Krishna Kant Singh

Stephane Redon

Single-pass Incremental Force Updates for Adaptively Restrained Molecular Dynamics

Keywords: Neighbor Lists, Incremental Force Update, Adaptively Restrained Molecular Dynamics

Adaptively Restrained Molecular dynamics (ARMD) allows users to perform more integration steps in wall-clock time by switching on and off positional degrees of freedoms. This article presents new, single-pass incremental force updates algorithms to efficiently simulate a system using ARMD. We assessed different algorithms for speedup measurements and implemented them in the LAMMPS MD package. We validated the single-pass incremental force update algorithm on four different benchmarks using diverse pair potentials. The proposed algorithm allows us to perform simulation of a system faster than traditional MD in both NVE and NVT ensembles. Moreover, ARMD using the new single-pass algorithm speeds up the convergence of observables in wall-clock time.

Introduction

Molecular Dynamics (MD) may serve as a computational microscope to decipher the structural and dynamical behavior of complex systems 1 . In particular, MD may provide atomistic descriptions over several time scales (e.g. femtoseconds to milliseconds) [2][3][4][5][6][7] . Unfortunately, simulating complex systems over experimental time scales is still computationally challenging 8 .

Numerous attempts have been made to either speed up simulations or enhance phasespace sampling, for example through the use of additional biased potentials, which help trajectories escape from local minima [8][9][10][11] . Complex systems can also be simplified through coarse-grained simulations [12][13][14][15] . Other strategies involve modifications of the integration time step, which may be increased either by increasing the mass associated with the fastest mode present in the system, by repartitioning the mass, or by creating virtual sites 16,17 . MD simulations have also been accelerated by optimized algorithms that perform mathematical operations faster, sometimes on special-purpose supercomputers 18 .

In many simulations, most of the time is spent computing non-bonded forces, typically divided into long-range and short-range forces. Long-range interactions involve electrostatic interactions that are generally computed using specialized methods, including the reaction field method 19,20 , the cell-multipole method, the fast multipole method 21 , particle Mesh Ewald (PME) 22 or the Wolf method [23][24][25][26][27][28] . Short-range forces decay fast and are generally truncated at a cut-off distance r c . These forces are efficiently computed using neighbor lists.

With the Wolf method, long-range forces may also be computed using neighbor lists, and this method recently attracted a great deal of attention for computing properties of electrolytes and simulating biomolecules in explicit solvent 25 .

Adaptively Restrained Molecular Dynamics (ARMD) is a recent approach that attempts to perform more integration steps per wall-clock second by reducing the number of computations per time step 29 . In ARMD, positional degrees of freedom are adaptively turned on and off during the simulation, based on their instantaneous kinetic energy. When two particles are restrained, the forces they apply on each other do not need to be updated from one time to the next. As a result, incremental force update algorithms, i.e. force calculations that keep track of the total force applied on each particle and update forces involving active particles, may significantly speedup the simulation [29][30][31][START_REF] Singh | Modelling and Simulation in Materials Science and Engineering[END_REF] .

Up to now, however, such incremental algorithms have required two passes: one to subtract interactions involving from the previous time step, and one to add updated interactions for the current time step. As a result, previous incremental force update algorithms have been useful in situations where at least half of the simulated particles were restrained. In this article, we present a single-pass incremental force update algorithm for ARMD that overcomes this limitation. The proposed algorithms are implemented in the LAMMPS MD package and validated by performing simulations of diverse systems in both the NVE and NVT ensembles.

Adaptively Restrained Molecular Dynamics

For completeness, we briefly review the ARMD methodology 29 . In ARMD, the time evolution of a system containing N particles is obtained by the Adaptively Restrained (AR) Hamiltonian:

H AR (q, p) = 1 2 p T Φ(q, p)p + V (q), (1)
where Φ(q, p) is the inverse inertia matrix that adaptively imposes restraints during the simulation. Precisely, particles whose kinetic energy is smaller that a restrained-dynamics threshold (ε r i ) are restrained, and their positions remain unchanged. Particles whose kinetic energy is greater than a full-dynamics threshold ε f i are considered as active, and have normal dynamics. Transition particles have kinetic energies between ε r i and ε f i (in the context of force calculations, transitioning particles are mobile, and are thus treated as, and called, active particles). The status of each particle can evolve during the simulation, and the Adaptively Restrained Hamiltonian (1) ensures that stable simulations can be performed.

Equilibrium statistics can be recovered when performing AR Langevin dynamics [START_REF] Redon | [END_REF] . In order to compute position-dependent equilibrium averages, equation 1 can be re-written as:

H AR = 1 2 p T M -1 p + V (q) + 1 2 p T (φ(q, p) -M -1 )p H AR = H + V AR . (2)
In equation 2, V AR is a biased term that depends upon the momenta of particles and averages can be estimated 29 . In particular, correct position-dependent equilibrium averages can be estimated as 29 :

< A(q) > H AR = A(q)e -H(q,p) KT dq e -H(q,p) KT dq =< A(q) > H . (3)
From equation 3, it is to be noted that position-dependent averages from an ARMD and an MD simulations are the same.

Due to the biased term V AR , ARMD might not be directly useful to compute dynamical properties or properties that depend upon the fluctuations of kinetic energies.

Algorithms for AR Molecular Dynamics

Let C denote the complete list of particles, A denote the list of active particles, and R denote the list of restrained particles. Let S A (resp. S R ) denote the list of particles that switched to being active (resp. restrained) between the previous and current time step.

In ARMD, a system of N particles can be referred to as a combination of N A active and N R (= N -N A ) restrained particles. Thus, at any time-step, interactions between particles can be categorized as active interactions (interactions involving at least one active particle)

and restrained interactions (interactions between restrained particles only). Precisely, active interactions involve forces between either two active particles (F AA ) or one active particle and one restrained particle (F AR ), while restrained interactions involve forces between two restrained particles (F RR ). Figure 1 shows the three types of interactions on an example.

Most often, forces acting on particles only depend upon their relative positions. At a given time step, relative positions between restrained particles do not change; hence, restrained interactions and associated forces remain unchanged. This eliminates the need to compute F RR force components involving restrained interactions, and the F RR force components from the previous time step can be cached and reused. However, active particles update their positions, and the associated force components (F AA and F AR ) need to be updated at each time step.

Active neighbor lists

In order to compute forces due to active interactions, we previously introduced Active Neighbor Lists (ANLs) i.e. lists of active neighbor particles at a given time-step [START_REF] Singh | Modelling and Simulation in Materials Science and Engineering[END_REF] . The ANLs are constructed from Full Neighbor Lists (FNLs), and the ANL of an active particle i can be

described as AN L(i) = {∀j ∈ F N L(i) : j ∈ R ∪ (j ∈ A ∩ (j < i))}.
In other words, AN L(i) contains all the restrained neighbors of i, and the active neighbors of i with smaller indices (due to the condition j < i).

