Flying ice cube

In molecular dynamics (MD) simulations, the flying ice cube effect is a numerical integration artifact in which the energy of high-frequency fundamental modes is drained into low-frequency modes, particularly into zero-frequency motions such as overall translation and rotation of the system. The artifact derives its name from a particularly noticeable manifestation that arises in simulations of particles in vacuum, where the system being simulated acquires high linear momentum and experiences extremely damped internal motions, freezing the system into a single conformation reminiscent of an ice cube or other rigid body flying through space. The artifact is entirely a consequence of molecular dynamics algorithms and is wholly unphysical, since it violates the principle of equipartition of energy.

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Origin

The flying ice cube artifact arises from repeated rescalings of the velocities of the particles in the simulation system. Velocity rescaling is a means of imposing a thermostat on the system, forcing it to maintain a roughly constant temperature. These rescalings are traditionally done, as in the Berendsen thermostat, by multiplying the system's velocities by a factor α, which equals the ratio of the desired mean kinetic energy divided by the instantaneous amount of kinetic energy. This scheme fails, however, because the instantaneous kinetic energy is located in the denominator of the ratio α; fluctuations in the kinetic energy make positive second-order contributions to α, making its average value greater than one even when the instaneous kinetic energy has the proper mean. This causes the constant energy terms — such as those of overall translation and rotation — to grow continuously. Since these energies are constantly increasing, the same rescaling decreases the internal energies, diminishing the internal vibrations. This may be shown mathematically as well; the fluctuating internal kinetic energy has its highs and lows, but its highs are decreased more by velocity rescaling than its lows are increased, leading to a net decrease on average with every rescaling.

When the rotation and translation of the system center of mass are not periodically removed, a particularly noticeable form of the artifact occurs in which nearly all of the system's kinetic energy accrues to these two forms of motion, resulting in a system with essentially no energy associated with internal motions which therefore appears to move as a rigid body. This problem can arise in explicit solvent under unusual circumstances, particularly when the Berendsen barostat is used or when the simulation parameters do not respect conservation of energy, but the artifact occurs most visibly in simulations in vacuum.

Avoidance

The flying ice cube problem in its rigid-body form can be largely avoided by periodically removing the center-of-mass motions, although this does not necessarily cure the less blatant equipartition artifacts. In systems that are simulated as an isolated cluster, such as a single molecule in vacuum, both the translational and rotational motion about the center of mass should be removed; however, for systems in which there is sufficient friction to prevent substantial rotation and many closely spaced fundamental modes between which energy can be transferred - such as those using explicitly represented solvent under periodic boundary conditions - only the translational motion should be removed. Although it does not produce a perfectly continuous trajectory, periodic reassignment of velocities as in the Andersen thermostat method also minimize the problem. More conservatively, the rate of velocity rescaling can be reduced, the scale factor computed over a time-averaged sample of instantaneous temperatures, or an alternative means of thermostatting such as the Nose-Hoover thermostat can be used.

References

• Harvey SC, Tan RKZ, Cheatham TE. (1998). The flying ice cube: Velocity rescaling in molecular dynamics leads to violation of energy equipartition. J Comp Chem 19(7): 726-40. See here for complete derivation and examples.