# fix hyper/local command

## Syntax

```
fix ID group-ID hyper/local cutbond qfactor Vmax Tequil Dcut alpha Btarget
```

ID, group-ID are documented in fix command

hyper/local = 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 = estimated height of bias potential (energy units)

Tequil = equilibration temperature (temperature units)

Dcut = minimum distance between boosted bonds (distance units)

alpha = boostostat relaxation time (time units)

Btarget = desired time boost factor (unitless)

zero or more keyword/value pairs may be appended

keyword =

*lost*or*check/bias*or*check/coeff**lostbond*value = error/warn/ignore*check/bias*values = Nevery error/warn/ignore*check/coeff*values = Nevery error/warn/ignore

## Examples

```
fix 1 all hyper/local 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 local hyperdynamics (LHD) simulation. The role of this fix is to a select multiple pairs of atoms in the system at each timestep to add a local 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/global command, which can be user to perform a global hyperdynamics (GHD) simulation, by adding a global bias potential to a single pair 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 doc 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 LHD 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).

To understand this description, you should first read the description of the GHD algorithm on the fix hyper/global doc page. This description of LHD builds on the GHD description.

The definition of bonds, Eij, and Emax are the same for GHD and LHD. The formulas for Vij(max) and Fij(max) are also the same except for a pre-factor Cij, explained below.

The bias energy Vij applied to a bond IJ with maximum strain is

Vij(max) = Cij * Vmax * (1 - (Eij/q)^2) for abs(Eij) < qfactor = 0 otherwise

The derivative of Vij(max) with respect to the position of each atom in the IJ bond gives a bias force Fij(max) acting on the bond as

Fij(max) = - dVij(max)/dEij = 2 Cij Vmax Eij / qfactor^2 for abs(Eij) < qfactor = 0 otherwise

which can be decomposed into an equal and opposite force acting on only the two I,J atoms in the IJ bond.

The key difference is that in GHD a bias energy and force is added (on a particular timestep) to only one bond (pair of atoms) in the system, which is the bond with maximum strain Emax.

In LHD, a bias energy and force can be added to multiple bonds
separated by the specified *Dcut* distance or more. A bond IJ is
biased if it is the maximum strain bond within its local
“neighborhood”, which is defined as the bond IJ plus any neighbor
bonds within a distance *Dcut* from IJ. The “distance” between bond
IJ and bond KL is the minimum distance between any of the IK, IL, JK,
JL pairs of atoms.

For a large system, multiple bonds will typically meet this requirement, and thus a bias potential Vij(max) will be applied to many bonds on the same timestep.

In LHD, all bonds store a Cij prefactor which appears in the Vij(max) and Fij(max) equations above. Note that the Cij factor scales the strength of the bias energy and forces whenever bond IJ is the maximum strain bond in its neighborhood.

Cij is initialized to 1.0 when a bond between the I,J atoms is first
defined. The specified *Btarget* factor is then used to adjust the
Cij prefactors for each bond every timestep in the following manner.

An instantaneous boost factor Bij is computed each timestep for each bond, as

Bij = exp(beta * Vkl(max))

where Vkl(max) is the bias energy of the maxstrain bond KL within bond
IJ’s neighborhood, beta = 1/kTequil, and *Tequil* is the temperature
of the system and an argument to this fix.

Note

To run LHD, the input script must also use the fix langevin command to thermostat the atoms at the
same *Tequil* as specified by this fix, so that the system is running
constant-temperature (NVT) dynamics. LAMMPS does not check that this
is done.

Note that if IJ = KL, then bond IJ is a biased bond on that timestep,
otherwise it is not. But regardless, the boost factor Bij can be
