Flying ice cube
Encyclopedia
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 algorithm
s and is wholly unphysical, since it violates the principle of equipartition of energy
.
of the particles in the simulation system. The artifact will not occur if the center-of-mass velocity of the system is kept separate and apart from those velocities being rescaled. 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
.
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 Nosé-Hoover thermostat
can be used.
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
(MD) simulations, the flying ice cube effect is a numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
artifact in which the energy of high-frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
fundamental modes is drained into low-frequency modes, particularly into zero-frequency motions such as overall translation
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
and rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
of the system. The artifact derives its name from a particularly noticeable manifestation that arises in simulations of particles in vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
, where the system being simulated acquires high linear momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
and experiences extremely damped internal motions, freezing the system into a single conformation reminiscent of an ice cube
Ice cube
Ice cubes are small, roughly cube-shaped pieces of ice, conventionally used to cool beverages. Ice cubes are often preferred over crushed ice because they melt more slowly; they are standard in mixed drinks that call for ice, in which case the drink is said to be "on the rocks."Ice cubes are...
or other rigid body
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
flying through space. The artifact is entirely a consequence of molecular dynamics algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s and is wholly unphysical, since it violates the principle of equipartition of energy
Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...
.
Origin
The flying ice cube artifact arises from repeated rescalings of the velocitiesVelocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
of the particles in the simulation system. The artifact will not occur if the center-of-mass velocity of the system is kept separate and apart from those velocities being rescaled. Velocity rescaling is a means of imposing a thermostat
Thermostat
A thermostat is the component of a control system which regulates the temperature of a system so that the system's temperature is maintained near a desired setpoint temperature. The thermostat does this by switching heating or cooling devices on or off, or regulating the flow of a heat transfer...
on the system, forcing it to maintain a roughly constant temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
. These rescalings are traditionally done, as in the Berendsen thermostat
Berendsen thermostat
The Berendsen thermostat is an algorithm to re-scale the velocities of particles in molecular dynamics simulations to control the simulation temperature.-Basic Description:In this scheme, the system is weakly coupled to a heat bath with some temperature...
, 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
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
and translation
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...
of the system center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
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
Water model
In computational chemistry, classical water models are used for the simulation of water clusters, liquid water, and aqueous solutions with explicit solvent. These models use the approximations of molecular mechanics...
under unusual circumstances, particularly when the Berendsen barostat is used or when the simulation parameters do not respect conservation of energy
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
, but the artifact occurs most visibly in simulations in vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
.
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 frictionFriction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of 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
Periodic boundary conditions
In mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge...
- 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 Nosé-Hoover thermostat
Nosé-Hoover thermostat
The Nosé–Hoover thermostat is a deterministic method used in molecular dynamics to keep the temperature around an average. It was originally introduced by Nosé and developed further by Hoover...
can be used.