Spectral content of a single trajectory
speaker: Gleb Oshanin (Univ. Paris VI, Jussieu)abstract: Time-dependent processes are often analyzed using the power spectral density (PSD), which is calculated by making an appropriate Fourier transform of individual trajectories and finding the ensemble-averaged value. In some cases, however, one cannot create a statistical sample of a big enough size and hence it is of a great conceptual and practical importance to understand to which extent a relevant information can be gained from the PSD of a single trajectory, S(f,T). Here we focus on the behaviour of S(f,T), which is a random, realization-dependent variable, for a broad family of anomalous diffusion processes – the so-called fractional Brownian motion with Hurst-index H, and derive exactly its probability-density-function. We show that S(f,T) is proportional - up to a random numerical-factor with universal distribution which we determine - to the ensemble-averaged PSD. For subdiffusion (H<12) we find that S(f,T) ∼ Af{2H+1} with random-amplitude A. In sharp contrast, for superdiffusion (H>12) S(f,T) ∼ B T{2H-1}f2 with random amplitude B. Remarkably, for H>12 the PSD exhibits the same frequency-dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably for H>12 the PSD is ageing and it explicitly depends on the observation time T. Our predictions for both subdiffusion and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels, and by extensive simulations.
timetable:
Wed 19 Sep, 10:15 - 11:00, Aula Dini
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