Why, according to thermodynamics, is the following scenario impossible? You are sitting in a warm room. Suddenly, on its own all the heat energy in the room collects in one corner, leaving the rest of the room so cold that you are fozen to an icicle.

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Another way of looking at this is to consider what would happen once the process starts. First, for a little bit of heat to flow to the corner, heat would need to flow even though the temperature was the same.  The rate of heat flow is related to the temperature difference.  No difference means no heat flow.  Secondly once the corner got a little hotter than the rest of the room, if it could, then all the rest of the heat would have to flow from the colder room to hotter corner even though heat flows from hot to cold and not the other way.  The second law of thermodynamics is an expression of the rate and direction of heat flow I described above.

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