thought of an estimate of time boost currently being applied within a
local region centered on bond IJ. For LHD, we want this to be the
specified *Btarget* value everywhere in the simulation domain.

To accomplish this, if Bij < Btarget, the Cij prefactor for bond IJ is
incremented on the current timestep by an amount proportional to the
inverse of the specified *alpha* and the difference (Bij - Btarget).
Conversely if Bij > Btarget, Cij is decremented by the same amount.
This procedure is termed “boostostatting” in
(Voter2013). It drives all of the individual Cij to
values such that when Vij*max* is applied as a bias to bond IJ, the
resulting boost factor Bij will be close to *Btarget* on average.
Thus the LHD time acceleration factor for the overall system is
effectively *Btarget*.

Note that in LHD, the boost factor *Btarget* is specified by the user.
This is in contrast to global hyperdynamics (GHD) where the boost
factor varies each timestep and is computed as a function of *Vmax*,
Emax, and *Tequil*; see the fix hyper/global
doc page for details.

Here is additional information on the input parameters for LHD.

Note that the *cutbond*, *qfactor*, and *Tequil* arguments have the
same meaning as for GHD. The *Vmax* argument is slightly different.
The *Dcut*, *alpha*, and *Btarget* parameters are unique to LHD.

The *cutbond* argument is the cutoff distance for defining bonds
between pairs of nearby atoms. A pair of I,J atoms in their
equilibrium, minimum-energy configuration, which are separated by a
distance Rij < *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) R0ij 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 can be achieved because the Eij strain of some bond in
the system will (nearly) always exceed *qfactor*. A value of 0.3 for
*qfactor* is typically a reasonable value.

The *Vmax* argument is a fixed prefactor on the bias potential. There
is a also a dynamic prefactor Cij, driven by the choice of *Btarget*
as discussed above. The product of these should be a value 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 Cij**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 LHD that
guarantee an accelerated time-accurate trajectory of the system.

Note

It may seem that *Vmax* can be set to any value, and Cij will
compensate to reduce the overall prefactor if necessary. However the
Cij are initialized to 1.0 and the boostostatting procedure typically
operates slowly enough that there can be a time period of bad dynamics
if *Vmax* is set too large. A better strategy is to set *Vmax* to the
smallest barrier height for an event (the same as for GHD), so that
the Cij remain near unity.

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. See the discussion of
the *Btarget* argument below.

As discussed above, the *Dcut* argument is the distance required
between two locally maxstrain bonds for them to both be selected as
biased bonds on the same timestep. Computationally, the larger *Dcut*
is, the more work (computation and communication) must be done each
timestep within the LHD algorithm. And the fewer bonds can be
simultaneously biased, which may mean the specified *Btarget* time
acceleration cannot be achieved.

Physically *Dcut* should be a long enough distance that biasing two
pairs of atoms that close together will not influence the dynamics of
each pair. E.g. something like 2x the cutoff of the interatomic
potential. In practice a *Dcut* value of ~10 Angstroms seems to work
well for many solid-state systems.

Note

You must also insure that ghost atom communication is performed
for a distance of at least *Dcut* + *cutevent* where *cutevent* = the
distance one or more atoms move (between quenched states) to be
considered an “event”. It is an argument to the “compute
event/displace” command used to detect events. By default the ghost
communication distance is set by the pair_style cutoff, which will
typically be < *Dcut*. The comm_modify cutoff
command can be used to set the ghost cutoff explicitly, e.g.

