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# fix hyper/global command

## Syntax

    fix ID group-ID hyper/global cutbond qfactor Vmax Tequil

- ID, group-ID are documented in fix command

- hyper/global = style name of this fix command

- cutbond = max distance at which a pair of atoms is considered bonded (distance units)

- qfactor = max strain at which bias potential goes to 0.0 (unitless)

- Vmax = height of bias potential (energy units)

- Tequil = equilibration temperature (temperature units)

## Examples

    fix 1 all hyper/global 1.0 0.3 0.8 300.0

## Description

This fix is meant to be used with the hyper command to perform a bond-boost global hyperdynamics (GHD) simulation. The role of this fix is to a select a single pair of atoms in the system at each timestep to add a global bias potential to, which will alter the dynamics of the system in a manner that effectively accelerates time. This is in contrast to the fix hyper/local command, which can be user to perform a local hyperdynamics (LHD) simulation, by adding a local bias potential to multiple pairs of atoms at each timestep. GHD can time accelerate a small simulation with up to a few 100 atoms. For larger systems, LHD is needed to achieve good time acceleration.

For a system that undergoes rare transition events, where one or more atoms move over an energy barrier to a new potential energy basin, the effect of the bias potential is to induce more rapid transitions. This can lead to a dramatic speed-up in the rate at which events occurs, without altering their relative frequencies, thus leading to an overall increase in the elapsed real time of the simulation as compared to running for the same number of timesteps with normal MD. See the hyper page for a more general discussion of hyperdynamics and citations that explain both GHD and LHD.

The equations and logic used by this fix and described here to perform GHD follow the description given in (Voter2013). The bond-boost form of a bias potential for HD is due to Miron and Fichthorn as described in (Miron). In LAMMPS we use a simplified version of bond-boost GHD where a single bond in the system is biased at any one timestep.

Bonds are defined between each pair of atoms ij, whose \(R^0_{ij}\) distance is less than cutbond, when the system is in a quenched state (minimum) energy. Note that these are not “bonds” in a covalent sense. A bond is simply any pair of atoms that meet the distance criterion. Cutbond is an argument to this fix; it is discussed below. A bond is only formed if one or both of the ij atoms are in the specified group.

The current strain of bond ij (when running dynamics) is defined as

\[E_{ij} = \frac{R_{ij} - R^0_{ij}}{R^0_{ij}}\]

where \(R_{ij}\) is the current distance between atoms i and j, and \(R^0_{ij}\) is the equilibrium distance in the quenched state.

The bias energy \(V_{ij}\) of any bond between atoms i and j is defined as

\[V_{ij} = V^{max} \cdot \left( 1 - \left(\frac{E_{ij}}{q}\right)^2 \right) \textrm{ for } \left|E_{ij}\right| < qfactor \textrm{ or } 0 \textrm{ otherwise}\]

where the prefactor \(V^{max}\) and the cutoff qfactor are arguments to this fix; they are discussed below. This functional form is an inverse parabola centered at 0.0 with height \(V^{max}\) and which goes to 0.0 at +/- qfactor.

Let \(E^{max}\) be the maximum of \(\left| E_{ij} \right|\) for all ij bonds in the system on a given timestep. On that step, \(V_{ij}\) is added as a bias potential to only the single bond with strain \(E^{max}\), call it \(V^{max}_{ij}\). Note that \(V^{max}_{ij}\) will be 0.0 if \(E^{max} >= \textrm{qfactor}\) on that timestep. Also note that \(V^{max}_{ij}\) is added to the normal interatomic potential that is computed between all atoms in the system at every step.

The derivative of \(V^{max}_{ij}\) with respect to the position of each atom in the \(E^{max}\) bond gives a bias force \(F^{max}_{ij}\) acting on the bond as

\[F^{max}_{ij} = - \frac{dV^{max}_{ij}}{dE_{ij}} = \frac{2 V^{max} E-{ij}}{\textrm{qfactor}^2} \textrm{ for } \left|E_{ij}\right| < \textrm{qfactor} \textrm{ or } 0 \textrm{ otherwise}\]

which can be decomposed into an equal and opposite force acting on only the two ij atoms in the \(E^{max}\) bond.

The time boost factor for the system is given each timestep I by

\[B_i = e^{\beta V^{max}_{ij}}\]

where \(\beta = \frac{1}{kT_{equil}}\), and \(T_{equil}\) is the temperature of the system and an argument to this fix. Note that \(B_i >= 1\) at every step.

Note

To run a GHD simulation, the input script must also use the fix langevin command to thermostat the atoms at the same \(T_{equil}\) as specified by this fix, so that the system is running constant-temperature (NVT) dynamics. LAMMPS does not check that this is done.

The elapsed time \(t_{hyper}\) for a GHD simulation running for N timesteps is simply

\[t_{hyper} = \sum_{i=1,N} B-i \cdot dt\]

where dt is the timestep size defined by the timestep command. The effective time acceleration due to GHD is thus \(t_{hyper} / N * dt\), where N*dt is elapsed time for a normal MD run of N timesteps.

