\(\renewcommand{\AA}{\text{Å}}\)

# fix sgcmc command¶

## Syntax¶

```
fix ID group-ID sgcmc every_nsteps swap_fraction temperature deltamu ...
```

ID, group-ID are documented in fix command

sgcmc = style name of this fix command

every_nsteps = number of MD steps between MC cycles

swap_fraction = fraction of a full MC cycle carried out at each call (a value of 1.0 will perform as many trial moves as there are atoms)

temperature = temperature that enters Boltzmann factor in Metropolis criterion (usually the same as MD temperature)

deltamu = chemical potential difference(s) (N-1 values must be provided, with N being the number of elements)

Zero or more keyword/value pairs may be appended to fix definition line:

keyword =

*variance*or*randseed*or*window_moves*or*window_size**variance*kappa conc1 [conc2] ... [concN] kappa = variance constraint parameter conc1,conc2,... = target concentration(s) in the range 0.0-1.0 (*N-1*values must be provided, with*N*being the number of elements)*randseed*N N = seed for pseudo random number generator*window_moves*N N = number of times sampling window is moved during one MC cycle*window_size*frac frac = size of sampling window (must be between 0.5 and 1.0)

## Examples¶

```
fix mc all sgcmc 50 0.1 400.0 -0.55
fix vc all sgcmc 20 0.2 700.0 -0.7 randseed 324234 variance 2000.0 0.05
fix 2 all sgcmc 20 0.1 700.0 -0.7 window_moves 20
```

## Description¶

New in version 22Dec2022.

This command allows to carry out parallel hybrid molecular dynamics/Monte Carlo (MD/MC) simulations using the algorithms described in (Sadigh1). Simulations can be carried out in either the semi-grand canonical (SGC) or variance constrained semi-grand canonical (VC-SGC) ensemble (Sadigh2). Only atom type swaps are performed by the SGCMC fix. Relaxations are accounted for by the molecular dynamics integration steps.

This fix can be used with standard multi-element EAM potentials (pair styles eam/alloy or eam/fs)

The SGCMC fix can handle Finnis/Sinclair type EAM potentials where \(\rho(r)\) is atom-type specific, such that different elements can contribute differently to the total electron density at an atomic site depending on the identity of the element at that atomic site.

If this fix is applied, the regular MD simulation will be interrupted in
defined intervals to carry out a fraction of a Monte Carlo (MC)
cycle. The interval is set using the parameter *every_nsteps* which
determines how many MD integrator steps are taken between subsequent
calls to the MC routine.

It is possible to carry out pure lattice MC simulations by setting
*every_nsteps* to 1 and not defining an integration fix such as NVE,
NPT etc. In that case, the particles will not move and only the MC
routine will be called to perform atom type swaps.

The parameter *swap_fraction* determines how many MC trial steps are carried
out every time the MC routine is entered. It is measured in units of full MC
cycles where one full cycle, *swap_fraction=1*, corresponds to as many MC
trial steps as there are atoms.

The parameter *temperature* specifies the temperature that is used
to evaluate the Metropolis acceptance criterion. While it usually
should be set to the same value as the MD temperature there are cases
when it can be useful to use two different values for at least part of
the simulation, e.g., to speed up equilibration at low temperatures.

The parameter *deltamu* is used to set the chemical potential difference
in the SGC MC algorithm (see Eq. 16 in Sadigh1). By
convention it is the difference of the chemical potentials of elements
B, C …, with respect to element A. When the simulation includes
N elements, N-1 values must be specified.

The variance-constrained SGC MC algorithm is activated if the keyword
*variance* is used. In that case the fix parameter *deltamu* determines
the effective average constraint in the parallel VC-SGC MC algorithm
(parameter \(\delta\mu_0\) in Eq. (20) of Sadigh1). The parameter *kappa* specifies the variance constraint
(see Eqs. (20-21) in Sadigh1).

The parameter *conc* sets the target concentration (parameter
\(c_0\) in Eqs. (20-21) of Sadigh1). The atomic
concentrations refer to components B, C …, with A being set
automatically. When the simulation includes N elements, N-1
concentration values must be specified.

There are several technical parameters that can be set via optional flags.

*randseed* is expected to be a positive integer number and is used
to initialize the random number generator on each processor.

*window_size* controls the size of the sampling window in a parallel MC
simulation. The size has to lie between 0.5 and 1.0. Normally, this
parameter should be left unspecified which instructs the code to choose
the optimal window size automatically (see Sect. III.B and Figure 6 in
Sadigh1 for details).

The number of times the window is moved during a MC cycle is set using
the parameter *window_moves* (see Sect. III.B in Sadigh1 for details).

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

No information about this fix is written to restart files.

The MC routine keeps track of the global concentration(s) as well as the number of accepted and rejected trial swaps during each MC step. These values are provided by the sgcmc fix in the form of a global vector that can be accessed by various output commands components of the vector represent the following quantities:

1 = The absolute number of accepted trial swaps during the last MC step

2 = The absolute number of rejected trial swaps during the last MC step

3 = The current global concentration of species

*A*(= number of atoms of type 1 / total number of atoms)4 = The current global concentration of species

*B*(= number of atoms of type 2 / total number of atoms)…

N+2: The current global concentration of species

*X*(= number of atoms of type*N*/ total number of atoms)

# Restrictions¶

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

At present the fix provides optimized subroutines for EAM type
potentials (see above) that calculate potential energy changes due to
*local* atom type swaps very efficiently. Other potentials are
supported by using the generic potential functions. This, however, will
lead to exceedingly slow simulations since it implies that the
energy of the *entire* system is recomputed at each MC trial step. If
other potentials are to be used it is strongly recommended to modify and
optimize the existing generic potential functions for this purpose.
Also, the generic energy calculation can not be used for parallel
execution i.e. it only works with a single MPI process.

# Default¶

The optional parameters default to the following values:

*randseed*= 324234*window_moves*= 8*window_size*= automatic

**(Sadigh1)** B. Sadigh, P. Erhart, A. Stukowski, A. Caro, E. Martinez, and L. Zepeda-Ruiz, Phys. Rev. B **85**, 184203 (2012)

**(Sadigh2)** B. Sadigh and P. Erhart, Phys. Rev. B **86**, 134204 (2012)