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where Xi is the value during the i-th measurement whose time duration is very short – so short, in fact, that during the i-th measurement the system can be considered to be in only one microscopic state. Then we can partition the sum as X obs 1 ( Number of times state ν is observed in the N observations) X N where Xv value of property X in microstate v. The term in the squared brackets is the probability or weight for finding the system during the course of the measurements in state ν. Remember, we believe that after a long time, all states are visited (ergodic hypothesis). We give the probability or fraction of time spent in state ν the symbol Pν and write Xobs PX X The averaging operation (i.e., the weight summation over Xν), indicated by the pointed brackets, X , is called an ensemble average. An ensemble is the assembly of all possible microstates – all states consistent with the constraints with which we characterize the system macroscopically. The idea that we observe the ensemble average, X , arises from the view in which measurements are performed over a long time, and that due to the flow of the system through phase space, the time average is the same as the ensemble average*. The equivalence of a time average and an ensemble average, while sounding reasonable, is not at all trivial. Dynamical systems that obey this equivalence are said to be ergodic or satisfy the ergodic hypothesis. These concepts can be sum up into two postulates. * Maxwell (1860) and Boltzmann (1872) showed that one can compute a macroscopic property of a system X as the time average and latter Gibbs (1890) showed that one could replace the time average with an ensemble ‘average’ of X. 35PDF Image | MOLECULAR SIMULATION STUDIES IN THE SUPERCRITICAL REGION
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