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Quantropy | | John C. Baez
; Blake S. Pollard
; | | Date: |
4 Nov 2013 | | Abstract: | There is a well-known analogy between statistical and quantum mechanics. In
statistical mechanics, Boltzmann realized that the probability for a system in
thermal equilibrium to occupy a given state is proportional to exp(-E/kT) where
E is the energy of that state. In quantum mechanics, Feynman realized that the
amplitude for a system to undergo a given history is proportional to exp(-S/i
hbar) where S is the action of that history. In statistical mechanics we can
recover Boltzmann’s formula by maximizing entropy subject to a constraint on
the expected energy. This raises the question: what is the quantum mechanical
analogue of entropy? We give a formula for this quantity, and for lack of a
better name we call it "quantropy". We recover Feynman’s formula from assuming
that histories have complex amplitudes, that these amplitudes sum to one, and
that the amplitudes give a stationary point of quantropy subject to a
constraint on the expected action. Alternatively, we can assume the amplitudes
sum to one and that they give a stationary point of a quantity we call "free
action", which is analogous to free energy in statistical mechanics. We compute
the quantropy, expected action and free action for a free particle, and draw
some conclusions from the results. | | Source: | arXiv, 1311.0813 | | Services: | Forum | Review | PDF | Favorites | |
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