We are led to believe that the "arrow of time" is determined by the direction in which entropy increases. Entropy is a measure of disorder, but disorder is very much in the eye of the beholder. Is there not some experiment that reduces to its most basic elements the reason that we can remember the past but not the future. Why is it that events in the past seem to leave a clearer imprint on the present than events in the future? Why is it trivial to store a bit of information about a past event but not about a future event? Maybe there's something fundamental about the behavior of a bistable memory latch?
I'm looking for the very simplest possible thought experiment that captures the essense of this phenomenon.
[Edit] This topic and the entropy arguments keep bothering me so I'll add a bonus in the hope of getting more attention. I see there's also a Wikipedia article and this includes a reference to a toy model about fleas and cats but again it reduces to a low entropy initial condition. There is also a statement that the past is what we can remember and the future is what we can influence. Is this some kind of perverse symmetry that we fail to recognise? - but I'm looking for physics not philosophy!
Brownian motion is about as simple as it gets. When I measure the position of a particle, it tells me something about its position, say, 1 second earlier. But it seems to tell the the same amount of information about where it will be 1 second later. So it's a simple example of a irreversible dissipative system that has memory - but it's completely time-symmetric.
Another observation concerns Markov chains. If I make the process go backwards instead of forwards, the labels change. But this seems to be because we label the arrows with conditional probabilities. If we were to label the arrows with absolute probabilities the labels would be unchanged when we reversed time. So this is just a curiosity.
[Edit #2] Maybe the most trivial example is thermal inertia. The temperature of a large thermal mass will lag the temperature of its environment. The current temperature of the mass has information about the temperature of the environment in the recent past, but it says nothing about the future environment. I suppose this simply says heat flows from hot to cold - which comes back to entropy.
[Edit #3] The idea that thermal inertia provides a fundamental example of the asymmetry of time seems to crumble once it's examined closely. The temperature of a thermal mass (in a gas) strictly does not give information about the recent temperature of the gas, it gives information about the average kinetic energy of the atoms that recently hit the mass. It gives exactly the same amount of information about the kinetic energy of the atoms bouncing off the object in the near future.
[Edit #4] Maxwell's demon seems to have some relevance here. The demon can switch on or off a large energy barrier separating the two compartments. The usual focus is on the information needed to switch the barrier at the correct instants. But the sudden change in the energy landscape is of interest in its own right. A trivial situation would be a single atom bouncing randomly between two states when a barrier is suddenly introduced. This captures one bit of information and stores it as long as the barrier is maintained. This seems like the essence of memory. The introduction of the barrier is time asymmetric. But, the removal of the barrier is the exact equivalent in backwards time. Also, the use of the words introduction and removal are problematic here because they define the direction of the time asymmetry.
particle in box with sliding partition
This picture shows the sort of answer I am after. This is an open irreversible system. Here we have a single particle in a box and everything is in thermal equilibrium. There is a partition that is positioned to be either outside the box or to be inside the box (dividing the box into two separate volumes). The partition changes position once. We can consider the time axis to be divided into two portions. In one portion of time, the particle can move between the two sides of the box. In the other portion of time, the particle is always in either the left or right side of the box. Note that, so far, we have been careful to say nothing to imply which way time is flowing.
In one interpretation, the partition is intially open (time flows from left to right). So the system can be viewed as forming a permanent memory of where the particle was around the time the position of the partition changes.
In the other interpretation, the partition is intially closed (time flows from right to left). So the system can be viewed as having permanent intention to influence where the particle will be around the time the position of the partition changes.
There are still the usual Maxwell demon problems here too. The sliding partition can be arranged to not require energy to move between the two positions, but to reliably open (or close) the latch that releases (or captures) the partition will require an energy source significantly larger than thermal energy. But the problem of figuring which half of the partitioned box the particle is in seems to go away. The particle isn't going anywhere, so we can make multiple noisy measurements until we're convinced of which half it's in.
I realize this doesn't answer the question, but it perhaps provides some food for thought.