Analytical solution to transport in Brownian ratchets via the gambler's ruin model

Explaining the Transition from Diffusion Limited to Reaction Limited Surface Assembly of Molecular Species through Spatial Variations

Surface assembly is often decomposed into two classes: diffusion and reaction limited processes. The transition between the two cases is complex because the dynamics are so different. In this article, we simulate, explain, and experimentally discuss the evolution of the spatial distribution for surface assemblies with diffusion limited and reaction limited processes. Explicitly, we demonstrate that diffusion limited and reaction limited processes show some temporal differences, but more importantly, we show that the spatial arrangements are different enough to discriminate between the two cases. Using fundamental properties, such as the diffusion constant, we calculate the evolution of the spatial profile and derive from physical, heuristic models the assembly rate for reaction and diffusion limited processes based on the individual particle's interactions with the surface. Finally, we confirm the spatial profile differences between diffusion and reaction limited cases by experimentally measuring the surface assembly between two molecules of similar size, but having different assembly routes. Unique to our description is that we have derived and simulated everything through the particle picture in place of ensemble descriptions such as the diffusion equation, and we show the equivalence between our heuristic formulas and those derived from the diffusion equation.

Quantum ratchet in two-dimensional semiconductors with Rashba spin-orbit interaction

Ratchet is a device that produces direct current of particles when driven by an unbiased force. We demonstrate a simple scattering quantum ratchet based on an asymmetrical quantum tunneling effect in two-dimensional electron gas with Rashba spin-orbit interaction (R2DEG). We consider the tunneling of electrons across a square potential barrier sandwiched by interface scattering potentials of unequal strengths on its either sides. It is found that while the intra-spin tunneling probabilities remain unchanged, the inter-spin-subband tunneling probabilities of electrons crossing the barrier in one direction is unequal to that of the opposite direction. Hence, when the system is driven by an unbiased periodic force, a directional flow of electron current is generated. The scattering quantum ratchet in R2DEG is conceptually simple and is capable of converting a.c. driving force into a rectified current without the need of additional symmetry breaking mechanism or external magnetic field.

Efficiency of rejection-free methods for dynamic Monte Carlo studies of off-lattice interacting particles

We calculate the efficiency of a rejection-free dynamic Monte Carlo method for d-dimensional off-lattice homogeneous particles interacting through a repulsive power-law potential r(-p). Theoretically we find the algorithmic efficiency in the limit of low temperatures and/or high densities is asymptotically proportional to rho(p+2)/2T(-d/2) with the particle density rho and the temperature T. Dynamic Monte Carlo simulations are performed in one-, two-, and three-dimensional systems with different powers p, and the results agree with the theoretical predictions.

Biased random walks on a lattice: exact numerical method to study the effect of alternating fields in disordered and asymmetric systems of obstacles

The migration of a particle in a system of obstacles under the action of an external field is often modeled using lattice Monte Carlo algorithms. For example, such simulation methods have been used to study the electrophoresis of charged molecules in sieving gels and the separation of particles using ratchet systems. In the case of constant fields or low-frequency alternating fields, the Monte Carlo simulation method can be mapped onto a numerical or algebraic matrix problem that can be solved exactly. In this Rapid Communication, we generalize this matrix approach to treat periodic time-dependent fields. The evolution of the spatial distribution function during a period is computed using a sequence of transfer matrices, and a steady-state closure relation allows us to calculate the exact mean velocity of the particle during a complete cycle. As an example, we examine the properties of a simple spatially asymmetric ratchet system in the presence of periodic alternating fields (symmetric and asymmetric) as well as random telegraph signals.