Exact formula for currents in strongly pumped diffusive systems

We analyze a generic model of mesoscopic machines driven by the nonadiabatic variation of external parameters. We derive a formula for the probability current; as a consequence we obtain a no-pumping theorem for cyclic processes satisfying detailed balance and demonstrate that the rectification of current requires broken spatial symmetry.Comment: 10 pages, accepted for publication in the Journal of Statistical Physic

Design and Synthesis of Nonequilibrium Systems

The active transport of ions and molecules across cell membranes is essential to creating the concentration gradients that sustain life in all living organisms, be they bacteria, fungi, plants, animals or Homo sapiens. Nature uses active transport everywhere for everything. Molecular biologists have long been attracted to the study of active transport and continue to this day to investigate and elucidate the tertiary structures of the complex motor proteins that sustain it, while physicists, interested in nonequilibrium statistical mechanics, have developed theoretical models to describe the driven ratcheting motions that are crucial to its function. The increasingly detailed understanding that contemporary science has acquired relating to active transport, however, has yet to lead to the design and construction of artificial molecular motors capable of employing ratchet-driven motions that can also perform work against concentration gradients. Mechanically interlocked molecules (MIMs) in the form of pseudo- and semirotaxanes are showing some encouraging signs in meeting these goals. This review summarizes recent progress in making artificial molecular motors that can perform work by “pumping” tetracationic rings into high-energy states. The launching pad is a bistable [2]rotaxane whose dumbbell component contains two electron-donating recognition sites, one, a tetrathiafulvalene (TTF) unit, which interacts more strongly with the ring component, cyclobis(paraquat-p-phenylene) (CBPQT4+), containing two electron-accepting bipyridinium units, than does the other 1,5-dioxynaphthalene (DNP) unit. Switching can be induced electrochemically by oxidizing the TTF unit to a TTF•+ radical cation, whereupon Coulombic repulsion takes care of moving the ring to the DNP unit. Reduction of the radical cation resets the switch. Molecular switches operate at, or close to, equilibrium. Any work done during one switching event is undone during the reset. Molecular motors, on the other hand, rely on a flux of energy, and a ratchet mechanism to make periodic changes to the potential energy surface of a system in order to move molecules uphill to higher energy states. Forging a path from molecular switches to motors involved designing a molecular pump prototype. An asymmetric dumbbell with a 2-isopropylphenyl (neutral) end and a 3,5-dimethylpyridinium (charged) end with a DNP recognition site to entice CBPQT4+ rings out of solution exhibits relative unidirectional movement of the rings with respect to the dumbbell. Redox chemistry does the trick. During the oxidative cycle, the rings enter the dumbbell by passing over the neutral end onto the recognition site; in the reduction cycle, much of the recognition is lost and the rings find their way back into solution by leaving the dumbbell from the charged end. This on-one-end, off-the-other process can be repeated over and over again using light as the energy source in the presence of a photosensitizer and a compound that shuttles electrons back and forth. Although this prototype demonstrates ratchet-driven translational motion, no work is done. A ring enters the dumbbell from one end and leaves from the other end. Another deficiency of the prototype is the fact that, although the recognition site is muted on reduction, it retains some attraction for the ring. What if the recognition site was attractive initially and then became repulsive? This question was answered by turning to radical chemistry and employing the known stabilization behavior of a bipyridinium radical cation and the bisradical dication, generated on reduction of the CBPQT4+ ring, to pluck rings out of solution and thread them over the charged end of the pump portion of a semidumbbell. On subsequent oxidation, the pump is primed and the rings pass through a one-way door, given a little thermal energy, onto a collecting-chain where they find themselves accumulating where they would rather not be present. In this manner, an artificial molecular pump mimics the pumping machinery commonplace in biological systems. Looking beyond this state-of-the-art artificial molecular pump, we discuss, from a theoretical standpoint, the measures that would need to be taken in order to render its operation autonomous

Current reversal with type-I intermittency in deterministic inertia ratchets

The intermittency is investigated when the current reversal occurs in a deterministic inertia ratchet system. To determine which type the intermittency belongs to, we obtain the return map of velocities of particle using stroboscopic recording, and numerically calculate the distribution of average laminar length ${}$. The distribution follows the scaling law of ${} \propto {\epsilon}^{-1/2}$, the characteristic relation of type-I intermittency.Comment: 4 pages, 7 figure

Mirror symmetry breaking through an internal degree of freedom leading to directional motion

We analyze here the minimal conditions for directional motion (net flow in phase space) of a molecular motor placed on a mirror-symmetric environment and driven by a center-symmetric and time-periodic force field. The complete characterization of the deterministic limit of the dissipative dynamics of several realizations of this minimal model, reveals a complex structure in the phase diagram in parameter space, with intertwined regions of pinning (closed orbits) and directional motion. This demonstrates that the mirror-symmetry breaking which is needed for directional motion to occur, can operate through an internal degree of freedom coupled to the translational one.Comment: Accepted for publication in Phys. Rev.

