Universality of anomalous diffusion in extremely disordered systems

Abstract

The universal time-dependence of the mean-square displacement for motion in a random energy landscape with equal minima is evaluated analytically and numerically in the percolation path approximation (PPA), which was recently shown by extensive computer simulations in two and three dimensions [Dyre and Schr~ler, cond-mat/9601052] to be more accurate than the standard effective medium approximation (EMA). According to the PPA the universal mean-square displacement in dimensionless units as function of time varies as 1 / In 2 ( t-~ ) for t ~ 0. This implies a quite different short-time behavior than predicted by the EMA, where the universal mean-square displacement varies as 1/ln(t -t) at short times [Dyre and Jacobsen, Phys. Rev. E 52 (1995) 2429].

Mechanism of Charge Transport along Columnar Stacks of a Triphenylene Dimer

“…To reach this goal, the explicit form of the transition rate between neighboring localization sites must be specified. This can be done by invoking different theoretical models, − schematically illustrated in Figure . These can differ in the shape postulated for the 1-D potential energy profile and/or in the particular mechanism of the transition between sites.…”

“…Theoretically the incoherent jumplike motion is treated as the random walk of a particle in a rough potential energy profile (for a review, see refs −13 and references therein). Several analytical approaches − and the Monte Carlo simulation technique enable one to study the 1-D motion for different model profiles (a smooth potential locally perturbed by random traps and/or random barriers) and for various mechanisms of site-to-site transitions (thermally activated jumps or tunneling). − In all cases the net result of 1-D energy disorder has been shown to be a sublinear dependence of the mean-square displacement on time. Consequently, the incoherent 1-D transport becomes non-Gaussian, i.e., dispersive.…”

Single molecule charge transport: from a quantum mechanical to a classical description

“…where f(m) describes the initial distribution of the charge, a is the distance between neighboring monomer units, (n À m)a is the distance between the orbitals localized on monomer units n and m, and c n (t,m) is the coefficient of the orbital on monomer unit n at time t for the case of the hole wavefunction being initially localized on monomer unit m. According to the work of Kubo, the frequency-dependent (one-dimensional) mobility of charge carriers is given by [126][127][128] …”

“…An implicit convergence factor exp(Àot) (lim o -0) is understood in the integral. 127,128 For normal Gaussian diffusion the mean squared displacement of charge carriers moving along an infinitely long one-dimensional chain increases linearly with time: View Online where D is the diffusion constant. In this special case, the mobility is frequency independent, and eqn (27) reduces to the Einstein relation…”

“…According to the work of Kubo, the frequency-dependent (one-dimensional) mobility of charge carriers is given by [126][127][128]…”