Two-pass incremental force update algorithm

In MD, the forces acting on all particles are typically computed once, after updating particles positions. In contrast, in ARMD, only the positions of active particles are updated, and forces are updated based on the new positions of active particles, thereby eliminating the need to compute all forces. In our previous approach, active interactions were incrementally updated using an algorithm that involved two force update steps 29,30,[START_REF] Singh | Modelling and Simulation in Materials Science and Engineering[END_REF] :

1. First pass or subtraction step: This pass removes the force increments involving active particles based on the old positions, by computing the force increments due to active particles and subtracting them from the total force. This incremental force update algorithm requires force increments to be calculated twice in a single integration step, limiting the usefulness of ARMD to cases where at least 50% of particles are restrained [START_REF] Singh | Modelling and Simulation in Materials Science and Engineering[END_REF] . Furthermore, if all particles are active, the proposed incremental force algorithm would be twice times slower than the normal MD algorithm.

Single-pass incremental force updates

In order to overcome the aforementioned limitation, we have designed a single-pass incremental force computation algorithm.

In ARMD, the force F i acting on a particle i can be written as 1), where F i AA refers to forces between two active particles, F i AR refers to forces between an active and a restrained particle and F i RR refers to the force between two restrained particles.

F i = F i AA + F i AR + F i RR (figure
Assume there are no switching particles between the n-th and (n + 1)-th time step. Using the two-pass incremental force computation algorithm, the position of an active particle i is updated, and the total force F i = F i AA + F i AR applied to it is updated during the addition step. Since the new position is not updated after the addition step, though, the next substraction step substracts the same force increments, yielding F i = 0 after the subtraction step. Therefore, in case of an active particle, the subtraction step can be replaced by assigning zero to the associated force.

In case of a restrained particle i, the total force applied to it is F i = F i AR +F i RR . As noted above, F i RR can be cached and reused, so that we only have to compute F i AR . In singlepass algorithm, force on the particle i is assigned as F i = F i RR before computing the force component, unlike the previous algorithm that adds and removes F i AR force component (as the F i RR force component remains unchanged and could be retained from the previous time step). Also, this algorithm eliminates the need to carry out subtraction step and subtraction step can be removed with algorithm 1. However, the time evolution of the system switches the state of the particles, force components associated with these switched particles need to be computed, before applying algorithm 1. Particles can switch their states in two ways:

1. Switching to active state

Switching to restrained state

It is likely that the switching in the state of particles may cause the change in the type of interaction (to an active or restrained interaction) and hence the associated force components.

An active interaction associated with pair i -j can switch to a restrained interaction in the following two ways:

• Assuming at time step n both particles i and j are active and they switch to a restrained state at n + 1 time step, the F AA force component associated with the pair i -j would now switch to the F RR force component. As ANLs avoid computation of F RR force component, this switched force component needs to be computed and stored for next time steps.

• On the other hand, if we assume particles i and j to be active and restrained respectively at a time step n (associated force component has type F AR ), and at n + 1 time step the particle i changes to a restrained state, this switching updates the interaction type from active to restrained and associated force component from F AR to F RR . As mentioned previously, the updated F RR force component needs to be computed and stored for the next time-steps.

Similarly, a restrained interaction belonging to a pair i -j can switch to an active interaction in the following ways:

• In case, particle i or j switches their state from restrained to an active state, force component F RR associated with this pair switches to F AR . This updated F AR force component needs to be computed and subtracted from the forces of restrained neighbors of the this switched particle i.

• Another possibility wherein both particles i and j switch from active to restrained state, the associated F RR force component switches to the F AA force components. This type of switching does not require any force computation as zero can be assigned for the forces associated with i and j.

In order to compute the switched force components (F RR or F AR ), we used a modified active neighbor list AN L . The AN L of a switched particle i is an extracted AN L that contains only restrained neighbors. Algorithm 2 gives a description of extracting the AN L from the AN L. Algorithm 3 shows the pseudo-code to compute switched force components.

Line 3 of algorithm 3 computes F RR force components and stores them in F + and line 9

computes F AR force components and stores then in F -for the switched particles.

Algorithm 1 gives the pseudo-code to perform the ARMD integration step using the ANLs and incremental force update algorithm. In this algorithm, before computing forces based on the new positions, zeros (line 3) and the force components F RR (line 6) are assigned to the forces corresponding to active and restrained particles respectively.

Algorithm 1: SingleP assIncrementalF orceComputation()

1 for i ∈ C do 2 if i ∈ A then 3 F + i ← 0 4 end 5 else 6 F + i ← F + i -F - i 7 end 8 F - i ← 0 9 end 10 F ← F + + ComputeF orces(A, AN L)

Analysis

In this section, we assess different algorithms for constructing ANLs and incrementally update forces. We compare these algorithms with brute-force MD algorithms which have a running time equal to τ M D = N * τ F N L + N * τ F , where N is the total number of particles, τ F N L is the time needed to build the FNL of one particle, and τ F is the time required to

Algorithm 2: ExtractAN L (i) AN L (i) ← ∅ for j ∈ AN L(i) do if (i ∈ R and j ∈ R) or(i ∈ A and (j ∈ R and j / ∈ S)) then AN L (i) ← AN L (i) ∪ j end end
Algorithm 3: SwitchedF orces()

for i ∈ S R do
ExtractAN L (i) SingleP assIncrementalF orceComputation() compute the total force acting on one particle due to its neighbors. To simplify the analysis, we consider the timing required to construct the FNL rather than the HNL.

F + ← F + + ComputeF orces(i, AN L (i)) ClearAN L(i) end for i ∈ S A do BuildAN L(i) ExtractAN L (i) F -← F -+ ComputeF orces(i, AN L (i)) 10 end 10
Algorithm 4: ARMD integration step 1 Update momenta 2 Update A, R, S A , S R 3 if (U pdateN eeded) then 4 for i ∈ C do 5 if i ∈ A then 6 Build F N L(i) and AN L(i) simultaneously

Time complexity

In this algorithm, the FNLs of all particles and the ANLs of active particles were constructed at the same time. The ANLs of particles that become active were constructed using the FNL of the corresponding particle, as building the ANLs from scratch is relatively more time consuming. The force components F AR or F RR for switched particles were then computed in two steps (Algorithm 3). The first step involves extracting AN L from ANL, and the second step involves computing specific force components with AN L .

The computation time for algorithm 4 is given by:

τ ARM D1 = N * τ F N L + N A * τ F + N SA * τ F AN L + N SA * τ SF + N SR * (τ SF + τ CL ) + N S * τ Ex (4)
τ Ex : Time to build the AN L for one particle.

τ F AN L : Time to construct the ANL from the FNL. Algorithm 4 is more efficient than classical MD when τ ARM D1 < τ M D , i.e.:

N A * τ F + N SA * τ F AN L + N SA * τ SF + N SR * (τ SF + τ CL ) + N S * τ Ex < N * τ F . (5)
To simplify the analysis, we assume the following worst cases:

Assumption 1: All switching particles switch to an active state, making it necessary to construct AN Ls for all switched particles: N SR = 0 and N S = N SA .

