Why does time appear to flow?

And why that's just an illusion...

This is the mystery: we can walk forwards and backwards in any direction in space, we can even stand still. But we cannot do this in time. Time just flows, it passes us, sometimes slowly, sometimes more quickly. We can't make it stand still, nor can we actually go backwards (certain drug-indosed experiences or unprecedented breakthroughs in physics aside). What's going on?

Well, we know all the laws of physics (without exception) follow the CPT symmetry (charge, parity, time). In simplistic terms, the laws work equally backswards in time as they do forwards. If you saw a film of atoms bouncing off each other in a gas, you couldn't tell whether it was shown backwards or not. But our daily world doesn't work like this. When you see a film of billiard balls bouncing off each other, they eventually come to stand still. If you saw a still billiard ball suddenly starting to move and accelerating, you'd know this film is in reverse. This effect is one example of increasing entropy, often observed in our daily lives as friction or heat.

Now here is an interesting fact. Imagine you try to record something, to form a memory. You might choose to write it down with a pen on paper. If this behaved like the gas, the ink would just jump right off the paper. You couldn't keep it there. This is because physics demands that both energy and momentum are conserved. But if you include entropy, the etching off the word onto the surface actually creates heat. That's where the surplus energy and momentum go.

It can actually be shown that the creation of memory requires a minimum of heat dissipation, or to be more precise, it's the erasing of prior memory to free up space that cause this requirement. But first let's define entropy. It is actually related to the number of microstates that are virtually indistinguishable for a given macrostate, in fact it is the logarithm of the number of microstates. Put differently, it is the number of yes/no (binary) questions that would need to be asked to fully identify the microstate for a given macrostate. For example, if we say a deck of card is fully ordered. There is only one possibly microstate. Boltzmann's formula for entropy S is S = k ln Ω, which Ω the number of microstates and k just a constant factor. So in this example S = k ln 1 = 0.

A randomly shuffled deck of 52 cards, on the other hand has 52! possible microstates. That's approximately 8 x 10^67, a huge number. S becomes k * ln(52!) = 157k. The number of yes/no questions for such a deck is log2(52!) = around 226 yes/no questions. This is because 52! expressed in binary is slightly smaller than 2^226. Then one could simply ask, is the first bit a 1, is the second bit a 1 and so forst 226 times? Translated to cards, one would need to define a particular coding for each card (e.g. Ace of Clubs = 0, King of Clubs = 1 etc.) and then assign the 52! permutations of this list a numeric value based on this coding. This numeric value converted to binary would then be the code to be determined via yes/no questions.

Now, let's say we want to store a memory in this deck. For example, we want the top card to be the Queen of Spades. There is a 1 out of 52 chance that this is the case in a random deck. But if we want to apply that permanently, we effectively reduce the entropy of the deck to 51! possible combinations. Δ Entropy becomes k ln (52! / 51!) = roughly 5.7 bits, i.e. approximately 6 fewer yes/no questions. So to make space for the Queen of Spades we first need to erase these 6 bits. You might remember these little boards given to children, you could write on them and then you could move a lever and wipe it clean. It's the same with an eraser to get rid of pencil marks on paper, or a piece of cloth to wipe a white board or a black board. You always need a minimum amount of externel energy to erase something. The consequence of this erasure is then extra heat added to the environment. Physicists have calculated the minimum amount of heat dumped in the environment to erase one bit and called it Landauer's principle. It is simply the fact that like with the billard ball the extra energy needs to go into the environment as heat. It can then be shown that the extra entropy in the environment is at least as big or bigger than the reduction of entropy in our memory, in this case our deck of cards.

So now we can work with two principles: a) forming a memory always increases entropy and b) forming a memory cannot be done instantenously. The second principle results as any physical process requiring the rearrangement of information cannot be infinitely fast, because information itself cannot be transmitted infinitely fast. (Note that people sometimes talk about quantum entanglement as a process that proceeds faster than the speed of light. You can however show that it is not possible to transmit information that way.)

Given these two principles, memory must always be something in the past. So even though the laws of physics are time-reversible, we cannot form a memory of the future because that would require that the entropy in the future is lower than right now. And the more time the memory formation takes, the further back in time it must go.

Now to the final step in this puzzle. The experience of time must relate to change. If everything stands entirely still, we wouldn't experience time at all. (Never mind that nothing in the brain would work either.) Instead we experience time as something dynamic, something moving, something where the before and after are different. We know that a movie can just be a series of still pictures, but each picture is different, and our brains compute this as movement.

In order to assess the speed of that change, we compare it with something else that is changing. But that something else is a bit more special, we need it to be a repeating change. It needs to be circular in the sense that it keeps returning to an original state. It must have a frequency. Examples are the pendulum of clock swinging or the vibrations of a quartz crystal in a digital clock. Whatever it is, at the heart of any clock there must be a vibration, a repeating change. We can count the vibrations to measure the passing of time. And as before these vibrations cannot be infinitely fast.

So in order to measure the passing of time, we need to remember something. It doesn't matter if this memory is in our brain (counting seconds) or externally, the time shown on the face of a clock. This memory needs to be from the side of change with lower entropy and some amount of vibrations must have passed. In fact we will then need to compare two memories with each other, one more recent to one from earlier. It is this comparison of memories that our consciousness interprets as the flowing of time.

If we are slow in forming memories, more change happens in the outside world, and it feels like time is passing quickly. If we are fast in forming memories, e.g. when stressed or under the influence of adrenaline, or indeed when the brain isn't busy with other stuff (which we call being bored), then not much change happens in the external world between two instances of memory and time feels like passing slowly. It's all relative.

So time isn't truly flowing, we just experience it that way, as we are ourselves are subject to just the same laws of physics. It would necessarily be the same for any other living creature or artificially designed object that can sense its environment, memorize what has been sensed, and run comparisons between such memorizations.

That doesn't mean that change isn't real. It's the topic for another essay, but change is likely what is fundamental. But change isn't flowing. Change just is. Similarly the past and the future just are. We just cannot access it consciously due the limitations of any computing device like our brain.

Does it mean that it's pointless to fret about the future, to make decisions, to learn from the past - if everything already is and always has been? No, not at all, we are a computing device precisely such that we bring about change. Our decisions still are part of it. And we are not that special type of change that is ever repeating, but we are part of that change that increases entropy. So we have the ability to replace whatever top card there is in the deck with the Queen of Spades.