Physics Reports 361 (2002) 57–265 www.elsevier.com/locate/physrep Brownian motors:noisy transport far from equilibrium Peter Reimann Institut fur Physik, Universitat Augsburg, Universitatsstr. 1, 86135 Augsburg, Germany Received August 2001; Editor:I : Procaccia Contents 1. Introduction 59 4.6. Biological applications:molecular 1.1. Outline and scope 59 pumps and motors 130 1.2. Historical landmarks 60 5. Tilting ratchets 133 1.3. Organization of the paper 61 5.1. Model 133 2. Basic concepts and phenomena 63 5.2. Adiabatic approximation 133 2.1. Smoluchowski–Feynman ratchet 63 5.3. Fast tilting 135 2.2. Fokker–Planck equation 67 5.4. Comparison with pulsating ratchets 136 2.3. Particle current 68 5.5. Fluctuating force ratchets 137 2.4. Solution and discussion 69 5.6. Photovoltaic e5ects 143 2.5. Tilted Smoluchowski–Feynman ratchet 72 5.7. Rocking ratchets 144 2.6. Temperature ratchet and ratchet e5ect 75 5.8. In=uence of inertia and Hamiltonian 2.7. Mechanochemical coupling 79 ratchets 148 2.8. Curie’s principle 80 5.9. Two-dimensional systems and entropic 2.9. Brillouin’s paradox 81 ratchets 150 2.10. Asymptotic analysis 82 5.10. Rocking ratchets in SQUIDs 152 2.11. Current inversions 83 5.11. Giant enhancement of di5usion 154 3. General framework 86 5.12. Asymmetrically tilting ratchets 156 3.1. Working model 86 6. Sundry extensions 159 3.2. Symmetry 90 6.1. Seebeck ratchets 160 3.3. Main ratchet types 91 6.2. Feynman ratchets 162 3.4. Physical basis 93 6.3. Temperature ratchets 164 3.5. Supersymmetry 100 6.4. Inhomogeneous, pulsating, and memory 3.6. Tailoring current inversions 107 friction 165 3.7. Linear response and high temperature 6.5. Ratchet models with an internal degree limit 108 of freedom 168 3.8. Activated barrier crossing limit 109 6.6. Drift ratchet 169 4. Pulsating ratchets 111 6.7. Spatially discrete models and 4.1. Fast and slow pulsating limits 111 Parrondo’s game 173 4.2. On–o5 ratchets 113 6.8. In=uence of disorder 175 4.3. Fluctuating potential ratchets 115 6.9. ECciency 176 4.4. Traveling potential ratchets 121 7. Molecular motors 178 4.5. Hybrids and further generalizations 127 7.1. Biological setup 179 0370-1573/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII:S 0370-1573(01)00081-3 58 P. Reimann / Physics Reports 361 (2002) 57–265 7.2. Basic modeling-steps 181 9.2. Genuine collective e5ects 223 7.3. Simpliÿed stochastic model 184 10. Conclusions 232 7.4. Collective one-head models 190 Acknowledgements 234 7.5. Coordinated two-head model 199 Appendix A. Supplementary material 7.6. Further models for a single motor regarding Section 2:1:1 234 enzyme 200 A.1. Gaussian white noise 234 7.7. Summary and discussion 203 A.2. Fluctuation–dissipation relation 235 8. Quantum ratchets 205 A.3. Einstein relation 236 8.1. Model 206 A.4. Dimensionless units and overdamped 8.2. Adiabatically tilting quantum ratchet 209 dynamics 237 8.3. Beyond the adiabatic limit 215 Appendix B. Alternative derivation of the 8.4. Experimental quantum ratchet systems 217 Fokker–Planck equation 239 9. Collective e5ects 220 Appendix C. Perturbation analysis 239 9.1. Coupled ratchets 222 References 241 Abstract Transport phenomena in spatially periodic systems far from thermal equilibrium are considered. The main emphasis is put on directed transport in so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the presence of thermal noise and some prototypical perturbation that drives the system out of equilibrium without introducing a priori an obvious bias into one or the other direction of motion. Symmetry conditions for the appearance (or not) of directed current, its inversion upon variation of certain parameters, and quantitative theoretical predictions for speciÿc models are reviewed as well as a wide variety of experimental realizations and biological applications, especially the modeling of molecular motors. Extensions include quantum me- chanical and collective e5ects, Hamiltonian ratchets, the in=uence of spatial disorder, and di5usive transport. c 2001 Elsevier Science B.V. All rights reserved. PACS: 5.40.−a; 5.60.−k; 87.16.Nn Keywords: Ratchet; Transport; Nonequilibrium process; Brownian motion; Molecular motor; Periodic potential; Symmetry breaking P. Reimann / Physics Reports 361 (2002) 57–265 59 1. Introduction 1.1. Outline and scope The subject of the present review are transport phenomena in spatially periodic systems out of thermal equilibrium. While the main emphasis is put on directed transport, also some aspects of di1usive transport will be addressed. We furthermore focus mostly on small-scale systems for which thermal noise plays a non-negligible or even dominating role. Physically, the thermal noise has its origin in the thermal environment of the actual system of interest. As an unavoidable consequence, the system dynamics is then always subjected to dissipative e5ects as well. Apart from transients, directed transport in a spatially periodic system in contact with a single dissipation- and