Assumption 2: Computing a force component F AR or F RR for a particle takes as much time as computing the total force on this particle: τ SF = τ F .

Assumption 3: Even though a) computing the AN L of a particle from its F N L and b) extracting the AN L of a particle from its AN L are both much more efficient than computing the F N L of a particle, we assume that the sum of τ F AN L and τ Ex is equal to

τ F N L : τ F N L = τ F AN L + τ Ex .
Considering these three assumptions, inequality 5 holds when:

N A * τ F + N S * τ F N L + N S * τ F < N * τ F , (6)
i.e. when:

N A N + N S N (1 + τ F N L τ F ) < 1. ( 7
)

Optimization

Although Algorithm 4 computes the AN Ls and AN L s required to perform the singlepass force update algorithm, it computes the F N Ls of all particles when neighbor lists are updated from scratch. Since the F N L of a restrained particle is only useful when it switches to an active state, however, Algorithm 4 may be optimized by maintaining hasF N L, a list of particles for which the F N L has been computed, and using this list to determine when to update the various neighbor lists.

Algorithm 5 describes the optimized version. When a particle switches to a restrained state, the ANL of this particle is cleared, while retaining its FNL. When a particle switches to an active state and it does not have a FNL, its ANL and FNL are constructed on-the-fly (i.e. outside the neighbor list construction step). When a particle switches to an active state and already has an FNL (such as when a particle switches states multiple times between two neighbor list construction steps), the ANL of this particle is constructed from the existing FNL. Algorithm 6 shows how Algorithm 3 is modified in the optimized case, when hasF N L is available.

The computation time for Algorithm 5 is given by:

τ ARM D2 = N A * τ F N L +N A * τ F +N SA * τ F N L +(N SA -N SA ) * τ F AN L +N SA * τ F +N SR * (τ SF +τ CL )+N S * τ Ex , (8)
for i ∈ S R do
ExtractAN L (i))

F + ← F + + ComputeF orces(i, AN L (i))
ClearAN L(i) ExtractAN L (i)

end for i ∈ S A do if i ∈ hasF N L then BuildAN L(i)
15

F -← F -+ ComputeF orces(i, AN L (i)) 16 end 15
where N SA is the number of particles without a FNL that have switched to an active state.

In order to compare the time complexity of Algorithm 5 with the MD algorithm, we considered the same assumptions as in the Algorithm 4) (N SR = 0, τ F AN L + τ Ex = τ F N L and τ F = τ SF ), and added another worst-case scenario assumption, where all switching particles switch to an active state and do not have an FNL:

N S = N SA = N SA .
After substitution, Equation 8 changes to:

τ ARM D2 = N A * τ F N L + N A * τ F + N S * τ F N L + N S * τ F + N S * τ Ex . (9)
Since both N S and τ Ex are small, we may neglect their product. As a result, Algorithm 5 is more efficient than classical MD when N A + N S ≤ N , which is generally true for ARMD.

The difference in the computation times of Algorithms 4 and 5 is:

τ ARM D1 -τ ARM D2 = (N -N A -N SA ) * τ F N L . ( 10
)
If the sum of the number of active particles and the number of particles that switch to an active state is smaller than the total number of particles (which is true most of the time since the number of switching particles is small), then Algorithm 5 performs better than Algorithm 4.

Results and discussions

The algorithms introduced above were validated on the following benchmarks:

1. Systems of Lennard-Jones particles.

2. Simulation of Crystal NaCl.

3. High-velocity impact of nanodroplet.

4. Simulation of a polymer in the solvent.

We note that, since the ARMD methodology has been validated beforehand 29,31 , the aim of these benchmarks is to validate the single-pass incremental force update algorithms.

Systems of Lennard-Jones particles

We performed simulations of different numbers (500, 4000 and 108000) of Lennard-Jones particles using the AR integrator in the NVE ensemble. All simulations were carried out in reduced units (lj units) using the LAMMPS MD package. Particles were generated on a fcc lattice with density 0.8442 and initial velocities were assigned according to the Boltzmann distribution at temperature k B T = 1.44. For all simulations, we used a time-step of 0.005 and interactions beyond a distance of 2.5σ were ignored. Periodic boundary conditions were employed in all three directions.

We performed two series of benchmarks. In the first series, the percentage of restrained particles was constrained by assigning very high AR parameters to a specific number of particles (and zero AR values to the others), to ensure they would not switch their state and remain restrained. In the second series, we let particles switch their state and assigned all particles the same (lower) AR parameters.

We compared the run time of our algorithms with that of LAMMPS algorithms. The reference simulations were performed using LAMMPS neighbor lists and force algorithms, whereas ARMD simulations were performed using the ANLs and single-pass incremental force algorithms. The neighbor list was updated at every 20 time-steps. For speedup measurements, we computed the average time spent at each integration step and the time spent in the construction of the ANL as well. These values were then compared to the average time spent per integration step and per construction of neighbor list in the reference simulations.

Series 1 (constrained number of restrained particles): Figure 2 shows the achieved speedup in constructing the ANL vs. the percentage of restrained particles. The construction of both the ANL and the FNL required a 27r 3 c volume to be searched, in contrast to the HNL, which required half of that volume. As a result, the construction time for building the ANL or the FNL was twice that of the HNL. However, in order to have a speedup when constructing the ANL, at least 50% of the particles are required to be restrained. Also, we found that the obtained speedup in constructing the ANL was the same regardless of the number of particles present in the systems. In conclusion, we observed a 2X (resp. 4X) speedup in constructing the ANL while restraining 80% (resp. 90%) of the particles.

Figure 3 shows the achieved speedup per integration step with respect to the percentage of restrained particles in the system. Thanks to the new single-pass force update algorithm, there is no constraint on the minimum number of restrained particles. In fact, even with only 40% of particles restrained, we achieved a 1.5X speedup. With 60% of the particles restrained, our algorithm performed twice as fast as the LAMMPS algorithm. Furthermore, 3X to 5X speedups were observed when 80 -90% particles were restrained.

Finally, we measured the overall speedup which encompasses the time to built the NLs and the time to perform integration steps. Figure 4 shows the overall speedup vs. the percentage of restrained particles. For less than 50% of restrained particles, the new single-pass incremental algorithm is twice faster than the two-pass incremental algorithm. As expected, the single-pass incremental algorithm is always faster than the two-pass incremental algorithm due to the reduced number of force computations. When comparing with traditional MD and the number of restrained particles was less than 20%, however, no speedup was achieved. This is explained by the cost of constructing the ANL, which is approximately twice the cost of constructing the HNL. Overall, A similar trend to the speedup in Figure 3 Series 2 (identical AR parameters for all particles): In this series of benchmarks, particles were allowed to switch states during simulation. Table 1 shows that the average number of switched particles at each time step was much smaller than the number of particles in the system. Thanks to this, switched particles did not have much influence on the obtained speedup, which reproduced the patterns observed in Series 1, and the single-pass force update algorithm always performed better than the two-pass incremental algorithm.

