One problem with Boltzmann’s derivation of the second law of thermodynamics is that it “proves too much”. Because an analogous derivation also says that entropy “increases” into the past direction, not just into the future direction. So we should assume that the entropy is as its lowest right now (as you are reading these words), instead of in the beginning. It basically says that the past did look like the future, just mirrored at the present moment, e.g. we grow older both in the past and the future direction. Our memories to the contrary just emerged out of nothing (after we emerged out of a grave), just like we will forget them in the future.
This problem went largely unnoticed for many years (though some famous physicists did notice it, as Barry Loewer, Albert’s philosophical partner, points out in an interesting interview with Sean Carroll), until David Albert pointed it out more explicitly some 20 years ago. To “fix” the issue, we have to add, as an ad-hoc assumption, the Past Hypothesis, which simply asserts that the entropy in the beginning of the universe was minimal.
The problem here is that the Past Hypothesis can’t be supported by empirical evidence like we would naively expect, as its negation predicts that all our records of the past are misleading. So we have to resort to more abstract arguments in its favor. I haven’t seen such an account though. David Albert has a short footnote on how assuming a high entropy past would be “epistemically unstable” (presumably because the entropy being at its lowest “now” is a moving target), but that is far from a precise argument.
The assumptions made here are not time reversible as the macrostate at time t+1 being deterministic given the macrostate at time t, does not imply that the macrostate at time t is deterministic given the macrostate at time t+1.
So in this article the direction of time is given through the asymmetry of the evolution of macrostates.
Yup. Also, I’d add that entropy in this formulation increases exactly when more than one macrostate at time t maps to the same actually-realized macrostate at time t+1, i.e. when the macrostate evolution is not time-reversible.
One problem with Boltzmann’s derivation of the second law of thermodynamics is that it “proves too much”. Because an analogous derivation also says that entropy “increases” into the past direction, not just into the future direction. So we should assume that the entropy is as its lowest right now (as you are reading these words), instead of in the beginning. It basically says that the past did look like the future, just mirrored at the present moment, e.g. we grow older both in the past and the future direction. Our memories to the contrary just emerged out of nothing (after we emerged out of a grave), just like we will forget them in the future.
This problem went largely unnoticed for many years (though some famous physicists did notice it, as Barry Loewer, Albert’s philosophical partner, points out in an interesting interview with Sean Carroll), until David Albert pointed it out more explicitly some 20 years ago. To “fix” the issue, we have to add, as an ad-hoc assumption, the Past Hypothesis, which simply asserts that the entropy in the beginning of the universe was minimal.
The problem here is that the Past Hypothesis can’t be supported by empirical evidence like we would naively expect, as its negation predicts that all our records of the past are misleading. So we have to resort to more abstract arguments in its favor. I haven’t seen such an account though. David Albert has a short footnote on how assuming a high entropy past would be “epistemically unstable” (presumably because the entropy being at its lowest “now” is a moving target), but that is far from a precise argument.
The assumptions made here are not time reversible as the macrostate at time t+1 being deterministic given the macrostate at time t, does not imply that the macrostate at time t is deterministic given the macrostate at time t+1.
So in this article the direction of time is given through the asymmetry of the evolution of macrostates.
Yup. Also, I’d add that entropy in this formulation increases exactly when more than one macrostate at time t maps to the same actually-realized macrostate at time t+1, i.e. when the macrostate evolution is not time-reversible.