REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009 Artificial Brownian motors: Controlling transport on the nanoscale Peter Hänggi* Institut für Physik, Universität Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany and Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Singapore Fabio Marchesoni† Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy and School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea ͑Published 30 March 2009͒ In systems possessing spatial or dynamical symmetry breaking, Brownian motion combined with unbiased external input signals, deterministic and random alike, can assist directed motion of particles at submicron scales. In such cases, one speaks of “Brownian motors.” In this review the constructive role of Brownian motion is exemplified for various physical and technological setups, which are inspired by the cellular molecular machinery: the working principles and characteristics of stylized devices are discussed to show how fluctuations, either thermal or extrinsic, can be used to control diffusive particle transport. Recent experimental demonstrations of this concept are surveyed with particular attention to transport in artificial, i.e., nonbiological, nanopores, lithographic tracks, and optical traps, where single-particle currents were first measured. Much emphasis is given to two- and three-dimensional devices containing many interacting particles of one or more species; for this class of artificial motors, noise rectification results also from the interplay of particle Brownian motion and geometric constraints. Recently, selective control and optimization of the transport of interacting colloidal particles and magnetic vortices have been successfully achieved, thus leading to the new generation of microfluidic and superconducting devices presented here. The field has recently been enriched with impressive experimental achievements in building artificial Brownian motor devices that even operate within the quantum domain by harvesting quantum Brownian motion. Sundry akin topics include activities aimed at noise-assisted shuttling other degrees of freedom such as charge, spin, or even heat and the assembly of chemical synthetic molecular motors. This review ends with a perspective for future pathways and potential new applications. DOI: 10.1103/RevModPhys.81.387 PACS number͑s͒: 05.60.Ϫk, 47.61.Ϫk, 81.07.Ϫb, 85.25.Ϫj CONTENTS 2. Pulsated ratchets 399 3. Correlation ratchets 400 4. Further asymmetry effects 400 I. Introduction 388 a. Current offsets 400 A. Artificial nanodevices 389 b. Asymmetry induced mixing 401 B. Brownian motors 389 E. Efficiency and control issues 402 II. Single-Particle Transport 390 1. Optimization 402 A. Symmetric substrates 391 2. Vibrated ratchets 403 1. dc drive 391 III. Transport in Nanopores 404 2. ac drive 392 A. Ion pumps 404 3. Diffusion peak 393 B. Artificial nanopores 405 4. Single-file diffusion 393 C. Chain translocation 405 B. Rectification of asymmetric processes 393 D. Toward a next generation of mass rectifiers 406 C. Nonlinear mechanisms 394 1. Zeolites 406 1. Harmonic mixing 394 2. Nanotubes 407 2. Gating 395 IV. Cold Atoms in Optical Lattices 408 3. Noise induced transport 395 A. Biharmonic driving 408 a. Noise mixing 395 B. Multifrequency driving 409 b. Noise recycling 396 C. More cold atom devices 410 D. Brownian motors 396 V. Collective Transport 410 1. Rocked ratchets 397 A. Asymmetric 1D geometries 411 1. Boundary effects 411 2. Asymmetric patterns of symmetric traps 411 *[email protected] B. 2D lattices of asymmetric traps 412 †[email protected] C. Binary mixtures 413 0034-6861/2009/81͑1͒/387͑56͒ 387 ©2009 The American Physical Society 388 Peter Hänggi and Fabio Marchesoni: Artificial Brownian motors: Controlling … VI. Microfluidics 414 ing any spontaneous or natural process. Notably, within A. Transporting colloids 415 the whole virtual factory of all natural processes, the B. Transporting condensed phases 416 first law takes on the role of an account clerk, keeping C. Granular flows 418 track of all energy changes, while the second law takes VII. Superconducting Devices 419 on the role of the director, determining the direction and A. Fluxon channels 420 action of all processes. B. Fluxon rectification in 2D arrays 422 The fathers of thermodynamics developed their laws 1. Symmetric arrays of asymmetric traps 422 having in mind macroscopic systems that they could de- 2. Asymmetric arrays of symmetric traps 423 scribe in terms of state ͑i.e., average͒ quantities such as C. Anisotropic fluxon rectifiers 423 pressure and temperature, a reasonable assumption VIII. Quantum Devices 424 when dealing with the monster steam engines of Victo- A. Quantum dissipative Brownian