Sampling the chemical potential in NVT simulations

This problem has been brilliantly reviewed by Kumar in a recent book and hence we summarize only the most sahent features here. For small molecule systems, sampling of the chemical potential rests on the Widom test particle insertion method  

Once the first polymer is generated on the lattice, a starting point for the second polymer is selected out of the remaining L - N sites with equal probability, and the further N - 1 monomers of the second polymer are placed on the lattice according to the same method as described above. The excluded volume interaction is taken into account with respect to the first chain and the already grown parts of the second chain. This procedure is continued until the desired number of chains on the lattice has been reached. 

The fraction of successful construction attempts is not an exponentially decreasing function of the number of chains jV, but stays approximately constant at unity until a critical value that depends on N and The 

Since one knows at each step of the construction of a configuration the probability for selecting a lattice site for the next monomer, one can multiply all these single step probabilities in order to obtain the probability of constructing the multiple chain configuration v. Then the partition function Z is estimated from a sampling of the inverse of 

From the partition function the free energy Fjy follows and hence all thermodynamic quantities of interest can be estimated (entropy, chemical potential, osmotic pressure...). Ottinger applied this technique to test the osmotic equation of state for dilute and semidilute polymer solutions for N 60. Extension of this technique to off-lattice systems has also been made.  

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