Jaynes, E. T. (1957)
Jaynes, E. T. (1957) Information Theory and Statistical Mechanics. Physical Review, 106(4), 620–630. https://doi.org/10.1103/PhysRev.106.620
Jaynes pointed out in this paper that we are solving an insufficient reason problem. What we could measure is some macroscopic quantity, from which we derive other macroscopic quantities. That being said, we know a system with a lot of possible microscopic states, \(\{ s_i \}\) while the probabilities of each microscopic state \(\{p_i \}\) is not known at all. We also know a macroscopic quantity \(\langle f(s_i) \rangle\) which is defined as
The question that Jaynes asked was the following.
How are we supposed to find another macroscopic quantity that also depends on the microscopic state of the system? Say \(g(s_i)\).
It is a quite interesting question for stat mech. In my opinion, it can be generalized. To visualize this problem, we know think of this landscape of the states. Instead of using the state as the dimensions, we use the probabilities as the dimensions since they are unknown. In the end, we have a coordinate system with each dimension as the value of the probabilities \(\{p_i\}\) and the one dimension for the value of \(\langle g \rangle (p_i)\) which depends on \(\{p_i\}\). Now we constructed a landscape of \(\langle g \rangle (p_i)\). The question is, how does our universe arrange the landscape? Where are we in this landscape if we are at equilibrium?
In Jaynes’ paper, he mentioned several crucial problems.
We write down a function, which is the average of the balls.
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