The answer is perhaps somewhat technical, but hopefully this shows the argument why heat flows from negative to positive temperature.
Entropy is essentially the (logarithm of) the number of states a system can be in, without changing the macroscopic observables. These states have all the same probability. The second law of thermodynamics is then simply a consequence of the number of allowed transitions of the system. And temperature is the change of the number of states if energy is added to the system. That heat flows from the lower (positive) energy to the higher is then a consequence of calculating the probabilities, as is the observation that heat flows always from a negative to a positive temperature system.
Perhaps a example will make this somewhat clearer:
Think of a chain of 20 capacitors, each can be charged or uncharged and I call the 10 left capacitors my left subsystem, and the 10 on the right the right subsystem. Initially there is 1 charged capacitor in the left and 3 in the right. Then the probability that after one charge moves there are two charged capacitors in each subsystem is 9/16 ( since 9 of the uncharged capacitors are on the left). In this case the temperature is positive in both subsystems. (The number of possible configurations of the left 10 capacitors is higher for two charged ones ( 5*4) than for one charged ( 5).
The negative temperature case would in this analogy be, if in one subsystem there are more than 5 charged capacitors. Then there are more charged than uncharged capacitors, the number of allowed states would decrease if I add additional charge. ( I can distribute 9 charges in 10 different ways, but 10 charges just in one way.) But nothing happens about the argument of transition probabilities. If there are 7 charged capacitors on the left and 2 on the right, then after moving a charge the probability that there are 6 and 3 charged ones is 8/11 ( 8 of the 11 uncharged capacitors are on the right).
In the case of the boiling water and the negative (close to zero) system it is the same, it is about counting possible states. And since in one the number of states increases if I add energy ( the temperature is positive) and in the other the number of states increases if I remove energy ( the temperature is negative), both have a preference for transferring energy from the negative energy system to the positive one.
Entropy is essentially the (logarithm of) the number of states a system can be in, without changing the macroscopic observables. These states have all the same probability. The second law of thermodynamics is then simply a consequence of the number of allowed transitions of the system. And temperature is the change of the number of states if energy is added to the system. That heat flows from the lower (positive) energy to the higher is then a consequence of calculating the probabilities, as is the observation that heat flows always from a negative to a positive temperature system.
Perhaps a example will make this somewhat clearer: Think of a chain of 20 capacitors, each can be charged or uncharged and I call the 10 left capacitors my left subsystem, and the 10 on the right the right subsystem. Initially there is 1 charged capacitor in the left and 3 in the right. Then the probability that after one charge moves there are two charged capacitors in each subsystem is 9/16 ( since 9 of the uncharged capacitors are on the left). In this case the temperature is positive in both subsystems. (The number of possible configurations of the left 10 capacitors is higher for two charged ones ( 5*4) than for one charged ( 5).
The negative temperature case would in this analogy be, if in one subsystem there are more than 5 charged capacitors. Then there are more charged than uncharged capacitors, the number of allowed states would decrease if I add additional charge. ( I can distribute 9 charges in 10 different ways, but 10 charges just in one way.) But nothing happens about the argument of transition probabilities. If there are 7 charged capacitors on the left and 2 on the right, then after moving a charge the probability that there are 6 and 3 charged ones is 8/11 ( 8 of the 11 uncharged capacitors are on the right).
In the case of the boiling water and the negative (close to zero) system it is the same, it is about counting possible states. And since in one the number of states increases if I add energy ( the temperature is positive) and in the other the number of states increases if I remove energy ( the temperature is negative), both have a preference for transferring energy from the negative energy system to the positive one.