Feedback-controlled diffusion: From self-trapping to true self-avoiding walks

We study the asymptotic behavior of a Brownian particle under the influence of a dynamical feedback by numerical simulations and analytical considerations. The feedback is controlled by a memory coupling of strength λ. Whereas a negative memory strength yields a true self avoiding walk, a positive memory leads to a self-trapping of the particle. The localization is manifested by a constant mean square displacement in the long time limit which appears after an initial diffusive regime. The probability distribution function of the trapping distance shows an exponential decay. The numerical simulations are compared with an analytical modeling.