Dynamics of Annealed Systems under External Fields: CTRW and the Fractional Fokker-Planck Equations

We consider the linear response of a system modelled by continuous-time random walks (CTRW) to an external field pulse of rectangular shape. We calculate the corresponding response function explicitely and show that it exhibits aging, i.e. that it is not translationally invariant in the time-domain. This result differs from that of systems which behave according to fractional Fokker-Planck equations

Quantum walk approach to search on fractal structures

We study continuous-time quantum walks mimicking the quantum search based on Grover's procedure. This allows us to consider structures, that is, databases, with arbitrary topological arrangements of their entries. We show that the topological structure of the database plays a crucial role by analyzing, both analytically and numerically, the transition from the ground to the first excited state of the Hamiltonian associated with different (fractal) structures. Additionally, we use the probability of successfully finding a specific target as another indicator of the importance of the topological structure.Comment: 15 pages, 14 figure

Dynamical Scaling Behavior of Percolation Clusters in Scale-free Networks

In this work we investigate the spectra of Laplacian matrices that determine many dynamic properties of scale-free networks below and at the percolation threshold. We use a replica formalism to develop analytically, based on an integral equation, a systematic way to determine the ensemble averaged eigenvalue spectrum for a general type of tree-like networks. Close to the percolation threshold we find characteristic scaling functions for the density of states rho(lambda) of scale-free networks. rho(lambda) shows characteristic power laws rho(lambda) ~ lambda^alpha_1 or rho(lambda) ~ lambda^alpha_2 for small lambda, where alpha_1 holds below and alpha_2 at the percolation threshold. In the range where the spectra are accessible from a numerical diagonalization procedure the two methods lead to very similar results.Comment: 9 pages, 6 figure

Does strange kinetics imply unusual thermodynamics?

We introduce a fractional Fokker-Planck equation (FFPE) for Levy flights in the presence of an external field. The equation is derived within the framework of the subordination of random processes which leads to Levy flights. It is shown that the coexistence of anomalous transport and a potential displays a regular exponential relaxation towards the Boltzmann equilibrium distribution. The properties of the Levy-flight FFPE derived here are compared with earlier findings for subdiffusive FFPE. The latter is characterized by a non-exponential Mittag-Leffler relaxation to the Boltzmann distribution. In both cases, which describe strange kinetics, the Boltzmann equilibrium is reached and modifications of the Boltzmann thermodynamics are not required

Space Fortress Game Training and Executive Control in Older Adults: A Pilot Intervention

We investigated the feasibility of using the Space Fortress (SF) game, a complex video game originally developed to study complex skill acquisition in young adults, to improve executive control processes in cognitively healthy older adults. The study protocol consisted of 36 one-hour game play sessions over 3 months with cognitive evaluations before and after, and a follow-up evaluation at 6 months. Sixty participants were randomized to one of three conditions: Emphasis Change (EC)--elders were instructed to concentrate on playing the entire game but place particular emphasis on a specific aspect of game play in each particular game; Active Control (AC)--game play with standard instructions; Passive Control (PC)--evaluation sessions without game play. Primary outcome measures were obtained from five tasks, presumably tapping executive control processes. A total of 54 older adults completed the study protocol. One measure of executive control, WAIS-III letter-number sequencing, showed improvement in performance from pre- to post-evaluations in the EC condition, but not in the other two conditions. These initial findings are modest but encouraging. Future SF interventions need to carefully consider increasing the duration and or the intensity of the intervention by providing at-home game training, reducing the motor demands of the game, and selecting appropriate outcome measures

Lie theory and separation of variables. 5. The equations iUt + Uxx = 0 and iUt + Uxx −c/x2 U = 0

A detailed study of the group of symmetries of the time-dependent free particle Schrödinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well-defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x2 is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrödinger equation, several of which are new

Precise Asymptotics for a Random Walker's Maximum

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[M_n] of the walk up to n steps behaves asymptotically for large n as, E[M_n]/\sigma=\sqrt{2n/\pi}+ \gamma +O(n^{-1/2}), where \sigma^2 is the variance of the step lengths. While the leading \sqrt{n} behavior is universal and easy to derive, the leading correction term turns out to be a nontrivial constant \gamma. For the special case of uniform distribution over [-1,1], Coffmann et. al. recently computed \gamma=-0.516068...by exactly enumerating a lengthy double series. Here we present a closed exact formula for \gamma valid for arbitrary symmetric distributions. We also demonstrate how \gamma appears in the thermodynamic limit as the leading behavior of the difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker after n steps. An application of these results to the equilibrium thermodynamics of a Rouse polymer chain is pointed out. We also generalize our results to L\'evy walks.Comment: new references added, typos corrected, published versio

Efficiency of quantum and classical transport on graphs

We propose a measure to quantify the efficiency of classical and quantum mechanical transport processes on graphs. The measure only depends on the density of states (DOS), which contains all the necessary information about the graph. For some given (continuous) DOS, the measure shows a power law behavior, where the exponent for the quantum transport is twice the exponent of its classical counterpart. For small-world networks, however, the measure shows rather a stretched exponential law but still the quantum transport outperforms the classical one. Some finite tree-graphs have a few highly degenerate eigenvalues, such that, on the other hand, on them the classical transport may be more efficient than the quantum one.Comment: 5 pages, 3 figure