Reweighting

Recently, Orkoulas and Panagiotopoulos [161] have shown that it is possible to use histogram reweighting and multicanonical simulations, starting with individual simulations near the critical point, to map out the liquid-vapour coexistence curve in a very efficient way. 

Ferrenberg A M, Landau D P and Swendsen R H 1995 Statistical errors in histogram reweighting Phys. Rev. E 51 5092-100... 

Having obtained a set of histograms like Fig. 12 for different system sizes (using histogram reweighting methods [175] in larger systems), the proce-... 

A useful method of weighting is through the use of an iterative reweighted least squares algorithm. The first step in this process is to fit the data to an unweighted model. Table 11.7 shows a set of responses to a range of concentrations of an agonist in a functional assay. The data is fit to a three-parameter model of the form... 

A more general version of the canonical reweighting scheme in (3.5), in which the value of any order parameter is reweighted to different temperatures, is given by ... 

A brief sketch of the derivation of the WHAM equations follows we note that a detailed explanation is available in the book by Frenkel and Smit [14], Consider the canonical reweighing (3.5). Our goal will be to combine the histograms pi(U) from several runs at different temperatures T to predict the distribution of potential energies at a new temperature T. Individually, each run enables us to reweight its histogram to obtain the distribution at T... 

Fig. 3.3. Typical results from a density-of-states simulation in which one generates the entropy for aliquid at fixed N and V (i.e., fixed density) (adapted from [29]). The dimensionless entropy. r/ In ( is shown as a function of potential energy U for the 110-particle Lennard-Jones fluid at p = 0.88. Given an input temperature, the entropy function can be reweighted to obtain canonical probabilities. The most probable potential energy U for a given temperature is related to the slope of this curve, d// /dU(U ) = l/k T, and this temperature-energy relationship is shown by the dotted line. Energy and temperature are expressed in Lennard-Jones units...

Panagiotopoulos, A. Z. Wong, V. Floriano, M. A., Phase equilibria of lattice polymers from histogram reweighting Monte Carlo simulations, Macromolecules 1998, 31, 912-918... 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

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. ... 

Fig. 8.2. Maxwell (left) and skewed momenta (right) distributions in two dimensions. If a slow direction is identified, the probability can be skewed along that direction such that it is more likely to kick the system along it exact kinetics is recovered by reweighting...

In summary, the Gibbs ensemble MC methodology provides a direct and efficient route to the phase coexistence properties of fluids, for calculations of moderate accuracy. The method has become a standard tool for the simulation community, as evidenced by the large number of applications using the method. Histogram reweighting techniques (Chap. 3) have the potential for higher accuracy, especially if... 

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