Note that in GHD, the boost factor varies from timestep to timestep. Likewise, which bond has \(E^{max}\) strain and thus which pair of atoms the bias potential is added to, will also vary from timestep to timestep. This is in contrast to local hyperdynamics (LHD) where the boost factor is an input parameter; see the fix hyper/local page for details.

Here is additional information on the input parameters for GHD.

The cutbond argument is the cutoff distance for defining bonds between pairs of nearby atoms. A pair of ij atoms in their equilibrium, minimum-energy configuration, which are separated by a distance \(R_{ij} < cutbond\), are flagged as a bonded pair. Setting cubond to be ~25% larger than the nearest-neighbor distance in a crystalline lattice is a typical choice for solids, so that bonds exist only between nearest neighbor pairs.

The qfactor argument is the limiting strain at which the bias potential goes to 0.0. It is dimensionless, so a value of 0.3 means a bond distance can be up to 30% larger or 30% smaller than the equilibrium (quenched) \(R^0_{ij}\) distance and the two atoms in the bond could still experience a non-zero bias force.

If qfactor is set too large, then transitions from one energy basin to another are affected because the bias potential is non-zero at the transition state (e.g. saddle point). If qfactor is set too small than little boost is achieved because the \(E_{ij}\) strain of some bond in the system will (nearly) always exceed qfactor. A value of 0.3 for qfactor is typically reasonable.

The Vmax argument is the prefactor on the bias potential. Ideally, tt should be set to a value slightly less than the smallest barrier height for an event to occur. Otherwise the applied bias potential may be large enough (when added to the interatomic potential) to produce a local energy basin with a maxima in the center. This can produce artificial energy minima in the same basin that trap an atom. Or if Vmax is even larger, it may induce an atom(s) to rapidly transition to another energy basin. Both cases are “bad dynamics” which violate the assumptions of GHD that guarantee an accelerated time-accurate trajectory of the system.

Note that if Vmax is set too small, the GHD simulation will run correctly. There will just be fewer events because the hyper time (\(t_{hyper}\) equation above) will be shorter.

Note

If you have no physical intuition as to the smallest barrier height in your system, a reasonable strategy to determine the largest Vmax you can use for a GHD model, is to run a sequence of simulations with smaller and smaller Vmax values, until the event rate does not change (as a function of hyper time).

The Tequil argument is the temperature at which the system is simulated; see the comment above about the fix langevin thermostatting. It is also part of the beta term in the exponential factor that determines how much boost is achieved as a function of the bias potential.

In general, the lower the value of Tequil and the higher the value of Vmax, the more time boost will be achievable by the GHD algorithm.

## Restart, fix_modify, output, run start/stop, minimize info

No information about this fix is written to binary restart files.

The fix_modify energy option is supported by this fix to add the energy of the bias potential to the global potential energy of the system as part of thermodynamic output. The default setting for this fix is fix_modify energy no.

This fix computes a global scalar and global vector of length 12, which can be accessed by various output commands. The scalar is the magnitude of the bias potential (energy units) applied on the current timestep. The vector stores the following quantities:

> 1. boost factor on this step (unitless)

> 2. max strain \(E_{ij}\) of any bond on this step (absolute value, unitless)

> 3. ID of first atom in the max-strain bond

> 4. ID of second atom in the max-strain bond

> 5. average # of bonds/atom on this step

> 6. fraction of timesteps where the biased bond has bias = 0.0 during this run

> 7. fraction of timesteps where the biased bond has negative strain during this run

> 8. max drift distance of any atom during this run (distance units)

> 9. max bond length during this run (distance units)

> 10. cumulative hyper time since fix was defined (time units)

> 11. cumulative count of event timesteps since fix was defined

> 12. cumulative count of atoms in events since fix was defined

The first 5 quantities are for the current timestep. Quantities 6-9 are for the current hyper run. They are reset each time a new hyper run is performed. Quantities 19-12 are cumulative across multiple runs (since the point in the input script the fix was defined).

For value 8, drift is the distance an atom moves between two quenched states when the second quench determines an event has occurred. Atoms involved in an event will typically move the greatest distance since others typically remain near their original quenched position.

For value 11, events are checked for by the hyper command once every Nevent timesteps. This value is the count of those timesteps on which one (or more) events was detected. It is NOT the number of distinct events, since more than one event may occur in the same Nevent time window.

For value 12, each time the hyper command checks for an event, it invokes a compute to flag zero or more atoms as participating in one or more events. E.g. atoms that have displaced more than some distance from the previous quench state. Value 11 is the cumulative count of the number of atoms participating in any of the events that were found.

The scalar and vector values calculated by this fix are all “intensive”.

No parameter of this fix can be used with the start/stop keywords of the run command. This fix is not invoked during energy minimization.

## Restrictions

This command can only be used if LAMMPS was built with the REPLICA package. See the Build package page for more info.

## Related commands

hyper, fix hyper/local

## Default

none

(Voter2013) S. Y. Kim, D. Perez, A. F. Voter, J Chem Phys, 139, 144110 (2013).

(Miron) R. A. Miron and K. A. Fichthorn, J Chem Phys, 119, 6210 (2003).