Friday, October 08, 2010 ... / / Arrow of time, part 101010

The past and the future play a fundamentally asymmetric role in logic. After more than 100 years, some people are still not getting it.

A backward guy is living his ordinary everyantiday backward life. Things get dramatic when he starts to antidrink and antieat. Some people still don't believe me but the events above don't naturally occur in the real world.

They imagine that to exchange the past and the future is as easy as to rotate a picture by 180°. But it's not.

This 101010th part of the neverending story will try to shrink the argument as much as possible.

Any evolution whose probability we want to calculate may be written as

Forward = A evolves into B
where "A" means that the physical system is found in one of the microstates (precisely determined positions and/or velocities of elementary particles, or whatever else can be maximally determined) A_1 or A_2 or A_3 ... or A_{N_a} where N_a is the number of microstates that we summarize as A.

Similarly, "B" means that the physical system is in one of the microstates B_1 or B_2 or B_3 ... or B_{N_b} where N_b is the number of microstates that we summarize as B.

The probability of evolution is calculated as
P(Forward) = 1/N_a Sum [i=1...N_a, j=1...N_b]
P(A_i evolves to B_j)
Note that to get the total probability, we're simply summing over possible final states B_j because the probability that "B_3 or B_5" appears as the final state is simply the sum of the partial probabilities for B_3 and B_5.

However, we're averaging over the initial states A_i. We know that one of the microstates A_i was realized in the initial state but we don't know which one. The sum of these "prior" probabilities of A_1, A_2, etc. has to equal one. Therefore, each of them has to be given the prior probability 1/N_a only. (We assume all the microstates to be equally likely, an assumption that can be relaxed if you need it.) The probability of the evolution from A_i is proportional to the prior probabilities of A_i, so we had to divide by N_a because the priors contain 1/N_a.

Now, assume that the physical system is T-reversal symmetric. And we also want to calculate the probability of the reverse process,
Backward = B evolves into A.
More generally, we should consider states A* and B* whose "velocities" are reverted relatively to A, B which is appropriate when the sign of time is inverted.

But for the sake of simplicity, let's assume that the velocities of both signs are equally represented in the ensemble A as well as B, so that the ensembles satisfy A=A*, B=B*. It wouldn't be hard to generalize the calculation for general states A, B.

What is the probability of the backward process? Well, it's given by the same formula as the forward process but A is exchanged with B:
P(Backward) = 1/N_b Sum [i=1...N_a, j=1...N_b]
P(B_j evolves to A_i)
Note that the coefficient in front of the sum is 1/N_b in this case rather than 1/N_a. So are the probabilities of forward and backward evolutions equal? Even if the microscopic laws are T-reversal-symmetric i.e. P(B_j evolves to A_i) is the same as P(A_i evolves to B_j), it's still true that
P(Backward) / P(Forward) = N_a/N_b.
Note that they are not equal if the numbers N_a, N_b differ. When we use the language of thermodynamics and the ensembles A, B are large, the numbers N_a, N_b are given by the entropies of the states A, B called S_a, S_b:
N_a = exp(S_a / k), N_b = exp(S_b / k)
where k is Boltzmann's constant which means that
P(Backward) / P(Forward) = exp[(S_a-S_b) / k].
If the entropy of B, S_b, is greater by 10^{26}*k than the entropy of A, which is typically the case in a macroscopic process (which adds a bit of entropy to each atom, and recall how large Avogadro's number is), then the backward probability is smaller than the forward probability by a factor of exp(10^{26}): for all practical (and most impractical) purposes, it is zero.

Note that I don't mean just 10^{26}: I mean exp(10^{26}) which is approximately the same thing as 10^{10^{26}}. This number quantifies the error that the people who don't understand the difference between the past and future are making in every single calculation of probability. A huge error, indeed.

This is no philosophy and there is no controversy here; it is an unquestionable conclusion of basic mathematical logic. A large entropy only increases the probability of a process if it is the final state that has a large entropy; because of the 1/N_a factor, a large entropy of the initial state is not favored by Nature in any way.

At any rate, there is a fundamental asymmetry between the initial states and the final states in mathematical logic that governs all calculations of probabilities that we ever perform in science, and outside science. Some naive "visualizations" of the Universe may look past-future symmetric but the full-fledged reality including all the required structure, including mathematical logic, is surely not past-future symmetric.

This asymmetry has nothing whatsoever to do with any assumptions about cosmology. It is a property of each cubic Planck length of space at each individual Planck time. The calculations above are valid anywhere and everywhere, whether or not there were many Big Bangs, one Big Bang, or no Big Bang.

I am completely convinced that every college freshman or sophomore who has gotten a grade better than F from mechanics, statistical physics, and quantum mechanics should feel absolutely certain about the rudimentary calculations above because the logical considerations needed to derive the conclusions above are needed in every portion of science.

And that's the memo.    snail feedback (5) :

All true within the limits E.t uncertainty and any theories of the 2T variety I guess. Comments?

reader Luboš Motl said...

My only comment is that your sequence of 15 words or so is too short to contain any useful information or insight, but long enough so that it can be seen that your ideas have nothing to do with the topic of this article.

I heard it attributed to Feynman that "time is what goes on when nothing else does."

reader Fábio Novaes said...

But how come we have entropy increasing in a closed universe with unitary evolution? Putting in another way, how come the second law came into being? Or should we take it for granted?

reader Luboš Motl said...

Dear Fabio,

yes, you should definitely take the second law for granted and it's true. However, it hasn't fell from the skies or a religious text. It can be easily and universally proven, and if you carefully read this very text of mine, you will see one proof. If you haven't gotten it, read it twice, five times, ten times, 50 times, until you get this simple point because it is both trivial and paramount.

It's being proved that the probability that A evolves into B is much less (exp(-S/k) times) less likely than the probability from B to A if the evolution would decrease entropy. Consequently, processes with macroscopically decreasing entropy are impossible in practice as their probability is astronomically lower than the probability of the proper processes that increase the entropy.

The second law doesn't contradict unitarity. The entropy is not an "actual" objective property of the single microstate state (one that evolves unitarily). It is the logarithm of the number of microstates that are macroscopically indinstinguishable (given a convention for "indistinguishable", but the details of such a convention only make a very small impact) from the given microstate. This number of macroscopically indistinguishable states is not preserved by unitary evolution. And indeed, it increases all the time (unless one is already at equilibrium where it is constant).

Best wishes
Lubos