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Section: New Results

Adaptively Restrained Particle Simulations

Participants : Svetlana Artemova, Stephane Redon.

Last year, we have introduced a novel, general approach to speed up particle simulations that we call Adaptively Restrained Particle Simulations (ARPS). This year we continued working on this approach. The obtained results have been published in Physical Review Letters [3] , and the patent describing the theoretical basis and the algorithms for the numerical realization of ARPS has been deposited.

Particle simulations are widely used in physics, chemistry, biology [13] , [14] , and even computer graphics [9] , and faster simulations (in particular ARPS) may result in progress on many challenging problems, e.g., protein folding, diffusion across bio-membranes, fracture in metals, ion implantation, etc.

ARPS rely on an adaptively restrained (AR) Hamiltonian used to describe a system of $N$ particles:

This Hamiltonian has an unusual inverse inertia matrix $\Phi (𝐪,𝐩)$, which is made a general function of phase-space coordinates. The precise form of this matrix can be chosen according to the system under study and the problem stated.

We have proposed a particular (diagonal) form of the inverse inertia matrix for the simulations in Cartesian coordinates. In this case, $\Phi$ adaptively switches on and off positional degrees of freedom of individual particles while letting particle momenta evolve. The decision whether the particle is restrained or not depends on the particle's momentum, and, precisely, on it's kinetic energy. Two user-defined thresholds regulate the amount of simplification of the particle’s motion. When a module of a particle's momentum becomes small enough (without necessarily becoming zero), the particle completely stops moving. Even when a particle is fully restrained, though, its momentum may continue to change, and its kinetic energy might become large enough again for the particle to resume moving. In general, ARPS restrain and release particles repeatedly over time.

This approach has numerous advantages: (a) it is mathematically grounded and is able to produce long, stable simulations; (b) it does not require modifications to the simulated interaction potential, so that any suitable existing force-field can be directly used with ARPS; (c) under frequently-used assumptions on the interaction potential, ARPS make it possible to reduce the number of forces that have to be updated at each time step, which may significantly speed up simulations; (d) when performing constant-energy simulations, ARPS allow users to finely and continuously trade between precision and computational cost, and rapidly obtain approximate trajectories; (e) the trade-off between precision and cost may be chosen for each particle independently, so that users may arbitrarily focus ARPS on specific regions of the simulated system (e.g., a polymer in a solvent); (f) most important, when performing Adaptively Restrained Molecular Dynamics (ARMD) in the canonical (NVT) ensemble, correct static equilibrium properties can be computed.

We have demonstrated the advantages of ARPS on several numerical experiments. For example, a planar collision cascade study in Fig. 7 shows how ARPS make it possible to smoothly trade between precision and speed of the simulation. Reference simulations were derived from the usual Hamiltonian $H(𝐪,𝐩)=\frac{1}{2}{𝐩}^T{𝐌}^{-1}𝐩+V\left(𝐪\right)$.

Figure 7. Simulating a collision cascade with controlled precision. Adaptively restrained simulations allow us to smoothly trade between precision and speed. Even for large speed-ups (up to 10x) the features of the shock are extremely well preserved.