Grandcanonical ensemble

To transform from the canonical to the grandcanonical ensemble with respect to guest molecules using the chemical potential of the guest species, the grand partition function, E, is written as... 

Certainly, restricting the window size limits order parameter fluctuations to far less than those explored in a grandcanonical simnlation and each subsimulations resembles more closely a simulation in the canonical ensemble than in the grandcanonical ensemble. We emphasize, however, that local density (order parameter) fluctuations are not restricted and that, ideally, configurations... 

Other quantities can be obtained from suitable derivatives of the free energy or direct statistical mechanics averages e.g. in the semi-grandcanonical ensemble of a symmetrical mixture (NA = NB = N) (where the chemical potential difference Ap = pA — pB per effective monomer is a given independent thermodynamical variable) the order parameter defined as the relative excess in the number of A-chains (nA) over the number of B-chains (nB) in the system... 

The first two equations show that the densities are given by the density of a single molecule in an external field (within the grandcanonical ensemble) while the next two equations relate the external fields to the local densities. Inserting the values at the saddle point in Eqs.(55) and (58) we obtain ... 

The surrounding of the bubble, the mother phase, is characterized by its pressure p = Pont and molar fraction jc. Since bubble (inner phase) and mother phase (outer phase) can exchange particles, the grandcanonical ensemble is appropriate and we employ chemical potentials ps(p, Jc) and pp(p, x) as to reproduce the pressure and composition of the mother phase. We use the total excess of polymer with respect to the mother phase... 

Similar to our SCF calculations we perform our simulations in the grandcanonical ensemble, i.e., we fix the volume V, temperature T, and chemical potentials fis and fip of both species, but the number of particles in the simulation cell flucm-ates. Particle insertions and deletions are implemented via the configuration bias MC method [104, 105,106,107, 108,109]. Additionally, the polymer conformations are updated by local monomer displacements and slithering snake movements [99]. 

The probability distribution Pmc of observing ns solvent particles and /> polymers in the simulation is related to the distribution function Pgc in the grandcanonical ensemble via ... 

Hence, the bias that the pre-weighting function imparts onto the distribution can be easily removed to calculate thermodynamic averages in the grandcanonical ensemble. Ideally, one chooses w(ns,np) = - In Pccins, np /us, jup), because all combinations of ns and np would be sampled with equal probability. In principle, this scheme would allow us to construct an entire isothermal slice of the phase diagram from a single simulation run. 

In the grandcanonical ensemble two phases coexist at fixed temperature T, if p(i)( coex TOex) (2), coex Using Eq. (156) One obtains the equal-... 

In view of these difficulties, we choose to investigate bubble formation in the grandcanonical ensemble. Thereby, we calculate the free energy of a bubble inside of a finite-sized simulation cell. This method does not provide direct information about the time evolution, because the particle numbers are not conserved and particles do not move according to realistic dynamics. Moreover, we also rely on the assumption... 

---