This book is about Brownian ratchets, devices that rectify microscopic fluctuations. It was written with two purposes in mind. First, to introduce new readers to the field. To this is devoted the first part of the book, which treats, with as few technicalities as possible, the general ideas as well as set the broad range of physical systems where the concept of ratchets may be of relevance. The second purpose of the book is to address researchers already active in the field, by providing them with a single source for a general and detailed theoretical analysis of ratchets in the different regimes (classical vs. quantum, stochastic vs. deterministic) as well as with a coverage of experimental realizations with different physical systems, each highlighting a specific unique feature of ratchets.

No attempt has been made to provide the reader with an exhaustive list of references. There are excellent review papers, cited in the book, that already serve this purpose. We limit our reference list to the material we used the most, and to some historically important references which, despite their age, are still irreplaceable.

The book is divided into three parts. Part I covers, at an informal level, much of the material that will then be examined in depth in the rest of the book. As such, it can be used by the reader with a broad interest in the topic, but no specific need to go any deeper. This part starts with a historical overview of the Brownian ratchets. Definitions, basic ideas and fundamental concepts are introduced: from the very definition of Brownian ratchet, to the fundamental limitations imposed by the second law of thermodynamics. The historically important Brillouin paradox and Feynman ratchets are reviewed, and used to highlight the main ideas behind the concept of ratchets. This part continues by examining three fundamental models of ratchet devices, used to illustrate the general operation of a Brownian ratchet, and how directed motion is related to two fundamental requirements: out-of-equilibrium settings and symmetry breaking. This part concludes with an overview of the general relevance of the concept of ratchets. It is shown that the concept applies to very diverse fields, from biological systems to game theory.

Part II offers a detailed theoretical anlysis of ratchets. The relationship between symmetry and transport is examined in detail, thus formalizing a necessary condition, symmetry breaking, for the generation of directed motion.

The first part of this book covered the main principles of ratchets, with particular attention to the requirement of out-of-equilibrium and symmetry-breaking settings. This was illustrated with specific reference to some fundamental models of ratchets.

This chapter aims to put on solid theoretical grounds the material presented so far, and to extend it along several directions. The relationship between symmetry and transport will be discussed in detail, also covering the case of higher dimensions and quasiperiodic drivings. These two cases are characterized by distinguishing features: new rectification mechanisms appear in systems with more than one dimension, and the use of quasiperiodic drivings results in an effective change in the number of degrees of freedom, thus changing the nature of the ratchet system.

Two different approaches to the symmetry analysis are covered in this chapter. In the first approach, the symmetries of the dynamical equation of motion are considered. This allows the identification of the symmetries which prevent the generation of directed motion, and thus the symmetry-breaking requirements to produce a ratchet current. In the second approach, the average current is considered as a function of the driving force. Symmetries directly relate the current to the driving. Such an approach allows not only to identify the symmetries which control the suppression of directed motion, but also to determine the generic functional form of the current as a function of the driving parameters. Not relying on the details of the dynamical equation of motion, the results of this approach are fairly independent of the specific details of the system at play, the type of non-linearity, or whether the system is classical or quantum.

Finally, the theoretical framework is further extended by considering non- Gaussian noise – and in particular Lévy noise – and feedback ratchets.

Brownian motion

Microscopic particles suspended in a liquid or in a gas undergo a random motion that is usually termed Brownian motion, in honor of Robert Brown who first – in 1827 – experimentally studied the zigzag motion of pollen grains suspended in water. Theoretically, the system can be described by a stochastic equation for the instantaneous velocity v(t) = dx/dt of the particle of mass m:

Here γ is a constant – the friction coefficient – which describes the damping of the particle's motion as a result of the interaction with the fluid in which it is immersed.

This book is written with the graduate student in mind. I had in mind to write a text that would introduce my students to the basic ideas and concepts behind many-body physics. At the same time, I felt very strongly that I would like to share my excitement about this field, for without feeling the thrill of entering uncharted territory I do not think one has the motivation to learn and to make the passage from learning to research.

Traditionally, as physicists we ask “what are the microscopic laws of nature?”, often proceeding with the brash certainty that, once revealed, these laws will have such profound beauty and symmetry that the properties of the universe at large will be self-evident. This basic philosophy can be traced from the earliest atomistic philosophy of Democritus to the most modern quests to unify quantum mechanics and gravity.