For the system containing 500 particles, with the single-pass incremental algorithm, a 2.1X to 3.5X speedup was observed with 81% to 86% particles restrained, and a maximum speedup of 4.5X was achieved when 92% of the particles were restrained. The AR parameters, the average number of switched particles and the average number of restrained particles can be found in Table 1.

For the system containing 4000 particles, a 3.8X speedup was attained when 91% of the particles were restrained, whereas in Series 1 the corresponding speedup was 4.5X with the same number of restrained particles. When 96% and 98.5% of the particles were restrained, we achieved a 6X and 8.9X speedup respectively. A 1.2X speedup was observed with the 57% of the particles as restrained particles. In another system containing 32000 particles, a 2X speedup was achieved with 46% of restrained particles. A 4X to 6X speedup was observed with 85% to 98% of particles restrained. For the system containing 108000 particles, a maximum speedup of 7.5X was observed with 97% of restrained particles. The reduction in overall speedups in Series 2 as compared to Series 1 is due to switched particles (building ANL and computing force components), and the fact that the observed percentages of restrained particles are averages.

Simulation of NaCl

In order to validate our algorithm on a computationally expensive potential, we simulated a system containing 8000 (4000 N a + and 4000 Cl -) NaCl particles. In this NaCl system, charged particles N a + and Cl -interact via electrostatic interactions. In MD, during force calculations, most of the computational time is used in calculating electrostatic forces. In the NaCl system, around 90% of the total time is spent in the computation of electrostatic interactions.

The NaCl system was simulated using the Tosi-Fumi (TF) potential augmented with a coulombic potential. The TF potential is given by equation 11, and this potential is broadly used in simulation of alkali halides.

U (r) = A exp σ -r ρ - C r 6 + D r 8 r < R c ( 11
)
The first term in equation 11 is the Born-Mayer exponential repulsive term and the second term involves 8, 6 Van der Waals attractive interaction. Parameters for TF potential are taken from reference 34 . Instead of calculating electrostatic forces based on Ewald summation method, we used Wolf summation. The Wolf method is computationally efficient (O(N ))

as compared to the Ewald-based methods. Furthermore, the Wolf method can use neighbor lists for computing electrostatic forces:

E W olf = 1 2 N i=1 j =i r ij <Rc q i q j erfc(αr ij ) r ij - q i q j erfc(αR c ) R c - erfc(αR c ) 2R c + α √ π N i=1 q 2 i 4π 0 (12)
In equation 12, erfc is the complementary error function, q i and q j represent the point charges on particles i and j, α is the damping parameter and R c is the cut-off radius. Details regarding the Wolf method can be found from references [23][24][25][26][27][28] . For the Wolf method, we used the same parameters as those mentioned in the literature 35 .

Cut-off distances of 7.5 Å and 15 Å were used for the TF potential and Wolf summation respectively. The system was simulated for 100000 time-steps using an integration time-step of 2f s in the NVE ensemble. In order to measure the speedup, we performed reference MD simulations as well as ARMD simulations with different AR parameters. For timing measurements, we ran 50 independent simulations (MD and different ARMD simulations)

for each combination of parameters and timings were averaged over these 50 simulations.

Different AR parameters (Table 2) give rise to different average numbers of active and switched particles. Figure 6 shows the speedup obtained with ARMD compared to MD.

We achieved a 2X speedup with 65% of restrained particles, and a 4X to 10X speedup was achieved with 82% to 96% of restrained particles. This benchmark shows that combining the Wolf method with ARMD significantly reduces the number of calculations, which may result in a significant speedup.

High-velocity impact of a nanodroplet

The high-velocity impact of a nanodroplet on a crystal surface changes the state of the crystal to an amorphous state, and this process is known as amorphization 36,37 . We performed this benchmark to show that ARMD simulations may offer important speedups on such processes.

Our model system contains three types of particles, namely 1) nanodroplet, 2) target slab and 3) boundary particles. The nanodroplet consists of 2891 identical particles that are spherically distributed in a hexagonal close-packed arrangement (blue particles shown in Figure 7); the target slab consist of 344,988 identical atoms on the fcc lattice (grey and red particles shown in Figure 7); the boundary of the slab remains fixed and has no velocity (green particles shown in Figure 7). Interactions among particles were computed using the Lennard-jones potential. All simulations were performed using a 1 f s integration time step and ran for 75 ps. The initial velocity of the nanodroplet was set to 4 km/s (in negative z direction). Due to the high velocity of the nanodroplet, the neighbor list was constructed every fifth time step.

In order to measure the amorphization process, we observed the radial distribution function (RDF) of the impact volume of the target slab (red particles in Figure 7) for different sets of AR parameters, and compared the obtained RDFs with the one obtained with a reference MD simulation. In this benchmark, most slab particles were initially restrained, and gradually started to switch to an active state after impact.

As shown in Figure 7, the crystalline structure at the impact area completely changes to a non-crystalline structure (amorphous state). This figure also shows that ARMD simulations allow us to trade between speed and accuracy. Lower values of AR parameters produce a trajectory similar to the MD trajectory, but higher values of AR parameters produce higher speedups. Figure 8 shows the obtained RDFs of the impact area (particles denoted as red in figure 7) at different time-steps. At the beginning of the simulation (at t = 0ps), the impact area had a crystalline structure. As the simulation proceeds, this crystalline structure started to deform and changed to an amorphous state (RDFs at t = 15, 30, 45, 60 and 75ps). During the amorphization process, the RDFs obtained from ARMD simulations mostly coincide with the RDF obtained using MD (Figure 8). The RDFs of the impact area at 75ps obtained using both ARMD and MD had their first peak at 2.35 Å and second peak at 4.5 Å. In order to measure the speed obtained by ARMD, we ran each simulation 50 times and computed the average time spent. We measured the speedup with respect to the reference MD simulation.

ARMD achieved 2.3-7X speedup with 56 -85% restrained particles as compared to MD.

Table 3 shows the speedup obtained with different AR parameters. This benchmark shows that ARMD saves wall-clock time while obtaining the structural properties of a specific part of the system much faster than classical MD, thanks to the automatic particle state switching resulting from the AR Hamiltonian 29 .

A single polymer chain in solution

This model is used to study the properties of a polymer in solution. This model is also a representative model of the biological coarse-grained model of protein in water. In this benchmark, we test the possibility for ARMD to obtain statistical averages faster than with MD. We performed ARMD of the system with different AR parameters and compared our results with those obtained using MD. A polymer chain of 30 monomers immersed in 4000 solvent particles was simulated in the NVT ensemble at temperature k B T = 1.2 with density ρ = 0.86, while using a time step of 0.001. Periodic boundary conditions were imposed in all the three directions. The FENE potential was used for the backbone (eq. 13) and truncated Lennard-jones (WCA) potential was used for pair interactions. Initial velocities were assigned according to the Boltzmann distribution.

U F EN E (r) = - K 2 R 2 0 ln 1 - r 2 R 2 0 (13)
The system was initially minimized and then equilibrated for 10000 steps. After the equilibration period, the system was simulated for 10 9 steps in the NVT ensemble. Initial velocities were assigned using Maxwell-Boltzmann distribution at temperature k B T = 1.2.

