One man's look at the arrow of time

This article by Dan Polansky looks at the problem of the arrow of time in physics.

One formulation of this problem is this: since physical laws are symmetric as for the arrow of time, why is it that we see arrow of time in events? Put differently, why is it that we can distinguish a movie played forward from a movie played backward?

A first response is this. Is it really true that all fundamental physical laws are symmetric? Among the currently recognized fundamental forces/interactions, there are repulsive forces. And a repulsive force does not seem to be symmetric as for the arrow of time. To see this, let us consider two material points located at a certain distance (not mass points since we make no assumption about mass and gravitational force) and let there be a single force only, a repulsive force. Let them be stationary, not moving. Let us press "play" on a simulation. Then, the repulsive force makes the two points move apart. At no point in time does this change: they will keep moving apart indefinitely. And then, anyone seeing a movie of the two points will be able to tell whether the movie is being played forward or backward. The situation changes with a mixture of attractive and repulsive forces, but it is not clear that the result will be symmetric in time given the presence of at least one repulsive force. (To be more specific, two electrons are subject to repulsive electromagnetic force. So are two protons; they are kept together in an atomic nucleus by an attractive force stronger than the repulsive electromagnetic force.)

The above paragraph is now criticized at the talk page, Talk:One man's look at the arrow of time. The criticism points out that the apparent detectability of the arrow of time (direction of the movie replayed) is there only if we know the initial condition of the system. As for a system that combines, say, the repulsive electromagnetic force on two electrons and the attractive gravitational force, this would require a closer look, but if we believe the sources that say that the fundamental laws are symmetric in time, one would assume the result to be that one cannot detect the direction of the movie either.

It can be admitted that simulating only the gravitational force for two points will lead to symmetrical behavior. This is so if we assume that the two points cannot touch, having no extension (extension would be explained by something like a repulsive force). Thus, initially, the two points do not move and are at a distance, then, they accelerate toward each other, and then they slow down again until they end up with zero velocity and their original locations swapped. Then the behavior gets cyclic. The behavior is perfectly symmetric in time. This reminds of a different thought experiment: one drills a hole through the Earth and lets a body drop into the hole. The body accelerates toward the center and then decelerates, ending up at zero velocity at exactly the opposite point. This, of course, assumes that the Earth is perfectly symmetric along the drilling axis and that the center of the Earth is solid (and therefore fit for drilling a hole), which is probably not the case.

The above consideration was for two points but we can also consider multiple points subject to gravitational force. This then seems related to the three-body problem (W: Three-body problem). One would have to figure out whether this is really symmetric in time or whether the points being more than two make a difference.

Another formulation of the problem of the arrow of time concerns only thermodynamic systems. This is technically a different problem. The formulation can be this: the microscopic laws of statistical thermodynamics are symmetric in time. How does it happen that the macroscopic classical thermodynamics has the second law? A response is this. Is it really true that the microscopic laws are symmetric, given repulsive forces are at play at microscopic level? But my understanding of the technicalities of this problem is possibly poor, and there could be something to render this response invalid. Be it as it may, a problem concerning a purely thermodynamic system cannot have a direct bearing on a cosmological problem of arrow of time since the world has the gravitational force, which is not represented in a purely thermodynamic model.

There is some talk about tendency of increase of entropy in relation to the arrow of time. First of all, this is merely a tendency, not an inexorable law of increase. Moreover, we need to consider which concept of entropy we are dealing with since there are multiple of them. In the context of thermodynamics, this could be thermodynamic entropy, either classical thermodynamic entropy or statistical thermodynamic entropy.

We can also investigate an increase of mixing entropy, a different concept. Let us consider a pack of cards. Let us consider the ordering of colors to define the mixing entropy (we do not consider the other values such as king vs. queen to avoid the charge that ordering of these is arbitrary). Let the cards be initially ordered by colors, that is, there is a sequence of cards of one color, then the second color, etc. Let us consider the mixing operation of swapping two randomly chosen cards. By iterating this operation, the system will tend to move away from the initial state. But if we wait long enough, it is very likely (nearly certain) to eventually return to the perfectly ordered state in terms of order of colors. The arrow of time is given in the thought experiment; it is not something that we detect by measuring the mixing entropy increase. But let us suppose that there is a mechamism that allows some entity to revert the arrow of time, in the sense that there will be a undo. But it is not known when or how often such an intervention occurs. And let us only be able to observe the sequence of the states of the system. Then, we could indeed try to infer that such intervention is happening by oserving a suspect decrease of entropy such as a move from a well mixed state to the perfectly ordered state by a minimum length sequence of swaps. Here, we still have the observer's arrow of time fixed, but we emulate something like another arrow of time by representing the undo intervention. Then, it would make sense to think of an observer trying to detect whether the arrow of time is being swapped. On the other hand, a particular sequence of swaps leading to a perfectly ordered state is as improbable as any other particular sequence of swaps. This could shed doubt on the observer's ability to detect the undo interventions. It is perhaps something of a puzzle.

We can consider another discrete model of entropy, that of Kolmogorov complexity of a sequence of zeros and ones. Let the system with a state of length n start at all zeros and let each change in state consist in a random flip of a randomly chosen bit. Then, as time passes, it seems that the system will tend to move away from all zeros or all ones. However, Kolmogorov complexity (the length of the program generating the sequence) stands in sharp contrast to Shannon entropy, which is probabilistic. It is not clear how a theorem about a tendency of increase of Kolmogorov complexity in the our system would look like (especially how to specify what "tendency" means), whether there is such a theorem and which sources cover it. There is a related stackexchange.com question.

Concerning time being part of space-time and therefore something like an analogue of a space coordinate, Popper points out that the axis of time is clearly given while no particular axis (there are three of them) of space is clearly given. This does not concern the arrow/direction of time, though, so is perhaps something of a tangent. Esfeld 2005 discusses Popper's views on the arrow of time.

My tentative take, one that I cannot formally substantiate, is this: the physical time has the same direction as the psychological time (but not the same rate of flow). There is no need to try to track the direction of time using some quantity or characteristic of the world. A characteristic of the word can have a tendency to change in a certain direction as the time passes, but this does not define the arrow of time, and merely tends to be associated with it.

But from a different angle, something like the initial question has not been answered: what is it about the world we are living in that makes movies played forward so different from those played backward? This question does not question the existence of the real arrow of time independent of states of the world; it nonetheless puzzles about the kind of changes that happen in time. This apparent phenomemon is detectable without considering chemistry, biology or sociology; physics alone will do. Since, we can observe a piece of rock fall onto the Moon, but we do not expect a rock lift from the surface of the Moon.

Further reading

• Arrow of time, wikipedia.org
• T-symmetry, wikipedia.org
• The Arrow of Time, Internet Encyclopedia of Philosophy
• Thermodynamic Asymmetry in Time, Stanford Encyclopedia of Philosophy