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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Entropy as a costant of motion

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**