# Metropolis method under the Gand Canonical Ensemble

I am writing code to simulate Langmuir adsorption using Metropolis method under the Gand Canoninal Ensemble.

Aided by the Wikipedia Article, I derived the corresponding equation, proposed a sample method and reproduced the Langmuir isotherm.

Many reliable sources propose a different sample mechanism, if I use that mechanism I get wrong results. I wish to know what I missing here.

In my approach, I choose if propose an adsorption or desorption according to the fraction of occupied sites, and simply use the acceptance probability

$$\alpha = \min[1, e^{\pm \beta (\mu-\epsilon)}]$$

where $$\epsilon$$ is the energy of each adsorbed molecule and $$\mu$$ the chemical potential of the ideal gas.

The way I understood the other approach is: to propose an adsorption or desorption with equal probability (1/2). Then use

$$\alpha_{\text{adsorpt}} = \min[1, \frac{zV}{N+1}e^{-\beta \epsilon }]$$

$$\alpha_{\text{desorpt}} = \min[1, \frac{N}{zV}e^{+\beta \epsilon }]$$

where $$z = \exp(\beta \mu)/\lambda^3$$. For example, eqs. 68 & 69 of this source (NB: PDF document).

I found the $$\alpha$$ to be similar, but I do not understand why use $$p=1/2$$ with those alpha independently of the fraction of occupied sites. I adapt my code to the last approach and I found results that do not coincide with the Langmuir equation.