The balls have to touch thermometer. Otherwise the thermometer observes nothing. Deriving the exact formula of this interaction is complex but the intuition makes sense.
Interaction with the thermometer at a certain energy level produces a macrostate at Temperature T that is high probability. You see this macrostate throughout your lifetime. But nothing precludes the low probability macrostate (where all balls are at another corner of the box) from occuring. It's just that microstate has such a low probability of occuring you never see it. When nothing touches the thermometer, that temperature reading is the absolute zero macrostate.
>Apparently you say that the entropy increases because the macrostate changes. But the macrostate is undefined until the balls hit the thermometer. But the entropy has to increase for the balls to move. It’s all a bit confusing, don’t you think?
No. The macrostate is never undefined. I defined it as the measurement on that thermometer. Whatever you read on that thermometer (let's assume mercury levels move instantaneously and not be pedantic) then THAT is the current macrostate. I chose this set of macrostates, thus I pick that definition. There is no notion of undefined, the thermometer always HAS a reading.
>Won’t the balls collide with the thermometer even if the entropy doesn’t change? It’s their movement that will take them there.
They can't collide with the thermometer if they're on the other side of the box.
>Consider your state - or any other state, for that matter - at t=0 and the state at t=1ps where the balls have advanced slightly. Is the macrostate different? How? Is the entropy different? How?
Macrostate doesn't change if they're still far away from the thermometer. Still absolute zero. Only the microstate changed. The absolute zero macrostate includes all configurations of particles that will cause the thermometer to read 0. The probability of this macrostate occuring is quite low.
Entropy in this state is low. It increases as more and more particles interact with the thermometer.
The most probable series of events is that particles on the left corner of the box will begin to spread more evenly around the box. More and more particles will begin interacting with the thermometer causing the temperature to rise until it reaches some equilibrium. What can occur is all particles can by sheer luck suddenly concentrate to another corner of the box, but this is a low probability event.
To bring it back around to the main point. All of this is independent of your knowledge of the microstate. Your knowledge does not effect the outcome.
