Metropolis Monte Carlo comparison with

Figure 52 A comparison of cation concentration profiles for a bulk 0.05 M 1 1 electrolyte with 1 and 2-A radius hard-sphere ions in the presence of a charged cylinder of radius 10 A with surface charge density = —0.01 eo/A calculated according to the PB equation (Eq. [389]) without activity corrections (dashed line), the PB equation with MPB activity corrections of Eqs. [429] and [430] (solid line), and Metropolis Monte Carlo simulations (filled circles —1 A open circles 2 A).

Well, we have a probability but what we need is a clear decision to be or not to be in state 2. This is where the Monte Carlo spirit comes in, see Fig. 7.12. By using a random number generator we draw a random number u from section [0,1] and... compare it with the number a. If m < a, then we accept the new conformation, otherwise conformation 2 is rejected (and we forget about it). The whole procedure is repeated over and over again drawing micro-modifications a new conformation comparison with the old one by the Metropolis criterion -> accepting (the new conformation becomes the current one) or rejecting (the old conformation remains the current one), etc. 

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