Shannon entropy, Fisher information and uncertainty relations for log-periodic oscillators

• We calculate the time-dependent Shannon entropy of three log-periodic oscillators.
• We calculate the time-dependent Fisher information (FI) for the same oscillators.
• Relations among the (FI) and the Stam and Cramer–Rao inequalities are discussed.

We calculate the time-dependent Shannon (SxSx and SpSp) entropy and Fisher (FxFx and FpFp) information of three log-periodic oscillators. We obtain a general expression for Sx,pSx,p and Fx,pFx,p in the state n=0n=0 in terms of ρρ, a c-number quantity satisfying a nonlinear differential equation. For two out of three oscillators Sx,pSx,p and Fx,pFx,p depend on time, but Sx+SpSx+Sp and FxFpFxFp do not. The other oscillator behaves as the time-independent harmonic oscillator where Sx,pSx,p and Fx,pFx,p are all constants. Relations among the Fisher information and the Stam and Cramer–Rao inequalities are also discussed.