# fix hyper/global command

## Syntax

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

-   ID, group-ID are documented in [fix](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

``` LAMMPS
fix 1 all hyper/global 1.0 0.3 0.8 300.0
```

## Description

This fix is meant to be used with the [hyper](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](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](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)](Voter2013ghd). The
bond-boost form of a bias potential for HD is due to Miron and Fichthorn
as described in [(Miron)](Mironghd). 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
::: title
Note
:::

To run a GHD simulation, the input script must also use the [fix
langevin](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.
::::

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](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](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) 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 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
::: title
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](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](restart).

The [fix_modify](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](thermo_style). The default
setting for this fix is [fix_modify energy no](fix_modify).

This fix computes a global scalar and global vector of length 12, which
can be accessed by various [output commands](Howto_output). 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](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](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](run) command. This fix is not invoked during [energy
minimization](minimize).

## Restrictions

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

## Related commands

[hyper](hyper), [fix hyper/local](fix_hyper_local)

## Default

none

------------------------------------------------------------------------

::: {#Voter2013ghd}
**(Voter2013)** S. Y. Kim, D. Perez, A. F. Voter, J Chem Phys, 139,
144110 (2013).
:::

::: {#Mironghd}
**(Miron)** R. A. Miron and K. A. Fichthorn, J Chem Phys, 119, 6210
(2003).
:::