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Introduction to Thermal Physics by Schroeder explains the fundamental assumption of statistical mechanics as,

Let's assume that, over long time scales,the energy in a system gets passed around randomly in such a way that all microstates are equally probable. So if you look at the system at any instant, you are equally likely to find it in any of the possible microstates.

While I understand the big picture of what this assumption is saying, I have questions about the precise details. Why is it important that we consider long time scales? And what does it mean by "equally likely"? Is it saying that if we looked at the system, say every 1 second, for the next 100 hours, each and every microstate will occur with the same frequency?

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2 Answers 2

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I think if you understand the big picture you've understood what Schroeder is trying to explain.

For your questions specifically, you start with the system in some state. So not all microstates are equally likely because your system is in some state in the beginning. What Schroeder is saying is that if you wait for some time (the length of time depending on the specifics of your system), the system could be in any possible microstate with equal probability.

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    $\begingroup$ "Is it saying that if we looked at the system, say every 1 second, for the next 100 hours, each and every microstate will occur with the same frequency?" Yes, it's saying that, but this is clearly not even close to true because there are so many microstates. More precisely, on those timescales (or over billions of years) many systems will explore a representative set of microstates, so that macroscopic observables will appear as though the system really had explored every microstate. $\endgroup$
    – David
    Aug 8, 2021 at 22:42
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    $\begingroup$ I never thought about it that way before, thanks David! $\endgroup$ Aug 8, 2021 at 23:06
  • $\begingroup$ Thank you for the answer. So to summarize all the answers that have been given: the text basically saying that if enough time passes, enough collisions have occured such that energy has spread out from the initial configuration to possibly any configuration, each with equal probability. Is that correct? $\endgroup$
    – sl2outnow
    Aug 12, 2021 at 9:22
  • $\begingroup$ Yes. Since yes is too short an answer for a comment, I'll add that one natural followup question is to ask what all the possible configurations are, and that depends on your environment. For example, is your system at thermal equilibrium with a heat reservoir? This is where you'll hear terms like microcanonical ensemble, grand canonical ensemble, etc. $\endgroup$ Aug 12, 2021 at 17:30
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It is important to consider long time scales because in short time scales, not all the microstates are available. The time scale here is specific to the system in question. So a long time scale for the system need not be a long time for you. If given the initial microstate, you wait a nanosecond and the particles in the system undergo 3 collisions per nanosecond, then the number of possible microstates has increased. But not all microstates are available. Only the ones 3 collisions away from the initial one. But if you wait over a long time scale, so many collisions have occurred that you can safely assume that the system could be in any microstate. That's what he meant by equally likely. If there are a million microstates corresponding to the macrostate, then the probability of you observing any microstate after a minute is one in a million.

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