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For many years, I leant strongly towards some form of non-analytic emergence principle to explain eg some of the wackier hypotheses of statistical mechanics that are simply not seen in classical mechanics, such as 'all the molecules of gas are in the corner of the box' type paradoxes.

I'm tending to drift away from this as I see these paradoxes being due to a) crude over-simplification of statistical mechanics by some rather than a real fault in the method, and b) an increasing awareness that quantum phenomena significantly modify the assumptions of randomicity by the creation of 'forbidden zones' due to destructive interference.

In particular, maximal quantum entanglement of the gas particles is often equated with the state of thermal equilibrium. This suggests that the system has shared memories of its recent particle collision history embedded within the affected fields, and if the Transactional Interpretation of QM has traction, then this memory extends at least a little into the future too. 

The upshot is that the molecules do not act independently of the bulk gas. And the bulk gas behaves accordingly. Quantum thermodynamics is a currently developing field that is attempting to address this. Perhaps the most appropriate methodology could be called 'holistic reductionism'? Is that an oxymoron?

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