Some comments on importance sampling: Why weight average sampling is superior to simple average method?

摘要

Integration by Monte Carlo simulation has been proven to be very efficient technique in past few decades, especially for the investigation of complex three dimensional systems. Free minimization drives the system towards equilibrium. One may sample the physical observables either by simple averaging or by importance sampling. Since, the kinetics in Molecular Dynamic (MD) simulation is based on Newton's law of motion, and the whole system is treated as one, sampling is restricted to simple average method. Sampling is done, when equilibrium is reached. If sampling is done after achieving equilibrium, the estimations done by importance sampling and average sampling yield same results, because the spins have unit probability to occupy their states, in that part of the configuration space. But, any early estimation requires the weight average of individual states (i.e. the probability of spins to exist in that state). These weight averages are used to perform importance sampling. Weight averaging of the states not only helps us to determine proximity from the equilibrium state, but leads to the production of reliable and accurate data, even in early stages of the simulation. Data from the simulation work of Ising ferromagnetic films using semi-infinite free boundary conditions are presented, in order to illustrate and highlight the genuineness of importance sampling using Monte Carlo simulation. It helps to explore efficient optimization of data, which has now a day become mandatory to save human and computational resources.