\(\renewcommand{\AA}{\text{Å}}\)

# 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 = N-1 chemical potential differences \(\mu_1-\mu_2, \ldots, \mu_1-\mu_N\) (N is the number of atom types)

- 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
          c_2, c_3,..., c_N = N-1 target concentration fractions
        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)
        atomic/energy yes/no
          yes = use the atomic energy method to calculate energy changes
          no = use the default method to calculate energy changes

## 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

Added 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 differences in the SGC MC algorithm (see Eq. 16 in Sadigh1). The N-1 differences are defined as \(\mu_1-\mu_2, \ldots, \mu_1-\mu_N\), where N is the number of atom types.

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 N-1 target atomic concentration fractions (parameter \(c_0\) in Eqs. (20-21) of Sadigh1) \(0 \le c_2, \ldots, c_N \le 1\), with \(c_1 = 1 - \Sigma_{i=2}^N c_i\). When the simulation includes N atom types (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).

The atomic/energy keyword controls which method is used for calculating the energy change when atom types are swapped. A value of no uses the default method, see discussion below in Restrictions section. A value of yes uses the atomic energy method, if the method has been implemented for the LAMMPS energy model, otherwise LAMMPS will exit with an error message. So far this has only been implemented for EAM type potentials.

## 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 = Current global concentration c_1 (= number of atoms of type 1 / total number of atoms)

- 4 = Current global concentration c_2 (= number of atoms of type 2 / total number of atoms)

- …

- N+2 = Current global concentration c_N (= number of atoms of type N / total number of atoms)

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

## Restrictions

This fix is part of the MC package. It is only enabled if LAMMPS was built with that package. Since it also contains specific support for EAM potentials it also requires installing the MANYBODY package. See the Build package page for more info.

This fix style requires an atom style with per atom type masses.

The fix provides three methods for calculating the potential energy change due to atom type swaps. For EAM type potentials, the default method is a carefully optimized local energy change calculation that is part of the source code for this fix. It takes advantage of the specific computational and communication requirements of EAM. Customizing the local method to handle other energy models such as Tersoff has been done in earlier versions of this fix, but these cases are not supported in the public LAMMPS code. For all other LAMMPS energy models, the default method calculates the total potential energy of the system before and after each atom type swap. This method does not depend on the details of the energy model and so is guaranteed to be correct. It is also orders of magnitude slower than the custom EAM calculation. In addition, it can not be used with parallel execution i.e. only a single MPI process is allowed. The third method uses the atomic/energy keyword described above. This allows parallel execution and it is also a local calculation, making it only a bit slower than a fully-optimized local calculation. So far, this has been implemented for EAM type potentials. It is straightforward to extend this to other potentials, requiring adding an atomic energy method to the pair style.

## Default

The optional parameters default to the following values:

- randseed = 324234

- window_moves = 8

- window_size = automatic

- atomic/energy = no

(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)