One of the main goal of MD is to compute the statistical properties of the system by calculating averages. Averages obtained by MD are time averages and, if simulation is long enough to be converged, these time averages are equal to ensemble averages (ergodic hypothesis). However, averages over a trajectory are subject to two types of errors: systematic bias and statistical errors. Systematic bias is due to the use of a discrete time step, and statistical errors occur due to the quality of sampling (and may be large if averages are obtained from an undersampled trajectory).

Statistical errors in time averages may be estimated by measuring the variance of an observable l. The variance can be measured either by correlation time analysis or by a block-averaging scheme [38][39][40][START_REF] Lyman | The journal of physical chemistry[END_REF] . In previous studies of ARMD, time correlation analysis was used to measure errors and variance [START_REF] Redon | [END_REF] , and we use the same approach for error estimations of ARMD trajectories in the present paper. The correlation time τ l of an observable l is the simulation time required for a trajectory to de-correlate its value from an initial value l 0 . Therefore, the correlation time for an observable provides an estimation of N ind l , the number of statistically independent values of the observable present in the trajectory: if t sim denotes the simulation length of a trajectory, then N ind l ∼ t sim /τ l . Larger values of N ind l suggests good sampling for the given observable. The time τ l depends upon the nature of observable 42 .

In ARMD, restraining positions of particles introduces additional correlation in the system, thus yielding a larger correlation time for a given observable, as compared to MD

(τ ARM D l ≥ τ M D l
). If ARMD simulations have the same length as MD simulations, then statistical errors are larger:

τ ARM D l ≥ τ M D l and t ARM D sim = t M D sim =⇒ N ind l ARM D ≤ N ind l M D
Fortunately, statistical errors of averages obtained by ARMD may be reduced, and more statistically independent values may be obtained, by performing simulations with longer lengths (more time steps), since each time step costs less in wall-clock time. Therefore, the overall speedup in the NVT ensemble is a function of the speedup obtained in wall-clock time at each time step, and the time required to attain a given precision in the estimation of a given observable [START_REF] Redon | [END_REF] . This speedup can be expressed as:

S = S algo σ 2 M D σ 2 ARM D (14)
where S algo represents the computational speedup at each time step adaptive algorithms (ANLs and single-pass incremental force update algorithms), σ 2 M D is the variance of a given observable when using MD, and σ 2 ARM D is the variance of the same observable when using ARMD.

In this benchmark of a polymer in solvent, we chose the end-to-end distance of the polymer as an observable, and computed the correlation times with different AR parameters.

Figure 10 shows the time correlation functions computed for end-to-end distances. As expected, the correlation function in the MD case is reduced in fewer time steps compared to the correlation functions in ARMD simulations. In wall-clock time, however, some ARMD trajectories decorrelate the end-to-end distance faster than MD, due to the reduction in the average cost of a time step.

Table 4 shows the averaged values of end-to-end distances and radius of gyration of the polymer chain. The averages obtained with MD and ARMD are approximately the same. This experimentally shows how, when the observable is converged, averages obtained with ARMD are the same as the ones obtained with MD (see 29 for a mathematical proof). Figure 11 shows the 2D projection of all trajectories as a function of end-to-end distance and radius of gyration of the polymer. Here as well we obtain unbiased position-dependent averages. This illustrates that ARMD samples the same conformational space as the MD in the NVT ensemble 29 .

Conclusions

We have presented a novel single-pass force update algorithm to speed up ARMD simulations.

Unlike our previous two-pass algorithms, the new algorithm may result in a speedup even when a small percentage of particles is restrained. We have validated the approach on several benchmarks, and have shown that the single-pass algorithm may be applied to computing electrostatic interactions with the Wolf method. We showed how ARMD may be used to converge a given position-dependent observable faster than with MD for systems where precise information is needed at a specific part of the system only. On the other hand, ARMD methodology might not be useful to study dynamical properties.

All algorithms have been implemented as a modified version of LAAMPS, which will be made available through SAMSON Connect (https://www.samson-connect.net). In the future, we would like to investigate the development of other incremental algorithms for long-range interactions, and apply the AR methodology to diverse systems which might benefit from restraining some particles, including ion channels, enzymatic reactions, defect and crack propagation, etc. Speed-up 500 partices using single-pass 4000 particles using single-pass 32000 particles using single-pass 108000 particles using single-pass 500 particles using two-pass 4000 particles using two-pass 32000 particles using two-pass 108000 particles using two-pass Speed-up 500 particles using single-pass 4000 particles using single-pass 32000 particles using single-pass 108000 particles using single-pass 500 particles using two-pass 4000 particles using two-pass 32000 particles using two-pass 108000 particles using two-pass

% < N R > % < N S > % < N R > % < N S > % < N R > % < N S > % < N R > % < N S > 0.

2 . 3 .

 23 Position update step: This involves updating the positions of active particles only, instead of updating the positions of all particles. Second pass or addition step: This step involves computing force increments based on the updated positions of active particles, which are then added to the forces obtained from the subtraction step.

τ

  SF : Time to compute specific force components (F AR or F RR ).

τ

  CL : Time to clear an ANL. N SA : Number of particles switching to an active state. N SR : Number of particles switching to a restrained state. N S : Total number of switched particles (N SA + N SR ).Note that Equation (4) does not include τ AN L , the time to compute an ANL, since, when neighbor lists are update from scratch (lines 3-12), Algorithm 4 computes FNLs and ANLs simultaneously.

Algorithm 5 :Algorithm 6 :

 56 ARMD integration step Update momenta Update A, R, S A , S R if (U pdateN eeded) then hasF N L ← ∅ for i ∈ A do Build F N L(i) and AN L(i) simultaneously hasF N L ← hasF N L ∪ {i} SwitchedF orces ()

end 10 else 11 Build

 11 F N L(i) and AN L(i) simultaneously 12 hasF N L ← hasF N L ∪ {i} 13 end 14

  was observed with the overall speedup: a 2X speedup was observed with 60% particles restrained and up to 5X speedup with 90% particles restrained (Figure fig:case1overallspeedup).

Figure 1 :

 1 Figure 1: Different types of force components present in the system. Green particles are active and blue particles are restrained. The force components F AA , F AR and F RR represent forces acting between active particles, forces between active and restrained particles, and forces between restrained particles.

Figure 2 :

 2 Figure 2: Speedup when constructing the ANL (compared to the HNL) as a function of the percentage of restrained particles.

Figure 3 :

 3 Figure 3: Series 1: Speedup per integration step.

Figure 4 :

 4 Figure 4: Series 1: Overall speedup obtained as a function of the percentage of restrained particles using the single-pass and two-pass incremental algorithms. The new single-pass algorithm always performs better than the two-pass algorithm.

Figure 5 :

 5 Figure 5: Series 2: Speedup achieved with the single-pass and two-pass incremental algorithms over MD as a function of the percentage of restrained particles.

