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There are roughly infinite states for many body particle systems. There is not enough time in universe or our life time to wait until the system passes through all the states and we sum over the observable to all states to find the expectation value of any observable. Instead we know that at thermal equilibrium, the system mostly wonder around similar states and we know that there are some states which have more probability of being in, and hence the expectation value is mostly decided by these Important states.
- So the idea is to sample the Important states only and find expectation value from these states. This process is called as Importance Sampling.
- Let us imagine we have some large set of states, out of these states we need to select the important states in which system has more probability to be in. This can be achieved by mimicking the mother nature which also select states on basis of Boltzmann probability distribution. If we use this idea then the estimator becomes :