We consider Lévy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Lévy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis's q-statistics is studied. 05.40.+j,05.60.+w,02.50.Ey,05.70.Ln
We consider the combined effects of a power law Lévy step distribution characterized by the step index f and a power law waiting time distribution characterized by the time index g on the long time behavior of a random walker. The main point of our analysis is a formulation in terms of coupled Langevin equations which allows in a natural way for the inclusion of external force fields. In the anomalous case for f < 2 and g < 1 the dynamic exponent z locks onto the ratio f /g. Drawing on recent results on Lévy flights in the presence of a random force field we also find that this result is independent of the presence of weak quenched disorder. For d below the critical dimension d c = 2f − 2 the disorder is relevant, corresponding to a non trivial fixed point for the force correlation function.
We consider Lévy flights characterized by the step index f in a quenched random force field. By means of a dynamic renormalization group analysis we find that the dynamic exponent z for f < 2 locks onto f , independent of dimension and independent of the presence of weak quenched disorder. The critical dimension, however, depends on the step index f for f < 2 and is given by d c = 2f − 2. For d < d c the disorder is relevant, corresponding to a non trivial fixed point for the force correlation function.Irrespective of the spatial dimension d, ordinary Brownian motion traces out a manifold of fractal dimension d F = 2 [17]. In the presence of a quenched disordered force field in d dimensions the Brownian walk is unaffected for d > d F , i.e., for d larger than the critical dimension d c = d F the walk is transparent and the dynamic exponent z locks onto the value 2 for the pure case. Below the critical dimension d c = 2 the long time characteristics of the walk is changed to subdiffusive behaviour with z > 2 [18,19,20]. In d = 1, r 2 (t) ∝ [log t] 4 , independent of the strength of the quenched disorder [21].Lévy flights constitute an interesting generalization of ordinary Brownian walks.
We study the dynamics of denaturation bubbles in double-stranded DNA. Demonstrating that the associated Fokker-Planck equation is equivalent to a Coulomb problem, we derive expressions for the bubble survival distribution W(t). Below Tm, W(t) is associated with the continuum of scattering states of the repulsive Coulomb potential. At Tm, the Coulomb potential vanishes and W(t) assumes a power-law tail with nontrivial dynamic exponents: the critical exponent of the entropy loss factor may cause a finite mean lifetime. Above Tm (attractive potential), the long-time dynamics is controlled by the lowest bound state. Correlations and finite size effects are discussed.
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