Performance optimization in two-dimensional Brownian rotary ratchet models.

With a model for two-dimensional (2D) Brownian rotary ratchets being capable of producing a net torque under athermal random forces, its optimization for mean angular momentum (L), mean angular velocity (ω), and efficiency (η) is considered. In the model, supposing that such a small ratchet system is placed in a thermal bath, the motion of the rotor in the stator is described by the Langevin dynamics of a particle in a 2D ratchet potential, which consists of a static and a time-dependent interaction between rotor and stator; for the latter, we examine a force [randomly directed dc field (RDDF)] for which only the direction is instantaneously updated in a sequence of events in a Poisson process. Because of the chirality of the static part of the potential, it is found that the RDDF causes net rotation while coupling with the thermal fluctuations. Then, to maximize the efficiency of the power consumption of the net rotation, we consider optimizing the static part of the ratchet potential. A crucial point is that the proposed form of ratchet potential enables us to capture the essential feature of 2D ratchet potentials with two closed curves and allows us to systematically construct an optimization strategy. In this paper, we show a method for maximizing L, ω, and η, its outcome in 2D two-tooth ratchet systems, and a direction of optimization for a three-tooth ratchet system.

Ordering dynamics of one-dimensional Bloch wall system and domain size distribution function.

The dynamics of domain size distribution in the ordering process for a one-dimensional classical anisotropic XY model is studied with a reduced equation of motion for the assembly of domain sizes. The system possesses two types of the domain wall structures, the Néel or Bloch walls, depending on the strength of magnetization anisotropy. In the Néel wall situation the neighboring walls interact with one another in only an attractive way. On the other hand, in the Bloch one, these walls interact in either an attractive or a repulsive way depending on their chiralities. For the Bloch wall situation, we found that the domain size distribution is characterized by solitonlike translational motion with a function form h(y-y(t)) and a characteristic domain size y(t) for the domain size y. This is in contrast to that in the Néel wall situation, which can be described as a scaling-type distribution function g[y/l(t)]/l(t), as was obtained by Nagai and Kawasaki, with a certain scaling length l(t). We discuss why such a solitonlike motion appears instead of the scaling-type distribution function, show a proof for the absence of the scaling-type distribution, a qualitative estimation for the distribution function in the Bloch wall situation, and an analysis for the realization probability of a specified twistness.

Magnetic walls in the anisotropic XY-spin system in an oscillating magnetic field.

Wall structures associated with dynamic phase transitions in the anisotropic XY -spin system in a temporally oscillating magnetic field h cos (Omegat) in a one-dimensional system are analyzed by using the time-dependent Ginzburg-Landau model. It is numerically confirmed that there exist two types of magnetic walls, i.e., the Néel and Bloch walls, and is found that the transition between the two walls can occur for changing h or Omega . The phase diagram for the stable regions of each wall is obtained by both numerical and analytical methods. Furthermore, the critical behavior of the modulus of the Bloch wall around the Néel-Bloch transition point is studied, and it is found that the transition can be either continuous or discontinuous with respect to h, depending on Omega .

Dynamic phase transitions in the anisotropic XY spin system in an oscillating magnetic field.

The Ginzburg-Landau model for the anisotropic XY spin system in an oscillating magnetic field below the critical temperature T(c), psi;(r,t)=(T(c)-T)psi-/psi/(2)psi+gammapsi(*)+ nabla (2)psi+h cos(Omegat) is both theoretically and numerically studied. Here psi is the complex order parameter and gamma stands for the real anisotropy parameter. It is numerically shown that the spatially uniform system shows various characteristic oscillations (dynamical phases), depending on the amplitude h and the frequency Omega of the external field. As the control parameter, either h or Omega, is changed, there exist dynamical phase transitions (DPT), separating them. By making use of the mode expansion analysis, we obtain the phase diagrams, which turn out to be in a qualitative agreement with the numerically obtained ones. By carrying out the Landau expansion, the reduced equations of motion near the DPT are derived. Furthermore, taking into account the spatial variation of order parameters, we will derive the analytic expressions for domain walls, which are represented by the Néel and Bloch type walls, depending on the difference of the coexistence of phases.

