When deriving the probability distribution for the isolated system, we were working in the dark – we assigned the probabilities devoid of information about the system. It is natural to ask: were we endowed with some data, how could we use it to revise our set of probabilities?
Let’s suppose that with each microstate we associate a value . The quantity is the value of some observable in the state . The physical meaning of can be anything you like, so long as it can be measured numerically. If is the probability that the system is in state , then the probability that has the value is . So the mean value of is
by definition of the probabilities . The mean value of the observable is that which we measure in the lab. If we can say with confidence that we know some property of the system through experimental evidence, we place a further constraint on the through the form of equation given above. We convey our knowledge about the system through constraints.
At last we’re ready to continue.