PX284 - B3 - assumptions

assumption 1

each of the possible microstates of an isolated system in thermal equilibrium is equally likely to occur

assumption 2

the system's dynamics are such that the microstate is constantly changing

assumption 3

given enough time, the system will explore all possible microstates, and spend equal time in each
this is known as the ergodic hypothesis

consequences

a system will appear to choose the macrostate associated with the maximum number of microstates
the measured value of an observable is equal to the average over all accessible microstates

example

three SHOs with $E = 2 ℏ ω$ (ignore ZPE)
the expected value of $E^{(1)} :$

⟨ E^{(1)} ⟩ = \sum_{i} E_{i}^{(1)} p_{i} = \frac{1}{Ω} \sum_{i} E_{i}^{(1)} = \frac{1}{6} (2 + 1 + 1 + 0 + 0 + 0) ℏ ω = \frac{2}{3} ℏ ω

where, $p_{i}$ is the probability of that microstate occurring, and $Ω$ is the number of microstates