Robust unidirectional rotation in three-tooth Brownian rotary ratchet systems.

We apply a simple Brownian ratchet model to an artificial molecular rotary system mounted in a biological membrane, in which the rotor always maintains unidirectional rotation in response to a linearly polarized weak ac field. Because the rotor and stator compose a ratchet system, we describe the motion of the rotor tip with the Langevin equation for a particle in a two-dimensional three-tooth ratchet potential of threefold symmetry. Unidirectional rotation can be induced under the field and optimized by stochastic resonance, wherein the mean angular momentum (MAM) of the rotor exhibits a bell-shaped curve for the noise strength. We obtain analytical expressions for the MAM and power loss from the corresponding Fokker-Planck equation, via a Markov transition model for coarse-grained states (six-state model). The MAM expression reveals a significant effect depending on the chirality of the ratchet potential: in achiral cases, the MAM approximately vanishes with respect to the polarization angle φ of the field; in chiral cases, the MAM does not crucially depend on φ, but depends on the direction of the ratchet; i.e., the parity of the unidirectional rotation is inherent in the ratchet structure. This feature is useful for artificial rotary systems to maintain robust unidirectional rotation independent of the mounting condition.

Frequency adaptation in controlled stochastic resonance utilizing delayed feedback method: two-pole approximation for response function.

Stochastic resonance (SR) enhanced by time-delayed feedback control is studied. The system in the absence of control is described by a Langevin equation for a bistable system, and possesses a usual SR response. The control with the feedback loop, the delay time of which equals to one-half of the period (2π/Ω) of the input signal, gives rise to a noise-induced oscillatory switching cycle between two states in the output time series, while its average frequency is just smaller than Ω in a small noise regime. As the noise intensity D approaches an appropriate level, the noise constructively works to adapt the frequency of the switching cycle to Ω, and this changes the dynamics into a state wherein the phase of the output signal is entrained to that of the input signal from its phase slipped state. The behavior is characterized by power loss of the external signal or response function. This paper deals with the response function based on a dichotomic model. A method of delay-coordinate series expansion, which reduces a non-Markovian transition probability flux to a series of memory fluxes on a discrete delay-coordinate system, is proposed. Its primitive implementation suggests that the method can be a potential tool for a systematic analysis of SR phenomenon with delayed feedback loop. We show that a D-dependent behavior of poles of a finite Laplace transform of the response function qualitatively characterizes the structure of the power loss, and we also show analytical results for the correlation function and the power spectral density.

Design of two-tooth unidirectional rotary-ratchet molecular machines driven by linearly polarized ac fields.

An artificial molecular rotor system mounted in a biological membrane, which can unidirectionally rotate in response to weak pumping from a linearly polarized ac field, is modeled. The dynamics of the rotor unit are described by the Langevin equation for a particle in a two-dimensional bistable potential with a two-tooth ratchet structure. This model reveals effects due to the two-dimensionality of the ratchet and the polarization of the applied field. First, we demonstrate that a unidirectional rotation appears with stochastic resonance exhibiting a bell-shaped peak for noise intensity in the mean angular momentum (MAM) of the rotor. An analytical expression for the MAM, (L), is obtained on the basis of a four-state Markov approximation. Second, a significant effect due to torsional nonlinearity (representing the ratchet-like structure) in the potential geometry is quantified: in the absence of torsion, the MAM depends on the polarization angle φ of the applied field as (L) sin(2φ), whereas in the presence of torsion, an additional bias appears in the MAM as (L)(bias + sin(2φ)). It is found that this effect can be used to make the rotor system robustly maintain rotation in a single direction independent of the mounting conditions. Possible designs for an artificial molecular rotor system using the torsion effect are discussed.