The dreams and aspirations of many-body physics interwine the reductionist approach with a complementary philosophy: that of emergent phenomena. In this view, fundamentally new kinds of phenomena emerge within complex assemblies of particles which cannot be anticipated from an a priori knowledge of the microscopic laws of nature. Many-body physics aspires to synthesize, from the microscopic laws, new principles that govern the macroscopic realm, asking:

This is a comparatively modern and far less familiar scientific philosophy. Charles Darwin was perhaps the first to seek an understanding of emergent laws of nature. Following in his footsteps, Ludwig Boltzmann and James Clerk Maxwell were among the first physicists to appreciate the need to understand how emergent principles are linked to microscopic physics. From Boltzmann's biography [1], we learn that he was strongly influenced and inspired by Charles Darwin. In more modern times, a strong advocate of this philosophy has been Philip W. Anderson, who first introduced the phrase “emergent phenomenon” into physics. In an influential article entitled “More is different,” written in 1967 [2], he captured the philosophy of emergence, writing:

On October 19, 2014, as I finished this book, a group of physicists convened at the University of Illinois Urbana-Champaign, home of BCS theory [1], to mark 60 years of progress in strongly correlated electron systems (SCES) and to celebrate the 90th birthday of David Pines. This occasion provided a great opportunity for us to reflect on the dizzying progress of the past 60 years of discovery in many-body physics: a period that spans from the Bohm–Pines plasmon theory of metals [2–4], the discovery of Landau Fermi-liquid theory [5], and BCS theory [1], to the modern era of topological order, strange quantum-critical metals, and high-temperature-superconductivity. This book has only scratched the surface of this period, leaving out some key aspects of the field that I perhaps will go into in a future edition. Yet still, almost everything in this book was unknown 60 years ago. The pioneers of early many-body physics in the 1950s simply could not have imagined the revolution of discovery that has since taken place. David Pines recalls thinking, as a graduate student in the early 1950s, that the physicists 20 years earlier had had all the luck. Yet Pines' thesis work with Bohm marked a beginning of 60 years of tremendous discovery.

You, the reader, might be tempted to think as David Pines did. However, at the “60 years of SCES” meeting in Urbana, many expressed remarkable optimism that the revolution is decidedly unfinished, and that the next 60 years has much in store. Like earthquakes, big scientific discoveries are unpredictable and come at varying intervals. Today, in the early twenty-first century, a large number of unsolved mysteries and problems in condensed matter physics provide the tremors that suggest that there is much yet to discover. I thought it would be interesting, if only as a historical record, to list a few of the challenges we all face:

1 Wanted: a broader understanding of the classes of emergent order in condensed Matter. Between the many decades of complexity that separate simple elements from primitive life surely lie many layers of emergent material behavior. The possibility of new forms of order, as challenging and surprising as superconductivity, cannot be ruled out.

This chapter continues our discussion of superconductivity, considering the effects of repulsive interactions and the physics of anisotropic Cooper pairing. According to an apocryphal story, Landau is reputed to have said that “nobody has yet repealed Coulomb's law” [1]. In the BCS theory of superconductors, there is no explicit appearance of the the repulsive Coulomb interaction between paired electrons. How then do real-world superconductors produce electron pairs, despite the presence of the strong interaction between them?

This chapter we will examine two routes by which Nature is able to satisfy the Coulomb interaction. In conventional superconductors, the attraction between electrons develops because the positive screening charge created by the ionic lattice around an electron remains in place long after the electron has moved away. This process that gives rise to a short-time repulsion between electrons is followed by a retarded attraction which drives s-wave pairing. However, since the 1980s physicists have been increasingly fascinated by anisotropic superconductors. In these systems, it is the repulsive interaction between the fermions that drives the pairing. The mechanism by which this takes place is through the development of nodes in the pair wavefunction – often by forming a higher angular momentum Cooper pair. The two classic examples of this physics are the p-wave pairs of superfluid 3He and the d-wave pairs of cuprate high-temperature superconductors.

In truth, the physics community is still trying to understand the full interplay of superconductivity and the Coulomb force. The discovery of room-temperature superconductivity will surely involve finding a quantum material where strong correlations within the electron fluid lead to a large reduction in the sum total of kinetic and Coulomb energy.

BCS theory with momentum-dependent coupling

We now illustrate these two different ways in which superconductors “overcome” the Coulomb interaction, by returning to the more generalized version of BCS theory with a momentum-dependent interaction:

Notice how we have deliberately included a + sign in front of the interaction HI, to emphasize its predominantly repulsive character.