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.