Boltzmann weights

The average of the step function, using the action for a Boltzmann weight can be pursued by standard statistical mechanics. It may require more elaborate sampling techniques such as the Umbrella sampling [20]. 

Eor many systems the ensemble that is used in an MC simulation refers to the canonical ensemble, (N, F/ 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 probabiUty that any system of N particles, in a volume H at temperature Tis found in a configuration x is proportional to the Boltzmann weighted energy at that state, E, and it is given by... 

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

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

The end result of this process is a series of energy minima for the molecule of interest. An appropriate weight can be assigned to each conformation using the usual Boltzmann weighting... 

This derivation addresses a potentially confusing point, namely that the final state at time t is not at equilibrium and may not have a well-defined temperature. As is clear from the derivation, temperature here is only a parameter that is once used to specify the initial condition, and then again in the Boltzmann weight of the integrated work. From this viewpoint, the reweighting is a convenient mathematical trick. ... 

With this distribution being Gaussian in bothp(0) and c/(0), and the work being linear in p(0) and q(0), the distribution of the work obtained for Boltzmann-weighted initial conditions is Gaussian as well, with a mean and variance of... 

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 ... 

We now briefly consider finite size effects at first-order phase transitions. The easiest case is transitions driven by the field conjugate to the order parameter in systems at T < T. Then Eq. (40) is easily generalized by introducing Boltzmann weight factors for the two states < according to their Zeemann energies... 

As the appropriate Boltzmann weights are included in the Metropolis Monte Carlo sampling technique, the average value of the polarizability, or any other property calculated from the MC data, is given as a simple average over all the values calculated for each configuration. 

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