A complementary relationship about anomalous diffusions under memory or memoryless damping

We report a complementary relationship about anomalous diffusions in memory and memoryless damping processes, namely, the product of the asymptotic position variances of a force-free particle subjected respectively to the same inertial and external noise is found to obey σi2(t)σe2(t)∼t2. We consider the noise spectrum having non-Ohmic power-law form, while such noise acts as an internal fluctuation source to drive a free particle, the motion can show subdiffusion or superdiffusion, then if the noise is used as an external one to induce the particle moving in a constant damping environment, in contrast, the particle should emerge superdiffusion or subdiffusion, respectively. The root is due to a fact that the former exhibits the fluctuation–dissipation theorem, when the noise correlation function has fast (slow) decaying behavior, the corresponding effective friction becomes weak (strong), so that the diffusion increases (decreases), however, the latter is only controlled by the correlation feature of noise. Two limit situations for the diffusing particle in elastic media, ballistic diffusion and local diffusion, are uniformly revealed. The presented result provides an approach to explore the noise and memory characteristics in generalized Brownian motion.