Noise-assisted classical adiabatic pumping in a symmetric periodic potential

We consider a classical overdamped Brownian particle moving in a symmetric periodic potential. We show that a net particle flow can be produced by adiabatically changing two external periodic potentials with a spatial and a temporal phase difference. The classical pumped current is found to be independent of the friction and to vanish both in the limit of low and high temperature. Below a critical temperature, adiabatic pumping appears to be more efficient than transport due to a constant external force.Comment: six pages, 3 figure

Effective rate equations for the over-damped motion in fluctuating potentials

We discuss physical and mathematical aspects of the over-damped motion of a Brownian particle in fluctuating potentials. It is shown that such a system can be described quantitatively by fluctuating rates if the potential fluctuations are slow compared to relaxation within the minima of the potential, and if the position of the minima does not fluctuate. Effective rates can be calculated; they describe the long-time dynamics of the system. Furthermore, we show the existence of a stationary solution of the Fokker-Planck equation that describes the motion within the fluctuating potential under some general conditions. We also show that a stationary solution of the rate equations with fluctuating rates exists.Comment: 18 pages, 2 figures, standard LaTeX2

Deterministic ratchets: route to diffusive transport

The rectification efficiency of an underdamped ratchet operated in the adiabatic regime increases according to a scaling current-amplitude curve as the damping constant approaches a critical threshold; below threshold the rectified signal becomes extremely irregular and eventually its time average drops to zero. Periodic (locked) and diffusive (fully chaotic) trajectories coexist on fine tuning the amplitude of the input signal. The transition from regular to chaotic transport in noiseless ratchets is studied numerically.Comment: 9 pages, 5 figures, to be published in Phys. Rev.

Breaking of general rotational symmetries by multi-dimensional classical ratchets

We demonstrate that a particle driven by a set of spatially uncorrelated, independent colored noise forces in a bounded, multidimensional potential exhibits rotations that are independent of the initial conditions. We calculate the particle currents in terms of the noise statistics and the potential asymmetries by deriving an n-dimensional Fokker-Planck equation in the small correlation time limit. We analyze a variety of flow patterns for various potential structures, generating various combinations of laminar and rotational flows.Comment: Accepted, Physical Review

Effect of Chaotic Noise on Multistable Systems

In a recent letter [Phys.Rev.Lett. {\bf 30}, 3269 (1995), chao-dyn/9510011], we reported that a macroscopic chaotic determinism emerges in a multistable system: the unidirectional motion of a dissipative particle subject to an apparently symmetric chaotic noise occurs even if the particle is in a spatially symmetric potential. In this paper, we study the global dynamics of a dissipative particle by investigating the barrier crossing probability of the particle between two basins of the multistable potential. We derive analytically an expression of the barrier crossing probability of the particle subject to a chaotic noise generated by a general piecewise linear map. We also show that the obtained analytical barrier crossing probability is applicable to a chaotic noise generated not only by a piecewise linear map with a uniform invariant density but also by a non-piecewise linear map with non-uniform invariant density. We claim, from the viewpoint of the noise induced motion in a multistable system, that chaotic noise is a first realization of the effect of {\em dynamical asymmetry} of general noise which induces the symmetry breaking dynamics.Comment: 14 pages, 9 figures, to appear in Phys.Rev.

Segregation of granular binary mixtures by a ratchet mechanism

We report on a segregation scheme for granular binary mixtures, where the segregation is performed by a ratchet mechanism realized by a vertically shaken asymmetric sawtooth-shaped base in a quasi-two-dimensional box. We have studied this system by computer simulations and found that most binary mixtures can be segregated using an appropriately chosen ratchet, even when the particles in the two components have the same size, and differ only in their normal restitution coefficient or friction coefficient. These results suggest that the components of otherwise non-segregating granular mixtures may be separated using our method.Comment: revtex, 4 pages, 4 figures, submitte