The winding angle distribution of an ordinary random walk

Self-repelling bipedal exploration process

A self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters β_{d} and β_{h}, so that the former controls the distance (d) between the legs positions, and the latter controls the statistics of self-crossing of the traversed paths. The probability measure for local movements is supposed to be the one for the "true self-avoiding walk" multiplied by a factor exponentially decaying with d. After a transient behavior for short times, a variety of behaviors have been observed for large times depending on the value of β_{d} and β_{h}. Our statistical analysis reveals that the system undergoes a crossover between two (small and large β_{d}) regimes identified in large times (t). In the small β_{d} regime, the random walkers (identified by the position of the legs of the spider) remain on average in a fixed nonzero distance in the large time limit, whereas in the second regime (large β_{d}), the absorbing force between the walkers dominates the other stochastic forces. In the latter regime, d decays in a power-law fashion with the logarithm of time. When the system is mapped to a growth process (represented by a height field which is identified by the number of visits for each point), the roughness and the average height show different behaviors in two regimes, i.e., they show a power law with respect to t in the first regime and logt in the second regime. The fractal dimension of the random walker traces and the winding angle are shown to consistently undergo a similar crossover.

Winding angle and maximum winding angle of the two-dimensional random walk

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

Winding angle and maximum winding angle of the two-dimensional random walk

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10 m steps, for m = 2, 3, 4, 5, 6, 7.

Topological energy storage of work generated by nanomotors

Most macroscopic machines rely on wheels and gears. Yet, rigid gears are entirely impractical on the nano-scale. Here we propose a more useful method to couple any rotary engine to any other mechanical elements on the nano- and micro-scale. We argue that a rotary molecular motor attached to an entangled polymer energy storage unit, which together form what we call the "tanglotron" device, is a viable concept that can be experimentally implemented. We derive the torque-entanglement relationship for a tanglotron (its "equation of state") and show that it can be understood by simple statistical mechanics arguments. We find that a typical entanglement at low packing density costs around 6kT. In the high entanglement regime, the free energy diverges logarithmically close to a maximal geometric packing density. We outline several promising applications of the tanglotron idea and conclude that the transmission, storage and back-conversion of topological entanglement energy are not only physically feasible but also practical for a number of reasons.

Vortex entanglement in disordered superconductors

Vortex entanglement and pinning in multiply connected disordered superconductors are studied. It is shown that the winding of vortices around repulsive obstacles is greatly enhanced by quenched columnar disorder and suppressed by point disorder, compared to the clean case. This leads to an additional contribution to the effective pinning force acting on vortices, which, unlike the conventional mechanisms of pinning, grows with temperature.