive estimates on the evolution of P is the existence of a spectral gap of the operator describing the dynamics once A ′ t and R ∂ x V i dµ t|x,i have converged (see ( 25)). We now justify the existence of such a spectral gap.

In order to do so, let us define the vector spaces

V l = v : T × {0, 1} → R ∀i ∈ {0, 1}, v i ψ ξ,I ∞ (x, i) ∈ L 2 T, ψ ξ,I ∞ (x, i) dx , 1 i=0 T v i (x) dx = l and W l = w ∈ V l ∀i ∈ {0, 1}, w i ψ ξ,I ∞ (x, i) ∈ H 1 T, ψ ξ,I ∞ (x, i) dx , 1 i=0 T w i (x) dx = l .
A function φ in V l (or in W l ) will also be considered as a vector valued function as φ :

T → R 2 x → (φ 0 (x), φ 1 (x)) . Notice that φ ∈ W 1 if and only if f := φ -ψ ξ,I ∞ ∈ W 0 .
Lemma 12 Recall the operator L = (L 0 , L 1 ), with L i defined as in (25),

L i φ = -∂ x ψ ξ,I ∞,i ∂ x φ i /ψ ξ,I ∞,i -λ(x)(φ i -φ 1-i ) .
Then i) The operator L is symmetric and positive definite with respect to the inner product

f, g = 1 i=0 T f i (x)g i (x) 1 ψ ξ,I ∞ (x, i) dx.
ii) L has a spectral gap θ > 0 in the sense that

inf f ∈W0,f =0 f, Lf f, f = θ > 0. ( 45
)
Proof : i) To show symmetry of the operator L, consider functions ϕ, φ ∈ W 0 . Now, using the fact that ∀x ∈ T \ E, ψ ξ,I ∞,0 (x) = ψ ξ,I ∞,1 (x),

1 i=0 T ϕ i L i φ 1 ψ ξ,I ∞,i dx = - 1 i=0 T ϕ i ∂ x ψ ξ,I ∞,i ∂ x φ i ψ ξ,I ∞,i 1 ψ ξ,I ∞,i dx + 1 i=0 T λ(x)ϕ i (φ i -φ 1-i ) 1 ψ ξ,I ∞,i dx = 1 i=0 T ∂ x ϕ i ψ ξ,I ∞,i ∂ x φ i ψ ξ,I ∞,i ψ ξ,I ∞,i dx + T λ(x)(ϕ 0 -ϕ 1 )(φ 0 -φ 1 ) 1 ψ ξ,I ∞,0 dx (46) = - 1 i=0 T φ i ∂ x ψ ξ,I ∞,i ∂ x ϕ i φ ∞,i 1 ψ ξ,I ∞,i dx + T λ(x)(φ 0 -φ 1 )(ϕ 0 -ϕ 1 ) 1 ψ ξ,I ∞,0 dx = 1 i=0 T φ i L i ϕ 1 ψ ξ,I ∞,i dx.
From (46), we conclude positive definiteness of L

1 i=0 T φ i L i φ 1 ψ ξ,I ∞,i dx = 1 i=0 T ∂ x φ i ψ ξ,I ∞,i 2 ψ ξ,I ∞,i dx + T λ(x)(φ 0 -φ 1 ) 2 1 ψ ξ,I ∞,0 dx > 0. (47)
Notice that the above is strictly positive for any φ ∈ W 0 , since φ, Lφ = 0 if and only if φ 0 = φ 1 = 0. ii) In fact, one can check that ∃κ > 0, such that ∀φ ∈ W 0 , φ = 0,

1 i=0 T φ i L i φ 1 ψ ξ,I ∞ dx ≥ κ 1 i=0 T |φ i | 2 + |∇φ i | 2 1 ψ ξ,I ∞ dx.
Therefore, by the Lax-Milgram theorem, L -1 is well defined from V 0 to W 0 and thus compact from V 0 to V 0 . From the symmetry and positive definiteness of L, and the fact that its inverse is a compact operator from V 0 to V 0 , it has a strictly positive and discrete spectrum. There exists a set of eigenvectors (v n ) n≥1 , orthonormal with respect to the inner product •, • , forming a basis of V 0 and W 0 , and associated to an increasing sequence of eigenvalues (σ n ) n≥1 , such that lim n→∞ σ n = ∞. In particular, there exists a spectral gap:

θ = σ 1 > 0. ♦ Remark 4 In the case where a function φ ∈ W 1 satisfies ∂ t φ = Lφ, a consequence of Lemma 12 is ∀t ≥ 0, φ(t, •) -ψ ξ,I ∞ 2 ≤ Ke -2θt , ( 48
)
where

• 2 = •, • and K = n≥1 φ(0, •) -ψ ξ,I ∞ , v n 2
. This is easily obtained by noticing that φψ ξ,I ∞ ∈ W 0 and therefore can be expressed in terms of the orthonormal eigenvectors

(v n ) n≥1 φ(t, •) -ψ ξ,I ∞ = n≥1 φ(0, •) -ψ ξ,I ∞ , v n v n e -σnt .
The result (48) follows immediately since ∀n ≥ 2, σ n ≥ σ 1 = θ. Notice in particular that the Fokker-Planck equation (40) satisfied by ψ ξ,I can be written as

∂ t ψ ξ,I (t, x, i) = L i ψ ξ,I + ∂ x R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ ψ ξ,I + ∂ x ((A ′ -A ′ t )ψ ξ,I .
We will show that the last two terms can be controlled by E m , see (49).

