Von Neumann Entropy time derivative(evolution)

[Astrocyte]

Homework Statement

When S= Tr(ρ lnρ), what is "dS/dt" ? rho is a density matrix.

Relevant Equations

S=Tr( ρlnρ), Tr(ABC)=Tr(CAB), Von Neumman equation.

I'm not sure about my proof. So please check my step. I used log as a natural log(ln).

Specially, I'm not sure about "d/dt=dρ/dt d/dρ=i/ħ [ρ, H] d/dρ" in the second line. and matrix can differentiate the other matrix? (d/dρ (dρ lnρ))

 

[jbsteele]

Let [A,B]=C, then lnAB=lnC.1. lnρ = ln exp(lnρ)2. d/dt=dρ/dt d/dρ=i/ħ [ρ, H] d/dρ3. d/dt(lnρ) = d/dρ (dρ lnρ) 4. d/dt(lnρ) = i/ħ d/dρ [ρ, H] lnρ5. d/dt(lnρ) = i/ħ [ρ, H] 6. d/dt(lnρ) = i/ħ [ρ,H] lnρ7. d/dt(lnρ) = i/ħ ln[ρ, H] 8. lnρ = ln e^(i/ħ ln[ρ, H]) 9. ρ = e^(i/ħ ln[ρ, H]) Is this proof correct?