## Tuesday, November 10, 2009

### Logical arrow of time and terminology

It's probably too short a time for a new comment about the arrow of time. But let me say a few words, anyway.

The people who are bothered by the second law of thermodynamics - by the lower entropy in the past - often say that it is "far more likely that the state in the past is a higher-entropy state than the present one".

Of course, this statement is incorrect because it directly contradicts the second law of thermodynamics. This law says that the entropy in the past was lower than the current entropy. So as long as the laws of physics are obeyed, it is impossible (or so unlikely that one can identify the probability with zero for all conceivable purposes) for the entropy to be macroscopically higher in the past.

But it may be interesting to look why those people end up with the flagrantly wrong proposition in more detail. They are thinking in terms of the picture above. These folks begin with the current, relatively low-entropy state with an egg. Its entropy equals "S1" and this state appears in the middle of the picture.

They say that if we evolve the present state into the future (the right side of the picture), there are many more higher-entropy states than lower-entropy states to end up with. Consequently, the future state will have a higher entropy than the present one. The egg will get spoiled and smelly. Let me stop with these disgusting details: the punch line is that the entropy increases in the future.

We agree. But these people like to add another statement.

They also say that if we evolve the present state into the past (the left side of the picture), the same argument applies and they deduce that the state in the past had a higher entropy, too. Well, except for one wrong word in this paragraph and two corresponding errors in the diagram we drew. The correct diagram should actually look like this:

What's the difference? Well, it's simple. The two arrows in the left half of the picture were redirected from the right to the left. Using the words of a capitalist pig who has described himself as "pretty much a convert,"
"that other way isn't really backwards in time, it's forward too!"
Why is the second picture correct and the first picture incorrect? You may only answer this question if you realize what the arrows are supposed to mean.
The arrows point from the initial data to the final calculated outcomes.
The calculations are meant to be based on the common evolution laws and on the reasoning that we routinely use to predict the future.

Because in both cases, the egg was used as an assumption (the initial state) of their argument why they end up with higher-entropy states, both arrows should start rather than end in the low-entropy state with the egg.

In the very same way, we must correct the corresponding linguistic bug. The high-entropy state in the left corner of the diagram is actually not a state that should be called the past: it should be called the future simply because it was calculated from - it has evolved from - the egg which is the past.

So as their algorithms to calculate the high-entropy states make very clear, they are dealing with a hypothetical Universe that starts in the middle and whose time continues in two directions. Clearly, their Universe differs from ours because our Universe did not start with a macroscopic egg and it probably doesn't have two semi-infinite temporal lines in which it is evolving - just one of them.

Their Universe also heavily breaks the time-translational symmetry because there exists a privileged moment - the middle of the picture - at which the sign of the time-derivative of the entropy flips.

Nevertheless, it is obvious that this is the Universe they are talking about. Because they assume that the egg is used as the initial state i.e. input data that are being manipulated to directly calculate the final state, the egg is the ultimate beginning of their Universe.

In their reasoning, the other states were calculated from this initial state in the middle. Because their calculation was supposed to mimic the actual evolution, we may also say that if the calculation is correct, all the states have evolved from the eggs in the middle. The eggs are their real Big Bang and the subsequent histories continue in both directions.

They may also be confused about the sign of their "t" coordinate. Their "t" coordinate in the left portion of the diagram may be lower than the "t" coordinate of the central eggs: the latter is probably "t=0". But the signs of the coordinates depend on our conventions. They don't have immediate physical consequences. We can always choose "-t" instead of "t" and predict the same physics.

However, the right definition of the past and the future is independent of these sign conventions for spacetime coordinates. The right definition says that
The future is evolving from the past (and the present).
Correspondingly, the calculations that are designed to theoretically mimic this evolution have the same arrow:
The future is calculated from the past (and the present) as long as we use the usual calculations that resemble the evolution.
It's important that you can't exchange the words "future" and "past" in the sentence above.

That doesn't mean that science can never say anything about the past, by manipulating with the present data or the data from a closer past. But this type of calculation is different from predictions of the future. It follows different formulae, too. They're the formulae of logical inference, e.g. Bayesian inference.

And as we have explained many times, the results of this inference - the retrodictions - always depend on our priors. So the knowledge of the present is enough to calculate the future (classically) or to predict the unique probabilities of various states in the future (quantum mechanically). But it is simply never enough to calculate the unique state or unique probabilities of various states in the past.

The reason has been explained many times. But we can say that at least in the macroscopic context (when some microscopic detailed information is being omitted, e.g. because it's unmeasurable), different initial states "A,B" in the past may evolve into the same final state "C" in the future.

Note that the rules how to evolve the past into the future may be completely known and well-defined.

However, they can't be easily reversed. If we know that the state in the relative future is "C", we can't automatically say whether the initial state in the relative past was "A" or "B". We can't even assign any "objective" probabilities to "A" and "B". If we nevertheless want to make a qualified guess about these probabilities, the qualified guess is (almost) never 50:50. In most cases, being more egalitarian doesn't make you more right.

The qualified guess must take all the available information about the "far past" state that led to "A" or "B" into account. Such additional information inevitably affects qualified estimates of the probabilities of "A" and "B". And regardless of the detailed situation, the qualified guess will always imply that the state that led to "C" had a lower entropy than "C".

Summing vs averaging

More quantitatively, as we have previously argued, the probability of evolution from a macroscopic state U (representing many U-microstates) in the past to another macroscopic state V (representing many V-microstates) in the future is obtained by summing over the final V-microstates, but averaging over the initial U-microstates.

This asymmetry of the rules of logic implies that the entropy of V will almost certainly exceed that of U. It also implies that we need to know the right priors to choose the proper "weighted average" of the initial U-microstates (the "egalitarian" average becomes really bad if "U" is composed out of macroscopically different states) but we don't need to know any priors about the V-microstates - these "priors" or "posteriors" really don't exist for a final state - because the probability is summed over them so all the weights are equal to one.

A small disclaimer

Finally, it is necessary to emphasize that throughout the several paragraphs above, I was using the correct, physically invariant definition of the "past" and the "future". For example, both high-entropy pictures on the diagram, the right one as well as the left one, were called "the future".