# Statistical Mechanics – Knowledge

When deriving the probability distribution for the isolated system, we were working in the dark – we assigned the probabilities $\{p_\alpha\}$ 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 $\alpha$ we associate a value $q_\alpha$. The quantity $q_\alpha$  is the value of some observable $q$ in the state $\alpha$. The physical meaning of $q$ can be anything you like, so long as it can be measured numerically. If $p_\alpha$ is the probability that the system is in state $\alpha$, then the probability that $q$ has the value $q_\alpha$ is $p_\alpha$. So the mean value of $q$ is

$\displaystyle \langle q \rangle=\sum_\alpha p_\alpha q_\alpha$

by definition of the probabilities $\{p_\alpha\}$The mean value of the observable $q$ 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 $\{p_\alpha\}$ through the form of equation given above. We convey our knowledge about the system through constraints.

At last we’re ready to continue.