# Entropy as a costant of motion

1. Dec 2, 2005

### davide

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?