Calculation of pressure in dynamic Monte Carlo methods

2 Calculation of pressure in dynamic Monte Carlo methods 

If a polymer solution is modeled by an assembly of self-avoiding walks on a lattice, a basic physical quantity is the osmotic pressure II. Carrying out a simulation with a fixed number JT of chains of length at a lattice of volume V with one of the dynamic algorithms described in Section 1.2.2, the osmotic pressure is not straightforward to sample. If one had methods that yielded the excess chemical potential and the Helmholtz free energy Fjr, one would find II from the thermodynamic relation 

This quantity describes the probability that a randomly chosen iV-mer, placed at random into a randomly chosen configuration of AT-mers on a lattice of volume F, does not overlap any of the chains. From eqs (1.18) and (1.29) one derives the relation for the excess chemical potential in terms of this insertion probability, 

In the thermodynamic limit, the summation over the number of chains can be replaced by a thermodynamic integration over the volume fraction (f of occupied sites j = NJT f V measuring lengths in units of the lattice spacing) to find p 4 , iV) = lim p, N, V) at fixed (f ] 

This result shows that the osmotic pressure can be obtained from a thermodynamic integration if the insertion probabiUty p f , N) is sampled over a range of values from = 0 to 4 — f - This method has been appUed in conjunction with some of the methods of the previous subsection where the estimation of the chemical potential via the insertion probability was discussed. 

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