Grand canonical approach

The practicality of (10) implies the feasibility of a Metropolis sampling procedure in the grand canonical ensemble, so that steps in the Markov chain 

McDonald and Singer were the first to base a Monte Carlo calculation on the formulation of Q as a one-dimensional integral. 

In an attempt to obtain information about y U) over wider ranges of energy, the same authors proposed a bounded random-walk procedure. At each step of the Markov chain the trial state is always accepted unless its energy exceeds a preset upper bound. All states with energy below the bound will be equally probable and the frequency with which a particular energy is encountered will be proportional to y U). The unknown proportionality constant is 

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Due to its more complicated structure the canonical expansion is not well adapted for a discussion of general features like the properties of the theory under renormalization. Thus in the sequel we exclusively work with the grand canonical approach. 

Garel, T., Monthus, C. and Orland, H. (2000). Copolymer at a Selective Interface and Two-dimensional Wetting a Grand Canonical Approach, Eur. Phys. J. B 17, pp. 121-130. 

The definition of the distribution function given above is valid in the canonical ensemble. This means that N is finite. Of course, N will, in general, be very large. Hence, g(ri,..., r/,) approaches 1 when aU the molecules are far apart but there is a term of order X/N that sometimes must be considered. This problem can be avoided by using the grand canonical ensemble. We will not pursue this point here but do wish to point it out. 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

However, if one focuses on the adsorption of a fluid in heterogenous matrices [32,33] and/or on the fluctuations in an adsorbed fluid, it is inevitable to perform developments similar to those above in the grand canonical ensemble. Moreover, this derivation is of importance for the formulation of the virial route to thermodynamics of partially quenched systems. For this purpose, we include only some basic relations of this approach. 

The same approach applies to alternate partition functions the WL output is used directly in the macroscopic probability scheme. For example, in the previous grand canonical scenario, the WL simulation would yield which would be substituted directly into the macrostate probabilities as Q exp(J ) for subsequent results generation. 

Indeed, there is such an approach to DFT that gives a physical justification to the above assumption of continuity with the only complication involved being the non-differentiability of Eo n) at an integer number of electrons n = N, a phenomenon known as DFT derivative discontinuity . The approach is based on an extension of the original Hohenberg-Kohn theorem [20] to the grand canonical ensemble first given by Mermin [21]. It... 

More realistic approaches should, of course, comprise solvent models that give rise to electrostatic interactions. Shelley and Patey [273] used grand canonical MC simulations to investigate the demixing transition in model ionic solutions where the solvent is explicitly included. Charged hard-sphere ions in neutral, dipolar, and quadrupolar hard-sphere solvents were consi-... 

The fraction / of charged monomers on an annealed polyelectrolyte is measured in a titration experiment. The simplest description of this experiment [90] is to assume that the polymer is homogeneous and to minimize its grand canonical free energy (its free energy at constant pH). In a mean field approach, the free energy of one chain is... 

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