The God of Low Entropy

Arun wants some justification for my claims in the Boltzmann's brain post (Ludwig's Revenge).

...pure statistics seems to imply that entropy should increase in the past as well as the future.

I definitely don't follow the argument.

Physically, if a low entropy state most likely arose from a high entropy state, then we should see a lot of that happening around us. We don't...

Both physically and statistically, a state of given entropy is much more likely to have arisen from a state of lower entropy (even though the number of states of high entropy which could have had a fluctuation to lower entropy is enormous, the probability of such fluctuations is even smaller - e.g., statistical partition functions wouldn't exist without that - the sums would diverge.)

Consider a simple universe consisting of ten distinguishable bits, with an evolution law such that at each time step exactly one randomly chosen bit is flipped. There are a total of 2^10 = 1024 such states, of which 10x9/2 = 45 of them have two bits set and the others all off. Define our coarse grained entropy to be the [UPDATE: the log of the number of microstates corresponding to a given] number of set bits. Thus our state is fairly low entropy.

What about the predecessor state? There are two (coarse grained) possibilities: the predecessor state (PS) had only one bit set (only ten ways to do this), or it had three bits set (10x9x8/(3x2) = 120 ways to do this). Not all of these states are eligible to be the PS, though. If the PS has one bit set, there are just two ways this can happen - the set bit has to be one of the two set in the current state and the switched bit is the other. If there were three bits set in the PS, two of them have to be the two set in the current state and there are eight possibilities for the third set bit, which must also be the switched bit in the transition. Thus, of the ten possible PS, two have lower entropy and eight have higher entropy.

The reasoning here is quite general, and is in fact identical to that leading to the second law of thermo, except for the time orientation. If the previous state is arbitrary, it is more likely to have higher entropy than lower.

So why don't we see that happening all the time? Because it works the same way into the future! For our mini-universe to evolve into the 3-bit set state the probability is 8/10, while it is only 2/10 for it to evolve into the lower entropy 1-bit state. If all possible future states are equally likely, then the future will likely have higher entropy. Ditto the past.

Maybe the following will clarify the point: However unlikely it is that a brain emerged fully formed from chaos (and it's very, very, very unlikely indeed), it is still more unlikely a priori that a whole cosmos just happened to be in an extremely low entropy state.

We don't know of any laws that specify that the universe started in a low entropy state, but none of our laws make sense unless it did. Some magic, some unknown law of nature, or some design did apparently start it off thus, and that's a puzzle.

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