Boltzmann weighted probability

The Metropolis prescription dictates that we choose points with a Boltzmann-weighted probability. The typical approach is to begin with some reasonable configuration qj. The value of property A is computed as the first element of the sum in Eq. (3.33), and then qi is randomly perturbed to give a new configuration qa. In the constant particle number, constant... 

The probability of achieving a viable configuration can be estimated by associating a Boltzmann weight to all the configurations according to their energies... 

It can be shownthat asymptotically (i.e. in the limit where the number M of generated configurations tends to infinity) this procedure generates system configurations x, with a probability proportional to the Boltzmann weight, P,(x) = exp [— Jf(x)//cBr]/Z. Thus thermal averages are just calculated as simple arithmetic averages ... 

For many systems the ensemble that is used in an MC simulation refers to the canonical ensemble, (N, V, T). This ensemble permits a rise and fall in the pressure of the system, P, because the temperature and volume are held constant. Thus, the probability that any system of N particles, in a volume V at temperature Tis found in a configuration x is proportional to the Boltzmann weighted energy at that state, Ex> and it is given by... 

A polar molecule such as HC1 possesses a permanent dipole moment // by virtue of the non-uniform electric charge distribution within the neutral molecule. The electrostatic energy between two interacting dipoles //1 and 112 is strongly dependent on their relative orientation. If all relative orientations are equally probable and each orientation carries the Boltzmann weighting factor e-cW r, the following expression is obtained ... 

The canonical probability distribution of potential energy Pnvt( o T) is then given by the product of the density of states n(E) and the Boltzmann weight factor Wb(E T) ... 

The trial move from o to n is then accepted with a probability equal to Min l,p /po, where po) denote the (Boltzmann) weights of, respectively, the trial state and the original state. 

For a system of N atoms or molecules, an instantaneous configuration represents a microstate in the phase space of the system (position and momentum), characterized by the set of state variables a (denoted as a for brevity). The probability for a system being in a given microstate is given by its Boltzmann weighting factor... 

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