Suspended in a vacuum you have a red cube at 100 degrees and a blue cube of equal size and shape at 0 degrees.
You want to warm up the blue cube and cool down the red cube. The naive thing to do is touch them together, letting thermal redistribution eventually bring both to 50 degrees.
Lets say that you can slice the cubes up into pieces and touch small pieces together. Is it possible to get the blue cube hotter than 50 degrees? How hot can it get? (In the end you have to put each cube back together and wait for each cube to resolve to its average temperature.)
You may assume that objects touched together will resolve to a temperature that is the weighted average of their individual temperatures. You may also assume that there is no heat lost to the environment.
This may be breaking the rules but I don't know if heat is quantized.
Suppose the red cube is 100 degrees because its mass is 1g and it contains 100 quantum calories. The blue has 1g but 0 calories. Think of these calories as being like electrons in a metal. They are indivisible and buzz around inside the cube. Temperature is measured in calories/gram.
When two pieces of cubes are brought together they reach equilibrium by adding their masses and calories together.
When they are split apart, if the masses are not divisible, then one extra calorie has to 'decide' which piece to go to. The choice is random, but proportional to the two sizes if they are unequal.
So now split each cube into 100 pieces and combine them in pairs, one of each color. We get pairs together that each contains 1 calorie and 1 gram. When we split them apart each blue has a 50-50 chance of getting the calorie.
So the 100 blue pieces are like 100 binomial trials, each with p=0.5 The expected temperature may be 50cal/gram but the probability if this is only about 8% There's a 46% chance of being above 50.
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Posted by Jer
on 2019-04-05 11:49:27 |
A trick solution?