```
comm_modify cutoff 12.0
```

This fix does not know the *cutevent* parameter, but uses half the
bond length as an estimate to warn if the ghost cutoff is not long
enough.

As described above the *alpha* argument is a pre-factor in the
boostostat update equation for each bond’s Cij prefactor. *Alpha* is
specified in time units, similar to other thermostat or barostat
damping parameters. It is roughly the physical time it will take the
boostostat to adjust a Cij value from a too high (or too low) value to
a correct one. An *alpha* setting of a few ps is typically good for
solid-state systems. Note that the *alpha* argument here is the
inverse of the alpha parameter discussed in
(Voter2013).

The *Btarget* argument is the desired time boost factor (a value > 1)
that all the atoms in the system will experience. The elapsed time
t_hyper for an LHD simulation running for *N* timesteps is simply

t_hyper = Btarget * N*dt

where dt is the timestep size defined by the timestep command. The effective time acceleration due to LHD is thus t_hyper / N*dt = Btarget, where N*dt is elapsed time for a normal MD run of N timesteps.

You cannot choose an arbitrarily large setting for *Btarget*. The
maximum value you should choose is

Btarget = exp(beta * Vsmall)

where Vsmall is the smallest event barrier height in your system, beta
= 1/kTequil, and *Tequil* is the specified temperature of the system
(both by this fix and the Langevin thermostat).

Note that if *Btarget* is set smaller than this, the LHD 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
*Btarget* you can use for an LHD model, is to run a sequence of
simulations with smaller and smaller *Btarget* values, until the event
rate does not change.

**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 the system’s
potential energy as part of thermodynamic output.

This fix computes a global scalar and global vector of length 23, which can be accessed by various output commands. The scalar is the magnitude of the bias potential (energy units) applied on the current timestep, summed over all biased bonds. The vector stores the following quantities:

- 1 = # of biased bonds on this step
- 2 = max strain Eij of any bond on this step (unitless)
- 3 = average bias potential for all biased bonds on this step (energy units)
- 4 = average # of bonds/atom on this step
- 5 = average neighbor bonds/bond on this step within
*Dcut* - 6 = fraction of steps and bonds with no bias during this run
- 7 = max drift distance of any atom during this run (distance units)
- 8 = max bond length during this run (distance units)
- 9 = average # of biased bonds/step during this run
- 10 = average bias potential for all biased bonds during this run (energy units)
- 11 = max bias potential for any biased bond during this run (energy units)
- 12 = min bias potential for any biased bond during this run (energy units)
- 13 = max distance from my sub-box of any ghost atom with maxstrain < qfactor during this run (distance units)
- 14 = max distance outside my box of any ghost atom with any maxstrain during this run (distance units)
- 15 = count of ghost neighbor atoms not found on reneighbor steps during this run
- 16 = count of lost bond partners during this run
- 17 = average bias coeff for lost bond partners during this run
- 18 = count of bias overlaps found during this run
- 19 = count of non-matching bias coefficients found during this run
- 20 = cumulative hyper time since fix created (time units)
- 21 = cumulative count of event timesteps since fix created
- 22 = cumulative count of atoms in events since fix created
- 23 = cumulative # of new bonds since fix created

The first quantities (1-5) are for the current timestep. Quantities 6-19 are for the current hyper run. They are reset each time a new hyper run is performed. Quantities 20-23 are cumulative across multiple runs (since the fix was defined in the input script).

For value 6, the numerator is a count of all biased bonds on every
timestep whose bias energy = 0.0 due to Eij >= *qfactor*. The
denominator is the count of all biased bonds on all timesteps.

For value 7, drift is the distance an atom moves between timesteps when the bond list is reset, i.e. between events. Atoms involved in an event will typically move the greatest distance since others are typically oscillating around their lattice site.

For values 13 and 14, the maxstrain of a ghost atom is the maxstrain of any bond it is part of, and it is checked for ghost atoms within the bond neighbor cutoff.

Values 15-19 are mostly useful for debugging and diagnostic purposes.

For values 15-17, it is possible that a ghost atom owned by another processor will move far enough (e.g. as part of an event-in-progress) that it will no longer be within the communication cutoff distance for acquiring ghost atoms. Likewise it may be a ghost atom bond partner that cannot be found because it has moved too far. These values count those occurrences. Because they typically involve atoms that are part of events, they do not usually indicate bad dynamics. Value 16 is the average bias coefficient for bonds where a partner atom was lost.

For value 18, no two bonds should be biased if they are within a
*Dcut* distance of each other. This value should be zero, indicating
that no pair of bonds “overlap”, meaning they are closer than *Dcut*
from each other.

For value 19, the same bias coefficient is stored by both atoms in an IJ bond. This value should be zero, indicating that for all bonds, each atom in the bond stores the a bias coefficient with the same value.

Value 20 is simply the specified *boost* factor times the number of
timestep times the timestep size.

For value 21, 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 22, 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 22 is the cumulative count of the number of atoms participating in any of the events that were found.

Value 23 tallies the number of new bonds created by the bond reset operation. Bonds between a specific I,J pair of atoms may persist for the entire hyperdynamics simulation if neither I or J are involved in an event.

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 fix is part of the REPLICA package. It is only enabled if LAMMPS was built with that package. See the Build package doc page for more info.