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Entropy as a costant of motion

  1. Dec 2, 2005 #1
    Entropy is supposed to increase in a system but is indeed a costant, when one considers the equation of motion generated by an hamiltonian function ( or operator); sometimes a lighter request is made, that there exist a monotone strictly increasing function of time but a theorem by poincaré forbid this possibility.
    the proof of the statement that entripy is a costant of motion is very simple;
    one has only to consider the gibbs entropy for a quantum ensamble <S>= Sp(-r ln r) (where r is the density matrix) and remembers the von neuman equation
    dr/dt = i [r, H]. The time total derivative of the entropy is then
    -Sp(i [r,H] + i [r, H] ln r)= 0 + i Sp(r H lnr - H r ln r) = i Sp (H lnr r - H r lnr )=
    i Sp (H r lnr - H r lnr)= 0. (* i have used the symmetric and cyclic propriety of the trace).

    The question is: where statistical mechanics come from, when entropy
    is a costant of motion?
  2. jcsd
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