Proceedings with Extended Abstracts (single PDF file) - Radio ...

Anomalous spectraTalkner [4] studied the phase relaxation functions for several random walk models. Welist in Table 1, several of those with a range of relaxation from algebraic to faster thanexponential. The model parameters associated with each curve can be chosen to fit thecorresponding spectra (evaluated analytically or numerically) to the experimental spectra.Random walk model α (t)Condition/Form2Processes with independent increments − 2k 2B tFractional Brownian motionLevy processesContinuous random walks with long rests and short jumpsContinuous random walks with short rests and arbitrary jumpse σeete2σ = γ + iβ2 2 γ−σk B t 0 < γ < 2γ γβ−( σk) t−α− f ( k ) tB0 < γ < 20 < γβ < 2ConclusionWe have investigated how the stochastic motion of one scatterer can be related to thescattered field spectra. We have shown that if a successful stochastic description ofparticle movements can be found, the backscatter spectra can be related to thecorresponding phase relaxation function of the stochastic or diffusion model. It wasshown that such connection could be established by invoking the fluctuation- dissipationtheorem and using the phase relaxation function in place of the generalized susceptance.While all these depends on finding a valid random walk model, we believe that once suchmodeling is successful, corresponding diffusional relaxation functions will have directand analytical correspondence to the radiowave backscatter spectra from turbulentmediums. We have presented several cases where the spectral density could display awide range of anomalous behavior from Lorentzian to Gaussian and sometimes slowerthan Lorentzian. While the rich spectral possibilities one may obtain from differentrandom walk models look promising, the most fundamental question remaining is thevalidity of the random walk model. It has to physically make sense. Nevertheless, webelieve that the theoretical connection between the spectral density of the scattered fieldfrom steady-state high-order statistical flows and the phase relaxation function establishesa sound framework for interpreting coherent backscatter spectra from refractive indexfluctuations in a scattering medium.References[1] R. Balescu, Anomalous transport in turbulent plasmas and continuous time random walks,Phys. Rev. E 51, 4807–4822, 1995[2] Herbert B. Callen and Richard F. Greene, On a theorem of irreversible thermodynamics, Phys.Rev. 86, 702–710, 1952[3] L. D. Landau, E. M. Lifshitz, Statistical physics 3 rd edition Part 1,Nauka, Moskow, 377,1976[4] P. Talkner, Anomalous diffusion and phase relaxation, Phys. Rev. E, 64,061101-1, 2001113