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.


One thought on “Statistical Mechanics – Knowledge

  1. Pingback: Statistical Mechanics – Closed Systems | Conversation of Momentum

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