Let's look at the two possible timelines here: At some point a particle pops up into existence in B, increasing B's pressure. Some time later, a particle from

Let's look at the two possible timelines here: At some point a particle pops up into existence in B, increasing B's pressure. Some time later, a particle from A disappears, decreasing A's pressure. This pressure differential would cause flow from B to A through X. Alternatively, a particle randomly disappears from B, decreasing B's pressure. Some time later, a particle appears in A, increasing A's pressure. This would cause flow from A to B through X. The more particles that cross from A to B via Y, increasing B's pressure, the more likely it is for particles to pass from B to A via Y, keeping these two effects equal. These two pressure differentials would cancel out for large numbers of particles. You've essentially just hooked up the system to itself, but at a future point in time. Unless the overall pressure of the system is changing, there is nothing to induce the flow. If you are increasing the pressure, valve Y is essentially just a standard one-way valve.