Figure 6 :

 6 Figure 6: Speedup achieved with ARMD for ANL construction and per time step in the NaCl benchmark.

Figure 7 :

 7 Figure 7: Cross-section view of the nanodroplet impact at 0, 15, 30, 45, 60 and 75 ps. Red particles belong to the impact area. Green particles represent the fixed boundary of the impact slab.

Figure 8 :

 8 Figure 8: Comparison of the RDFs of the impact area obtained with MD and ARMD. The RDFs obtained by ARMD with different AR parameters at different time-steps mostly coincide with the RDFs obtained by MD.

Figure 9 :

 9 Figure 9: Instantaneous temperature of the system with different AR parameters.

Figure 10 :

 10 Figure 10: Time correlation function of the end-to-end distance of the polymer. The left part shows that MD takes fewer iterations to decorrelate the end-to-end distance when compared to ARMD. The right part shows that, in wall-clock time, however, some ARMD simulations converge up to twice faster than a MD simulation.

Figure 11 :

 11 Figure 11: Projection of trajectories on the end-to-end distance (R) and radius of gyration (R g ) of the polymer (colors represent the number of conformations in each bin). This figure illustrates that ARMD simulations produce unbiased positional statistics.

  Figure 1 Krishna Kant Singh and Stephane Redon J. Comput. Chem.

  Figure 4 Krishna Kant Singh and Stephane Redon J. Comput. Chem.

  Figure 5 Krishna Kant Singh and Stephane Redon J. Comput. Chem.

  Figure 6 Krishna Kant Singh and Stephane Redon J. Comput. Chem.

  Figure 7 Krishna Kant Singh and Stephane Redon J. Comput. Chem.

Table 1 :

 1 AR parameters for Lennard-Jones systems. The average percentage of particles that switch states at each time step is very small, and does not significantly affect performance.

	5	8.21	0.015	36.34	0.0075	28.12	0.0011	26.97	0.0054
	1	26.47	0.024	56.94	0.0095	36.53	0.0091	45.73	0.0041
	1.5	57.48	0.034	82.015	0.0042	46.96	0.0084	67.99	0.0073
	2	60.54	0.029	84.27	0.0024	57.24	0.0021	70.99	0.0011
	2.5	68.04	0.025	87.23	0.0092	71.32	0.0062	77.81	0.0063
	3	81.25	0.021	91.08	0.0055	79.14	0.0055	87.16	0.0077
	3.5	86.81	0.018	96.23	0.0032	85.75	0.0031	93.36	0.0021
	4	92.42	0.014	96.23	0.0085	98.34	0.0022	97.15	0.0015

Table 2 :

 2 AR parameters used in the simulation of the N aCl system.

	r	f	% < N R > speedup
	1	2	56.4	2.3
	1	5	60.79	2.9
	2	5	61.87	3.1
	5	6	66.49	4.25
	10		72.49	4.7
	12		75.53	5.1
	14		79.34	5.9
	16		84.21	6.82

Table 3 :

 3 AR parameters used for performing ARMD simulations of the high-velocity impact of the nanodroplet. MD r / f .5/1 .75/1.0 1.0/1.2 1.2/1.5 2.5/2.75 3.0/3.75 3.1/3.75 3.5/4.0

	% < N R >			0	18	31	43	51	77.8	83	86	90
	R =< R 2 >	1 2		8.4404 8.481 8.634	8.3013	8.368	8.394	8.569	8.3919	8.556
	R G =< R 2 G >	1 2	3.523 3.547 3.585	3.520	3.55	3.535	3.546	3.534	3.569

Table 4 :

 4 Summary of statistical properties obtained by MD and ARMD for the polymer benchmark. R is the end-to-end distance of the chain and R G is the radius of gyration.Statistical averages obtained from ARMD and MD are similar.

Acknowledgments

We gratefully acknowledge funding from Rhone-Alpes Region through the ARC program, and the European Research Council through the ERC Starting Grant n. 307629.


Heyrman Eric
email: eric.heyrman@inrets.fr

Boer De
email: e.deboer@tudelft.nl

Enne

Gluenkin Werner
email: werner.gluenkin@bvfd.gr.ch

Towards a policy convergence in Europe? General approaches in the Netherlands, Switzerland and France

Everywhere policies are being developed to increase the access of the disabled to public transport. Comparative analysis of these policies is useful in two respects: content and process, i.e. what to achieve and how to achieve it. The first type of analysis is not unusual, the second type is. Given the general effort and the intensity of international contacts, it would not be strange to expect a convergence in policies but perhaps more so with regard to content than to process. Administrative traditions may differ widely between countries, There is extensive theorising on policy convergence. The essence of the theory is presented and the different policy levels at which convergence may occur are made operational for accessible transport policies.

The policies of three European countries (France, the Netherlands and Switzerland) are compared at the level of policy instruments. It shows that in all these countries a range of instruments is used. But there are remarkable differences in the relative importance of these instruments. Both France and Switzerland rely strongly on legal instruments (law and regulations), but the Netherlands more on the transfer of information (not enforcing but convincing). Both the Netherlands and Switzerland support their policies with subsidies for adapting the infrastructure. France seems to present it as a national duty which regional actors have to fulfil. The effectiveness of these strategies is not assessed.

KEY-WORDS:

Policy; convergence; accessibility; people with disabilities THE THEORY OF POLICY CONVERGENCE According to the works of Colin Bennett [Bennett, 1991, Bennett, Howlett, 1992], David Dolowitz and David Marsh, [Dolowitz, Marsh, 1996, Dolowitz, Marsh, 2000], the occurrences of policy transfer have increased for three reasons. First no nation can isolate its economy from global economic pressures. Moreover, the rapid growth in communication of all types makes the exchange of ideas and knowledge much easier. Lastly, international organizations, such as the European Union and the International Monetary Fund, advocate similar policies across diverse countries. Numerous actors try to learn from abroad:

-elected officials, -political parties, -bureaucrats and civil servants, -pressure groups, -policy entrepreneurs and experts, -trans-national corporations, -think tanks, -supra-national governmental institutions, -nongovernmental institutions, -consultants. Why? The legitimacy of elected officials is nowadays based on the effectiveness of policies and not only on their democratic election [Muller, 1992]. So they are encouraged to import a successful policy (a local or a national policy). Pressure groups compare national policies. They collect information and they highlight domestic delay in order to open an opportunity window and change their national policy. The European Union, a supra-national institution, aims at increasing the usability of transport for people with reduced mobility. Its policy fits with the general goals of the EU: creating a single European market and assuring the rights to travel inside its domestic territory for all people, including for people with disabilities. This objective has an economic side, since workers are the main target population. The European policy is based on the non-discrimination principle since 1997 and the treaty of Amsterdam. The EU uses various tools to harmonize technical standards and policies. It launched three COST actions on transport for people with reduced mobility (COST 322,COST 335 and COST 349). It adopted a directive on bus accessibility and a regulation on air travel. It published calls for proposals within the Sixth Framework Programme for Research and Technological Development which aim at comparing policies. Lastly, the EU subsidizes a few projects on accessible transport within the European fund for regional development and the European social fund. The European Conference of Ministers of Transport established in 1985 a working group on access and inclusion. Its work is another opportunity to compare national policies and to promote this social objective.