noise-generating thermal heat bath is ruled out by the second law of thermody- namics. The system has therefore to be driven away from thermal equilibrium by an additional deterministic or stochastic perturbation. Out of the inÿnitely many options, we will mainly focus on either a periodicdriving or a restricted selection of stochastic processes of prototypal simplicity. In the most interesting case, these perturbations are furthermore unbiased, i.e. the time-, space-, and ensemble-averaged forces which they entail are required to vanish. Physically, they may be either externally imposed (e.g. by the experimentalist) or of system-intrinsic origin, e.g. due to a second thermal heat reservoir at a di5erent temperature or a non-thermal bath. Besides the breaking of thermal equilibrium, a further indispensable requirement for directed trans- port in spatially periodic systems is clearly the breaking of the spatial inversion symmetry. There are essentially three di5erent ways to do this, and we will speak of a Brownian motor, or equiva- lently, a ratchet system whenever a single one or a combination of them is realized:First, the spatial inversion symmetry of the periodic system itself may be broken intrinsically, that is, already in the absence of the above mentioned non-equilibrium perturbations. This is the most common situation and typically involves some kind of periodic and asymmetric, so-called ratchet potential. A second option is that the non-equilibrium perturbations, notwithstanding the requirement that they must be unbiased, bring about a spatial asymmetry of the dynamics. A third possibility arises as a collective e5ect in coupled, perfectly symmetric non-equilibrium systems, namely in the form of spontaneous symmetry breaking. As it turns out, these two conditions (breaking of thermal equilibrium and of spatial inversion symmetry) are generically suCcient for the occurrence of the so-called ratchet e1ect, i.e. the emer- gence of directed transport in a spatially periodic system. Elucidating this basic phenomenon in all its facets is the central theme of our present review. We will mainly focus on two basic classes of ratchet systems, which may be roughly characterized as follows (for a more detailed discussion see Section 3.3):The ÿrst class, called pulsating ratchets, are those for which the above-mentioned periodic or stochastic non-equilibrium perturbation gives rise to a time-dependent variation of the potential shape without a5ecting its spatial periodicity. The second class, called tilting ratchets, are those for which these non-equilibrium perturbations act as an additive driving force, which is unbiased on the average. In full generality, also combinations of pulsating and tilting ratchet schemes are possible, but they exhibit hardly any fundamentally new basic features (see Section 3.4.2). Even within those two classes, the possibilities of breaking thermal equilibrium and symmetry in a ratchet system are still numerous and in many cases, predicting the actual direction of the transport is already far from obvious, not to speak of its quantitative value. 60 P. Reimann / Physics Reports 361 (2002) 57–265 In particular, while the occurrence of a ratchet e5ect is the rule, exceptions with zero current are still possible. For instance, such a non-generic situation may be created by ÿne-tuning of some parameter. Usually, the direction of transport then exhibits a change of sign upon variation of this parameter, called current inversions. Another type of exception can be traced back to symmetry reasons with the characteristic signature of zero current without ÿne-tuning of parameters. The understanding and control of such exceptional cases is clearly another issue of considerable theoretical and practical interest that we will discuss in detail (especially in Sections 3.5 and 3.6). 1.2. Historical landmarks Progress in the ÿeld of Brownian motors has evolved through contributions from rather di5erent directions and re-discoveries of the same basic principles in di5erent contexts have been made repeatedly. Moreover, the organization of the much more detailed subsequent chapters will not always admit it to keep the proper historical order. For these reasons, a brief historical tour d’horizon seems worthwhile at this place. At the same time, this gives a ÿrst =avor of the wide variety of applications of Brownian motor concepts. Though certain aspects of the ratchet e5ect are contained implicitly already in the works of Archimedes, Seebeck, Maxwell, Curie, and others, Smoluchowski’s Gedankenexperiment from 1912 [1] regarding the prima facie quite astonishing absence of directed transport in spatially asymmetric systems in contact with a single heat bath, may be considered as the ÿrst seizable major contribution (discussed in detail in Section 2.1). The next important step forward represents Feynman’s famous recapitulation and extension [2] to the case of two thermal heat baths at di5erent temperatures (see Section 6.2).