Linear response and stochastic resonance of subdiffusive bistable fractional Fokker–Planck systems

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

“…Similar expression for the spectrum has been obtained in Ref. [37]. For small frequencies ω ≪ λ 1/α 1 we can neglect the frequency when it appears together with the eigenvalues λ.…”

“…Such an expression for the autocorrelation function has been obtained in Ref. [37]. According to Wiener-Khintchine relations, the power spectral density is related to the autocorrelation function:…”

“…In the homogeneous systems the power spectral density (PSD) of the signals generated by time-subordinated Langevin equations has power-law dependency S(f ) ∼ f α−1 on the frequency as f → 0. [37]. Since 0 < α < 1, the power-law exponent 1 − α is smaller than 1.…”

Subdiffusion in a time-dependent force field

“…(17), let us carry out the Laplace transformation on Eqs. (14) and (17). As a result, we will obtain (18) …”

“…(1) by the location of the frac tional derivative operator. In some papers, this equa tion was used ad hoc to solve specific problems [16,17]. However, an analysis based on the CTRW model led the authors of [6,7] to conclude that it has no physical meaning.…”