$$\renewcommand{\AA}{\text{Å}}$$

# 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¶

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)