With these tools at hand, let us consider the time evolution of the functional P defined in (39)

1 2 dP dt = 1 i=0 T ψ ξ,I ψ ξ,I ∞ -1 ∂ t ψ ξ,I ψ ξ,I ∞ ψ ξ,I ∞ dx = 1 i=0 T ψ ξ,I ∂ t ψ ξ,I ψ ξ,I ∞ dx - 1 i=0 T ∂ t ψ ξ,I dx = 1 i=0 T ψ ξ,I ∂ t ψ ξ,I 1 ψ ξ,I ∞ dx.
Using equation (40) of Lemma 10, we get 1 2

dP dt = 1 i=0 T ψ ξ,I ∂ t ψ ξ,I 1 ψ ξ,I ∞ dx = - 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ 2 ψ ξ,I ∞ dx - 1 i=0 T λ(x)ψ ξ,I i (ψ ξ,I i -ψ ξ,I 1-i ) 1 ψ ξ,I ∞,i dx - 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ (A ′ -A ′ t )ψ ξ,I dx - 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ ψ ξ,I dx.
Notice that, by developing the sum and using the fact that ψ ξ,I ∞,0 = ψ ξ,I ∞,1 for λ(x) = 0, the second term may be replaced by

T λ(x) ψ ξ,I 0 -ψ ξ,I 1 2 1 ψ ξ,I ∞,0
dx. Finally by using Young's inequality on the last two terms, we obtain for a parameter α > 0 to be chosen later on, 1 2

dP dt = - 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ 2 ψ ξ,I ∞ dx - T λ(x) ψ ξ,I 0 -ψ ξ,I 1 2 1 ψ ξ,I ∞,0 dx + 1 4α 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ 2 ψ ξ,I dx + α T |A ′ -A ′ t | 2 ψ ξ dx + 1 4α 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ 2 ψ ξ,I dx + α 1 i=0 T R ∂ x V i ψ ψ ξ,I -R ∂ x V i ψ ∞ ψ ξ,I ∞ 2 ψ ξ,I dx.
Next, by Lemmas 6 and 11, 1 2

dP dt = - 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ 2 ψ ξ,I ∞ dx - T λ(x) ψ ξ,I 0 -ψ ξ,I 1 2 1 ψ ξ,I ∞ dx + 1 2α 1 i=0 T ∂ x ψ ξ,I ψ ξ,I ∞ 2 ψ ξ,I dx + 2α M 2 ρ E m + 2αR 2 E m .
Notice that, in the third term, ψ ξ,I ≤ ψ ξ ≤ M for M = ψ ξ (0, •) L ∞ and 1 ≤ ψ ξ,I ∞ /c for c = min Finally, by grouping terms together and using the fact that α may be chosen such that M /2αc < 1 (an appropriate choice for α is given later in the proof), we have (ψ ξ,I L i ψ ξ,I ) 1

ψ ξ,I ∞ dx + 4αR 2 E m ≤ -1 - M 2αc θP + 4αR 2 E m , (49)
where the last line is a result of (45), with f := ψ ξ,Iψ ξ,I ∞ . Notice that f ∈ W 0 since the normalization for ψ ξ,I is

1 i=0 T ψ ξ,I dx = 1.
To complete the proof of Theorem 1, we now need to study the system of inequalities (36) and (49).

Completing the proof

To show that E m decays exponentially fast, we study the system of two inequalities (36) and (49). Since, from (38), E c ≤ P , the system to be studied is

         dE m dt ≤ -2 ρ -R 2 ε E m + 2ρP + 1 4ε F 0 e -8π 2 t , dP dt ≤ 8αR 2 E m -2 1 - M 2αc θP. (50)
The parameters α > M 2c and ε > 0 remain to be chosen in order to obtain an exponential convergence with the best possible rate. To fix α, let us first study the eigenvalues of the matrix of coefficients, neglecting the terms in ε.

Lemma 13 Let us assume [H4]. The matrix

A = -ρ ρ 4αR 2 -1 -M 2αc θ
is negative definite and α may be chosen so that the eigenvalues -λ ± of A are such that -λ -≤ -λ + = -Λ(θ) < 0 where Λ : (θ min , ∞) → (0, ρ) is a positive, increasing function. We recall that θ min = 8 M R 2 c , where R = C + M ρ -1/2 . The function Λ is such that Λ → 0 as θ → θ min and Λ → ρ as θ → ∞. Moreover, Λ(ρ + 2θ min ) = ρ 2 .

