Ensemble weight function

The density of states in turn, is connected with the canonical ensemble partition function Z T) through the formal weighted sum over states,... 

To calculate the concentration or the number of molecules present in an open system at defined ju, V, T we must consider the grand canonical ensemble which is simply an ensemble in which each subsystem is itself a canonical ensemble. The grand partition function Z is simply the sum of the Boltzmann factors for each of the j canonical ensembles, weighted according to e / ... 

For the canonical ensemble, the systems are constrained to constant N and constant V, and are in contact with a reservoir at constant temperature T. Hence it is an N,V,T ensemble. Since all the ensembles we will consider are at constant N and V, the notation of constant N and V will be suppressed. The canonical weight function is... 

We can discuss MD data on a more sophisticated level by introducing the probability distribution P(X), for each variable X. For the energy E, for example, and for a general ensemble whose statistical weight function is W,... 

To construct an ensemble conjugate to the MD ensemble, we have to introduce a second parameter which gives 3 a nonzero average value. This is the generalized canonical ensemble, whose statistical weight function is... 

We note that Hoover defines y kT as the instantaneous kinetic energy per particle, KEqj /N, and for a "constant temperature simulation," Hoover constrains the kinetic energy to be strictly constant. This produces an ensemble whose weight function contains 6(KEqj - constant), and hence it is different from the MD ensemble. Several articles on nonequilibrium fluids appear in the January, 1984, issue of Physics Today, and of particular interest for nonequilibrium molecular dynamics are the articles by Evans, Hanley, and Hess, by Hoover, and by Alder and Alley. ... 

These conditions are necessary to reach the thermodynamic equilibrium and keep the system in the NVE or NVT ensemble. For simplicity, the weight functions are selected to be similar in form to the conservative force, that is ... 

One now has to correctly weigh the states generated by this biased insertion procedure when one calculates the chemical potential from eq. (1.20) since we generated states of the canonic ensemble modified with a weighting function, w, we have to correct for this weighting function as follows " ... 

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. 

The densities Pi are obtained from a set of degenerate KS wave functions and the w, are the corresponding weights. Without going into details we note that regular density functional theory can be extended to such ensembles. For our problems at hand, we can write down the energy expression as... 

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