Introduction

Curiously, and interestingly, there is an almost dialectical relationship between equilibrium and non-equilibrium systems. Almost all systems occurring in Nature are open systems coupled to external reservoirs such that the exchange of energy, particles, or other conserved quantities between the system and the reservoirs leads to currents through the system and which may drive the temporal evolution of the system. Such currents may be realised differently, for example as an electric current, or as a flow of particles, or heat conduction, or even as spin transport, to name just a few. On the microscopic level, such non-equilibrium effects manifest themselves as a breaking of detailed balance, which means that between pairs of microstates there is in general a non-vanishing flow of probability. Then one may also say that an intrinsic microscopic irreversibility will produce a macroscopic non-equilibrium behaviour.

In equilibrium systems, one considers situations where such currents have decayed away to zero such that internal probability currents are absent and the thermodynamic state variables of the system are fixed by the corresponding properties of the reservoirs. For the theoretical description of such systems, it has been understood for more than a century, mainly through the efforts of Boltzmann and Gibbs, how to formulate a general statistical description of equilibrium systems in terms of the probabilities of the micro-states and how to derive from this the thermodynamic behaviour at thermal equilibrium. Unfortunately, such a canonical and generally valid formalism does not yet exist for general non-equilibrium systems, in spite of recent efforts. In practise, for each non-equilibrium system afresh, first one has to find the probability distribution of the micro-states or some equivalent information. A common way of doing this is to write down either a Fokker-Planck equation or, more generally, a master equation for the probability distribution or alternatively a stochastic Langevin equation for some averages of physical observables. Such equations are in general no longer fully microscopic, but should rather be seen as some kind of coarse-grained description on a mesoscopic level, large enough with respect to the truly microscopic level which involves the details of the individual motion of atoms and molecules and yet still small compared to the scales of macroscopic observation. Hence the discussion of non-equilibrium behaviour will be almost always formulated in terms of phenomenological models. Of course it must then be understood to what extent such an approach is justified and useful and this will be one of the main themes of this book.