The goal of this contribution is to show whether the rise in international contacts has been translated in similar policies and whether a convergence of transport accessibility policy has occurred in Europe or not. To investigate this matter, we have chosen three countries for a policy comparison. Since the EU seems to be a key actor, we have chosen two member countries of the EU (the Netherlands and France) and a non member country (Switzerland), which is a CEMT member though. These countries are not similar in geographical size area, nor in population size or system of government as is shown in figure 1. This is not unlikely to cause a certain divergence in policies. All these countries have recently reformed their accessibility policy though, which makes a convergence analysis relevant. Our contribution is structured into two parts. In the first section we present the methodological grid and the different items to be examined in a policy comparison. The complexity of the matter is such that we had to focus on one item: the instruments used in the Dutch, Swiss and French policies.

The

Policy convergence: at what level?

Policy convergence means more than the development of similar policy contents. In fact, according to Patrick Hassenteufel [Hassenteufel, 2004], a policy convergence can be spotted at 7 different levels:

-Policy objectives: the convergence is of a cognitive order. It refers to the modes of problem construction and of definition of priority goals and of the policy legitimacy. -Policy contents: the convergence focuses on the link between objectives and instruments. -Policy instruments: the convergence is only on instruments and their use.

-Policy style: the convergence concerns decision making processes and the types of interaction between the policy actors -Policy recipients -Policy effects: the convergence focuses on policy outputs -Dominant actors: the convergence is more transversal. It focuses on actors who are key players in defining the problem, proposing solutions, making decisions and implementing policy.

Applying this methodological grid to the accessibility policy the following range of questions may be asked.

Policy objectives -What are the goals of the accessibility policy? -To integrate people with disabilities into society? -Non-discrimination? -To use the same services as able persons in the same conditions? -Social Mix?

-5 --Equal opportunities (to jobs, leisure facilities)? -To improve the mobility of people with disabilities? (to make the same number of travels a day as able people, whatever the means used?)

Policy contents -According to priority goals, a mode of transport may be emphasized:

-Mainstream public transport? Which types (heavy rail, subway, tram, bus)? -Specialized transport services? -Personal cars? The policy can be focused on new vehicles or new constructions only, but it may concern also the adaptation of existing public transport. Deadlines may be specified or not. According to the policy, accessible places could be geographically limited (like strategic transport networks, core network or main stations) or not.

Policy instruments (to be discussed in detail in the second part of this contribution). These instruments can take shape of:

-Laws and regulations, -Fiscal and economic instruments -Contractual and incentive instruments -Informative instrument and communications -Standards and best practices -Master plans Policy style (i.e. the process by which policy responses are developed). Is this process consensual or conflictuous? Incremental or rational? Anticipatory or reactive? Corporatist or pluralist?

Policy recipients -What is the target public of the accessibility policy? Which categories of people are mentioned as beneficiaries of the accessibility policy (in national legislations or in rules of transport companies for instance)? Which abilities of persons correspond with accessibility requirements and technical recommendations? Two categories of people must be investigated: people with disabilities and people with reduced mobility. As regards the category "people with reduced mobility", can elderly people, parents with a pram and people with luggage benefit from measures of the accessibility policy (financial aid or boarding devices for instance)?

Policy effects -This item requires defining clear policy objectives, which is pretty rare, and assessing the outcomes.

Various policy instruments

In the second part of our analysis we focus on the policy instruments used by the Netherlands, Switzerland and France. A policy instrument is defined by Pierre Lascoumes and Patrick Le Galès [Lascoumes, Le Galès, 2004] as "a device that is both technical and social, which organizes a specific relationship between the State and those it is addressed, according to the representations and meanings it carries" and as "a technical device with the general purpose of carrying a concrete concept of the polity/society relationship and sustained by a concept of regulation". You can see in table 2 the various types of instruments, their relation between polity and society and their legitimacy.

All these policy instruments are used by policy makers in the Netherlands, Switzerland and France, but they do not have the same significance in these countries nor are they defined by the same actors. provides that a national conference on the French disability policy is organized on the initiative of the Government every three years. The public and private actors working in favour of people with disabilities are invited to this conference to discuss orientations and means of the national disability policy. Accessibility is one the items debated by the participants.

In the Netherlands there is no general monitor but Dutch Rail and Regional authorities have to report on the development of accessible railway stations and bus stops Fiscal and economic instruments Financial incentives are instruments often used by policy makers. There are two recipients of these instruments: they are the actors which have to implement the accessibility policy at a local level and people with reduced mobility.

In France, there are no specific subsidises for accessibility. Since the enactment of the 2005-law, granting of a public subsidy to develop a collective transport system is subordinated to the taking into account of accessibility: only accessible systems will be subsidized by the central State.

In the Netherlands, the Dutch Ministry of Transport has two programs of making railway stations accessible and of increasing the platform height at bus stops. In the past it stimulated the introduction of low floor buses by subsidising the additional cost.

People with reduced mobility can benefit from specific tariff measures in France. This decision is on each transport authority's hands. In France several "Communes" had created "extra-municipal commissions of accessibility". The mission of these commissions was to discuss the problems which have occurred to people with disabilities in the communes and to elaborate programs of action.

Elaborating a charter is another tool used when there was no legislation and regulation with clear objectives. Several national actors discussed the content of a text which is not binding. Granting a charter is a moral commitment. The signatories promise to respect the charter. But a charter is not a contract so it can not be used to bring a suit against the partner who does not respect the charter. Two charters were elaborated in France in 2003before the enactment of the 2005law: the "Disability and Municipality Charter" and the "National Charter for Accessibility".

Informative instrument and communications

These instruments are used to inform the public and actors and to change the view of disability by the society. They may be:

-Awareness programs and training courses, -National or international competitions or rankings, -Guides to good practices, Indeed it is a common place to say that people with disabilities will not be well integrated into society until public opinion has become sensitive to their problems and until people have been trained to take into account the needs of people with disabilities. In France, this discourse is widespread. To make the implementation of the 2005-law easier and to change the common vision of disability, the ministry of social affairs launches a radio campaign on disability and people with disabilities in 2006. Moreover the 2005-law imposes an accessibility training course for architects and building trades' initial training courses. Besides numerous transport companies have already imposed to their new drivers a training course often run by representatives of people with disabilities. Lastly at a local level, some transport operators organize training courses for people with reduced mobility. The goals of these courses are to demystify transport networks and to show the functioning of access devices (ramps, lifts, sound announcements of bus stop, etc.). In the Netherlands, a several activities were undertaken: presentation of accessible transport systems [De Boer, 2003] and short courses for regional representatives of disabled organisations (2005).

