Along a given time-path, the average change in entropy is zero
Over the whole space of configurations of the universe, the average difference in entropy between a given state and the next state (according to the laws of physics) is zero. (Really this should be formulated in terms of derivatives, not differences, but you get the point).
This is definitely true, and this is an inescapable feature of any (compact) dynamical system. However, somewhat paradoxically, it’s consistent with the statement that, conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy.
In your time-diagrams, this might look something like this:
I.e when you occasionally swing down into a somewhat low-entropy state, it’s much more likely that you’ll go back to high-entropy than that you’ll go further down. So once you observe that you’re not in the maxentropy state, it’s more likely that you’ll increase than that you’ll decrease.
(It’s impossible for half of the mid-entropy states to continue to low-entropy states, because there are much more than twice as many mid-entropy states as low-entropy states, and the dynamics are measure-preserving).
“conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy”
I don’t think this statement can be true in any sense that would produce a non-symmetric behavior over a long time, and indeed it has some problem if you try to express it in a more accurate way: 1) what does “non-maximal” mean? You don’t really have a single maximum, you have a an average maximum and random oscillations around it 2) the “vast majority” of states are actually little oscillations around an average maximum value, and the downward oscillations are as frequent as the upward oscillations 3) any state of low entropy must have been reached in some way and the time needed to go from the maximum to the low entropy state should be almost equal to the time needed to go from the low entropy to the maximum: why shold it be different if the system has time symmetric laws?
In your graph you take very few time to reach low entropy states from high entropy—compared to the time needed to reach high entropy again, but would this make the high-low transition look more natural or more “probable”? Maybe it would look even more innatural and improbable!
This argument proves that
Along a given time-path, the average change in entropy is zero
Over the whole space of configurations of the universe, the average difference in entropy between a given state and the next state (according to the laws of physics) is zero. (Really this should be formulated in terms of derivatives, not differences, but you get the point).
This is definitely true, and this is an inescapable feature of any (compact) dynamical system. However, somewhat paradoxically, it’s consistent with the statement that, conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy.
In your time-diagrams, this might look something like this:
I.e when you occasionally swing down into a somewhat low-entropy state, it’s much more likely that you’ll go back to high-entropy than that you’ll go further down. So once you observe that you’re not in the maxentropy state, it’s more likely that you’ll increase than that you’ll decrease.
(It’s impossible for half of the mid-entropy states to continue to low-entropy states, because there are much more than twice as many mid-entropy states as low-entropy states, and the dynamics are measure-preserving).
“conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy”
I don’t think this statement can be true in any sense that would produce a non-symmetric behavior over a long time, and indeed it has some problem if you try to express it in a more accurate way:
1) what does “non-maximal” mean? You don’t really have a single maximum, you have a an average maximum and random oscillations around it
2) the “vast majority” of states are actually little oscillations around an average maximum value, and the downward oscillations are as frequent as the upward oscillations
3) any state of low entropy must have been reached in some way and the time needed to go from the maximum to the low entropy state should be almost equal to the time needed to go from the low entropy to the maximum: why shold it be different if the system has time symmetric laws?
In your graph you take very few time to reach low entropy states from high entropy—compared to the time needed to reach high entropy again, but would this make the high-low transition look more natural or more “probable”? Maybe it would look even more innatural and improbable!