Monte Carlo ensemble average sampling

Figure 1 Flowchart of the classic Metropolis Monte Carlo algorithm for sampling in the canonical ensemble. Note that samples of the property function f(x) are always accumulated for averaging purposes, irrespective of whether a move is accepted or rejected.

Thus, unlike molecular dynamics or Langevin dynamics, which calculate ensemble averages by calculating averages over time, Monte Carlo calculations evaluate ensemble averages directly by sampling configurations from the statistical ensemble. 

The expectation values on the right hand side of this equation depend only on the ensemble averages of position and momentum operators, which can be evaluated using the VQRS Monte-Carlo sampling scheme outlined above. 

The aim of Monte Carlo simulations is to arrive at a statistical average for properties of interest by using an algorithm that appropriately samples the equilibrium ensemble. The time evolution of the system, described by molecular dynamics, is not defined by Monte Carlo simulations, but in compensation the computation is simpler and more certain of describing an equilibrium state. Monte Carlo and molecular dynamics simulations use similar potential functions. 

There are two main approaches used to simulate polymer materials molecular dynamics and Monte Carlo methods. The molecular dynamics approach is based on numerical integration of Newton s equations of motion for a system of particles (or monomers). Particles follow dctcr-ministic trajectories in space for a well-defined set of interaction potentials between them. In a qualitatively different simulation technique, called Monte Carlo, phase space is sampled randomly. Molecular dynamics and Monte Carlo simulation approaches are analogous to time and ensemble methods of averaging in statistical mechanics. Some modern computer simulation methods use a combination of the two approaches. 

To model the interaction between CSP and analyte one must account for 1) the shapes of the two molecules in the binary complex, 2) the relative position of the two molecules, i.e. the analyte should be at its proper binding site on or around the CSP, and 3) the orientation of the two molecules with respect to each other. This is just a simple way of saying that some sort of ensemble average is needed wherein a molecular dynamics protocol must ensure adequate sampling of phase space, or, if using a Monte Carlo strategy, a sufficient number of important configurations must be sampled. 

In Monte Carlo simulations, ensembles of configurations of a system are generated using a Metropolis importance sampling scheme. Each configuration is sampled with the Boltzmann probability for the desired temperature of the system. From this set of configurations, or ensemble, average properties can... 

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