In the book 'Statistical Physics' by Daijiro Yoshioka, the author states without proof that almost all microscopic states will be realised as the state of the system changes temporally. I'm not entirely certain I managed to prove this using the Borel-Cantelli theorem as follows:
- We consider a system of solid, liquid, or gas enclosed by an adiabatic wall.
- We assume fixed Volume($V$),fixed number of molecules ($N$), and fixed but total energy($E$) with uncertainty $\delta E$.
- The total number of microscopic states allowed under the macroscopic constraints is given by
$$W=W(E,\delta E,V,N) \tag{1}$$ Assuming that each micro-state is realised with equal probability, we have for any micro-state $m_i$:
$$ P(m_i | t_n) = \frac{1}{W} \implies \sum_{n=1}^{\infty}P(m_i | t_n)=\infty \tag{2}$$
where I assume that the total number of micro-states is finite. We can then show using the Borel-Cantelli theorem that for any $m_i$, $E_n=\{m_i | t_n\}$ occurs infinitely often.
Note 1: There's something that bothers me about my proof. Somehow it assumes that the probability of transition between any two states,no matter how different, is always the same. For this reason, I now think that this proof might require more than the Borel-Cantelli theorem and that the 'equal probability' assumption might not hold for state transitions. It seems reasonable that some transitions would be more likely than others. Maybe I can use Borel-Cantelli for 'large-enough' time intervals. But, this would need careful justification as well.
Note 2: I think it might be helpful to use an integrable variant of the Borel-Cantelli theorem: https://mathoverflow.net/questions/258389/integrable-version-of-the-borel-cantelli-theorem