Creating a national or international competition or ranking is another possibility. These national or international competitions aim at encouraging the design, development and implementation of inclusive design. The competitions are instruments used to publicly spotlight best practices and to make other organizations aware of what can be achieved. The award ceremony is an opportunity for public entities to show their commitment for accessibility and to indicate to all other actors that improving accessibility is an absolute necessity. In 2003 the European Conference of Ministries of transport and the European Disability Forum gave an Access and inclusion Award for Transport Services and Infrastructure. The French Prime Minister created an "Accessibility and City" Award in 2004. The Netherlands organized a 1996-and a 1998-competition for systems of "supplementary transport" for people with mobility problems.

The guides to good practices are another instrument in this category: they are different from technical standards since their contents are not negotiated by associations of people with disabilities, by vehicle builders and by local authorities. They are elaborated by experts. These people "go to the market" and they highlight good practices. Guides to good practices are kind of integrated recommendations with an element of convincing: look how well it might be done. In France the Center for the study of urban planning, transport and public facilities (CERTU)which is a service of the French Ministry of Transporthas published such as guides to good practices on bus stops and consulting process.

Standards and technical recommendations

When a State uses these instruments, it wants to obtain a consensus between several points of view of the civil society without publishing a regulation. These instruments are sometimes called "soft laws".

A technical standard is negotiated between the central State, people with disabilities and people who will implement it. It is often adopted on a consensual basis. A technical standard is not a binding tool. It is used if the local actors want to. In France, the French standard Association (AFNOR), which is a state-approved organisation under the administrative supervision of the Ministry for Industry, has worked on sound traffic lights (standard S32-002), on warning tactile pavements (standard P98-351) and on taxis which carry at least one person with reduced mobility (standard XP R18-805). COLITRAH has published technical recommendations since 1992. Switzerland has also recommendations developed on behalf the Ministry of Transport. The height of railway platform is fixed on 55 cm for normal gauge and 36 cm for narrow gauge. The Netherlands follow the new European standard of 76 cm. The Dutch Ministry of Transport had detailed "suggestions" issued in 2000. The Netherlands developed in 2005-2006 a series of manuals for different elements and transport systems. The manuals were issued by the CROW institution, traditionally specialised in developing technical standards by creating expert consensus (the R stands for Regulation giving)

Planning tools This instrument is used to integrate various policy constraints and objectives, namely the economic constraint. So planning tool can be seen as a meta-instrument. In 2005 France has adopted a legal framework based on three master plans:

-All transport authorities and managers of main airports or main harbour stations have to prepare a master plan whose goal is to make accessible collective transport services they are responsible for. -Each mayor must establish a master plan of making street infrastructures and public installations accessible. -Urban mobility master plans have to integrate a particular appendix relating to accessibility. Associations of people with disabilities must be included during the preparation of urban mobility master plan or of a "street infrastructures plan" if they want to.

Both Switzerland and the Netherlands have operational plans for adapting rail infrastructures [BehiG-Umsetzungskonzept, 2006, Implementatieplan Toegankelijkheid, 2006].

CONCLUSION

Even if the Netherlands, Switzerland and France have a common policy objective (having public transport accessible to people with disabilities) and they each use almost all types of policy instruments, these countries have favoured one or two policy tools. This choice is probably linked to the history of each country: paths of dependence may have been created: the more a policy instrument is used the more efficient it is. The choice might be based on the administrative organization of the country and by the legal culture of its inhabitants. Is adopting law and regulations enough to orient behaviours? Or is it necessary to add an informative instrument to convince local actors of the legitimacy of the national law? The law can be sufficient in Switzerland, but not in France or in the Netherlands. Moreover different publics are targeted with the same type of policy instrument. The Netherlands for instance give priority to specialized transport services while France sooner supports personal car ownership.

The Swiss use of policy instruments differs from French and Dutch ones in two respects. Contrary to all expectations (and to the Netherlands and France) Switzerland uses very few informative instrumentswhich are often seen as modern tools. However, Switzerland prefers contractual and incentive instruments. That is less the case for the Netherlands and France. As for the others instruments types there is no real difference between member countries of the EU and Switzerland. It is difficult to determinate the effect of the membership of the EU versus of the CEMT on the choice of policy instrument.

  At a European level, the European Conference of Ministries of Transport has published several guides to good practices: Improving Transport Accessibility For All in 2006 and Improving Access to Public Transport in 2004. The Netherlands organized experiments during the 1990's with an evident demonstration function (reported 1998)

Table 1 -Different types of policy instruments, the corresponding role of the State, and the character of the legitimacy of each. Type of policy instrument Type of relation between polity and society Type of legitimacy

 1 Switzerland has adopted a law on equal treatment of the disabled in 2002, the Netherlands in 2003 and France in 2005. These laws require equal treatment in public transport as well, except for the Dutch one: the relevant section will come in to force no sooner than 2010. Switzerland has published regulations on a mode-by-mode principle. France is elaborating regulations to specify accessibility requirements. The Netherlands has no formal regulations other than European ones.Monitoring and controlling a policy implementation are traditional instruments used by the central State. In France the central State does not rely on its local services but on a new local commission to monitor the implementation of the accessibility policy. Representatives of Commune and of associations of people with disabilities are ex officio members of this commission. Moreover at a national level the ministerial delegate on accessibility, which is a department of the French ministry of transport, publishes an annual report on the state and the evolution of accessibility in France. Lastly article 3 of the 2005-law

	"Command and control" instruments: laws and regulations	A State which is the guardian of the Society.	Imposition of a general interest by elected officials or by high-ranking civil servants.
	Fiscal and economic instruments	A State which is a wealth producer. A State which redistributes wealth.	In search of collective utility Economic and social effectiveness.
	Agreement-based and	A State which stirs actors	In search of direct
	incentive instruments	into action.	commitment
	Informative instrument and communications	Audience democracy.	Explanation of decisions. Giving actors a more sense of responsibility.
			Scientific and technical
	Standards and technical recommendations	Adjustment inside the civil society.	legitimacy and bargaining (or/and competition). Pressure of economic
			mechanisms.
	Meta-instruments : master plans	A State which integrates all the constraints	Planning
	Command and control instruments

  It is not compensated by the central State.The tax instrument is used by the French State as well. The adaptation of personal cars entitles people with disabilities to a low VAT rate (only 5.5% against 19.6%). The Dutch social service and the social insurance may provide adapted cars, but since about 10 years it gives priority to providing special collective transport.Agreement-based and incentive instrumentsWhen a State used a contractual and incentive instrument, it wants to stimulate local actors to take action without paying for that or publishing a binding text. The central State often encourages local governments and associations of people with disabilities to make local policies. They negotiate goals, solutions, priorities, schedule and program of action at a local level. The central State incites the local governments to consult representatives of people with disabilities so that the local policies fit with their needs.

Consequently, we can not conclude to policy convergence in Europe. Studying the Dutch, the French and the Swiss use of policy instruments, one will note that different sets of instruments are thought to be useful for achieving identical goals. Convergence in policy instruments is only one of 7 types of policy convergence. We encourage the reader to examine all of 7 levels (objectives, content, instrument, style, recipients, effects, dominant actors). In this way one can draw the "identity card" of a national policy. It will demonstrate that different identities are possible.