Proof : In order to prove the negative definiteness of the matrix A, we show that for certain values of α > 0, tr(A) < 0 and det(A) > 0. In the following, we only consider positive values of α (which is imposed by the previous computations). We have The interval (α -, α + ) is indeed well defined and included in [0, ∞) since θ > θ min = 8 M R 2 /c (hypothesis [H4]). We seek an optimal α that minimizes eigenvalue -λ + and satisfies (51) and ( 52).

An analytical solution cannot be easily obtained. We choose α = α * := M c , which appears to be very close to the optimal choice, from numerical computations. Notice that α * satisfies (51) and (52) since, for α = α * , tr(A) = -ρθ/2 < 0 and det(A) = θρ/2 -4R 2 M ρ/c > 0.

The eigenvalues of the matrix are now given by

-λ ± = 1 2   -ρ + θ 2 ± ρ - θ 2 2 + 16R 2 M ρ c   < 0.
The rate of convergence of the system is given by the largest of the two eigenvalues -λ + . Let us introduce the function

Λ(θ) = - 1 2   -ρ + θ 2 + ρ - θ 2
≤ 2Y T AY + 2R 2 εE 2 m + 1 4ε F 0 e -8π 2 t E m ≤ -2Λ(θ) Y 2 2 + 2R 2 ε Y 2 2 + 1 4ε F 0 e -8π 2 t Y 2 ,
and as a result,

d Y 2 dt ≤ -2(Λ(θ) -R 2 ε) Y 2 + 1 4ε F 0 e -8π 2 t . ( 53
)
For arbitrary small ε > 0, let us consider λ ε = Λ(θ) -R 2 ε < Λ(θ). We may assume without loss of generality that λ ε = 4π 2 . Then, from (53), one gets:

E m ≤ Y 2 ≤ K ε e -2 min{λε,4π 2 }t , (54)
where

K ε = 2 max E 2 m (0) + P 2 (0), F 0 8ε|λ ε -4π 2 | ,
which concludes the proof of (27). The exponential convergence of the total entropy E results from the relation E = E M + E m , (54) and Lemmas 7 and 8. The Csiszar-Kullback inequality implies the same for ψ(t, •)ψ ∞ 2 L 1 . Finally, the convergence results on A ′ t are easily obtained from Lemma 6 and the fact that ψ ξ is bounded from below by a positive constant for times larger than an arbitrary small positive time, see the beginning of Section 3.3.2 in [START_REF] Lelièvre | Long-time convergence of an adaptive biasing force method[END_REF] for more details.

Figure 1 :

 1 Figure 1: Left: Contour plot of a 2-dimensional potential energy surface demonstrating the bichannel scenario. Right: In the bi-channel model the two channels are described by two potential functions V i : D → R, i ∈ {0, 1}. Exchange between the two channels is allowed only in regions T\E at a rate λ > 0.

2 ψ

 2 ξ,I ∞ dx + 4αR 2 E m .

  ρ > 0 iff α ∈ (α -, α + ), α ± = θc ± θ 2 c 2 -8 M θR 2 c 8R 2 c .(52)

  λ + = Λ(θ). It is easily shown that Λ is an increasing function of θ with Λ(θ) → 0 as θ → 8 M R 2 c , ρ as θ → ∞.Moreover, it is easy to check that Λ(ρ + 2θ min ) = ρ 2 , which concludes the proof. ♦We are now in position to complete the proof of Theorem 1. Let us define Y (t) = (E m (t), P (t)). Using (50) the fact that E m ≤ Y 2 (where Y 2 denotes the Euclidean norm of the twodimensional vector Y )

  x, y, i) dy.

	Measure	Definition	Description of measure
		1
	dµ t
		i=0
	dµ ξ t	ψ ξ (t, x) dx	Marginal measure in reaction coordinate
	dµ t|x,i	ψ(t, x, y, i) dy ψ ξ,I (t, x, i)	Measure conditioned to being at x and i
		1
		ψ(t, x, y, i)δ i dy
	dµ t|x	i=0	ψ ξ (t, x)	Measure conditioned to being at x
		1
			ψ ξ,I (t, x, i)δ i
	t|x dµ I	i=0

ψ(t, x, y, i)δ i dx dy Probability measure in domain D × {0, 1} dµ ξ,I t 1 i=0 ψ ξ,I (t, x, i)δ i dx Marginal measure in reaction coordinate and channel ψ ξ (t, x) Marginal in I, conditioned to being at x Table 1: A list of probability measures.

Acknowledgements

This work is supported in part by the Agence Nationale de la Recherche, under the grant MEGAS (ANR-09-BLAN-0216-01). The authors are grateful to Benjamin Jourdain, Eric Cancès, Andreas Eberle, Felix Otto and Gabriel Stoltz for helpful discussions.