How to relate chemical potential difference to Gibbs free energy with icet?

Take a look at the following tutorial page,
mchammer tutorial

Look at the first graph on chemical difference vs Pd concentration. Since the chemical potential of Ag is taken as 0, so the chemical potential derivative gives the chemical potential of Pd. Then I figour out the Gibbs free energy as the summation over the products of chemical potential and concentration, which is just the quantity shown in vertical axis. But we got confuzed, there is no minimum in the Gibbs ~ x curve. Any misunderstanding? I notice the size of the supercell contains just 108 atoms, is that enough? thanks a lot. We are waiting for explanation very anxiously.

In the tutorial the first plot, as you say, shows the free energy derivative vs x.
Note that this is the free energy derivative, so you would need to integrate this to obtain G vs x.
G vs x is not shown in the tutorial as far as I can tell.

In general 108 atoms is not enough to obtain fully converged results.
You can also take a look at the icet-paper where Ag-Pd system is studied but more details and converged results are provided. See for example Fig 6, where both the free energy derivative and the free energy is shown vs x.

thanks for you kind reply. but from the first code on the following page,
[Analyzing Monte Carlo simulations — icet documentation](https://Analyze Monte Carlo Data)

I copy the following snippet

if ensemble == ‘sgc’:
data_row[‘free_energy_derivative’] =
dc.ensemble_parameters[‘mu_Pd’] -
elif ensemble == ‘vcsgc’:
data_row[‘free_energy_derivative’] =
dc.get_average(‘free_energy_derivative_Pd’, start=equilibration)

free energy derivative is expressed as the difference of chemical potential of Ag and Pd. Is there an implied \Delta N =1 on the denominator such that (\mu_Pd - \mu_Ag) / \Delta N coming into the play? I have read some papers on semi grand canonical ensemble, but they always talk about chemical potential difference instead of free energy derivative. Do I misunderstand something?

Yes the free energy (per atom) derivative with respect to concentration is the difference in chemical potential in the SGC ensemble.

So you run many SGC simulations for multiple different values of mu. For each mu you obtain an average concentration x. This means you have mu vs x data, which is the free energy derivative vs x data you see in the tutorial. If you integrate this you obtain G vs x.

I dont know what you mean with Delta_N, but one needs to be careful with normalization. For example one needs to consider if the free energy is total free energy or free energy per atom, and if derivative is taken with respect to concentration or number of atoms etc.