transport 424 rian industry. A typical protein, however, is a few na- B. Josephson Brownian motors 425 nometers in size and consists of just a few tens of thou- 1. Classical regime 426 sands of atoms. As a consequence, the movements of a 2. Quantum regime 426 protein engine are swamped by fluctuations resulting C. Quantum dot ratchets 426 from the Brownian motion of its surroundings, which D. Coherent quantum ratchets 428 causes the energy of any of its parts to fluctuate continu- 1. Quantum ratchets from molecular wires 429 ally in units of kT, with k denoting the Boltzmann con- 2. Hamiltonian quantum ratchet for cold stant and T the temperature. The effects of such energy atoms 429 fluctuations were demonstrated by Yanagida and co- IX. Sundry Topics 430 workers ͑Nishiyama et al., 2001, 2002͒, who observed ki- A. Pumping of charge, spin, and heat 430 nesin molecules climbing the cytoskeletal track in a jud- B. Synthetic molecular motors and machines 430 dering motion made up of random hesitations, jumps, X. Concluding Remarks 432 and even backward steps. Similar results have been re- Acknowledgments 432 ported for a host of protein engines. The key question in References 433 modern thermodynamics is therefore how far energy fluctuations drive microengines and nanoengines beyond the limits of the macroscopic laws. I. INTRODUCTION A revealing experiment was performed by Busta- mante et al. ͑2005͒, who first stretched a single RNA Over the past two decades advances in microscopy molecule by optically tugging at a tiny plastic bead, at- and microscale control have allowed scientists and engi- tached to one end, and then released the bead, to study neers to delve into the workings of biological matter. the effect of random energy fluctuations on the molecule One century after Kelvin’s death, today’s researchers recovery process. By repeating many identical stretching aim to explain how the engines of life actually work by cycles, they found that the molecule “relaxation path” stretching thermodynamics beyond its 19th-century lim- ͑ ͒ was different every time. In fact, the bead was drawing its Haw, 2007 . useful energy from the thermal motion of the suspension Carnot realized that all engines transform energy fluid and transforming it into motion. However, by aver- from one form into another with a maximum possible aging over increasingly longer bead trajectories, that is, efficiency that does not depend on the technology devel- approaching a macroscopic situation, Bustamante et al. oped or on the fuel utilized, but rather on fundamental ͑2005͒ were able to reconcile their findings with the sec- quantities such as heat and temperature. Kelvin and ond law. These results led to the conclusion that the Clausius came up with two “rules of the engine,” later straightforward extension of the second law to micro- known as the first two laws of thermodynamics. The first scopic systems is unwarranted; individual small systems law states that energy cannot be destroyed or created do evolve under inherently nonequilibrium conditions. but only transformed; the second law sets fundamental However, a decade ago Jarzynski ͑1997͒ showed that limitations on the practical achievements of energy the demarcation line between equilibrium and nonequi- transformation. Just as the first law was centered on the ͑ ͒ librium processes is not always as clear-cut as we used to notion of energy from the Greek for “work capability” , think. Imagining a microscopic single-molecule process, the second law revolved around the new concept of en- Jarzynski evaluated not the simple average of the tropy ͑a term coined by Clausius, also from Greek, for ͑ ͒ ͒ 1 change of the random work of the underlying per- “change capability” . When expressed in such terms, turbed nanosystem, as it was pulled away from equilib- the second law states that entropy cannot decrease dur- rium according to an arbitrary protocol of forcing, but rather the average of the exponential of that tailored 1 nonequilibrium work. This quantity turned out to be the Rudolf Julius Emanuel Clausius throughout his life allegedly preferred the German word Verwandlungswert rather than en- same as for an adiabatically slow version of the same tropy, although his colleagues suggested that he choose a name process, and, most remarkably, equals the exponential of for his new thermodynamic quantity S ͑his chosen symbol for the system’s equilibrium free energy change. This result, labeling entropy, possibly in honor of Sadi Carnot͒ that also experimentally demonstrated by Bustamante et al. sounded as close as possible to the word energy. ͑2005͒, came as a great surprise because it meant that Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009