Ergodicity and finite time series

The general idea behind all Monte Carlo methodologies is to provide an efficient stochastic sampling of the configurational or conformational phase space or parts of it with the 

2 Conventional Markov-chain Monte Carlo sampling 

A Monte Carlo update corresponds to the discrete time step A to in the simulation process. In order to reduce correlations, typically a number of updates are performed between measurements of a quantity O. This series of updates is called a sweep and the time passed in a single sweep is At =ATAto if the sweep consists of N updates. Thus, if M sweeps are performed, the discrete time series is expressed by the vector (0(Tinit + At), 0(Tiiut + 2At),. .., 0(xaat + wAt),. .., 0(Ti it + MAt)) and represents the Monte Carlo trajectory. The period of equilibration Tjnjt sets the starting point of the measurement. For convenience, we use the abbreviation 0 = 0(Tjnjt -b wAt) and 

According to the theory of ergodicity, averaging a quantity over an infinitely long time series is identical to performing the statistical ensemble average  

M 1 (O). Note that the mean value will depend on the sample k, 

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