Having achieved this point we hope to have accomplished, at least partly, our main aim which was to convince the reader that once appropriate systems have been found (or built) they can present a very peculiar combination of microscopic parameters in such a way that quantum mechanics should be applied to general macroscopic variables to describe the collective effects therein. Moreover, the very nature of these macroscopic variables does not allow them to be treated in an isolated fashion. They must rather be considered coupled to uncontrollable microscopic degrees of freedom which is the ultimate origin of dissipative phenomena. The latter, at least in the great majority of cases, play a very deleterious role in the dynamics of the macroscopic variables and we hope to have introduced minimal phenomenological techniques in order to quantify this.

We have concentrated our discussions on questions originating from a few examples of superconducting or magnetic systems where quantum mechanics and dissipative effects coexist. In particular, superconducting devices which present the possibility of displaying several different quantum effects (quantum interference, decay by quantum tunneling, or coherent tunneling) are of special importance, as we will see below. Prior to development of the modern cryogenic techniques and/or the ability to build nanometric devices, it was unthinkable to imagine the existence of subtle quantum mechanical effects such as the entanglement of macroscopically distinct quantum states.

Superconductivity was discovered by Onnes (1911), who observed that the electrical resistance of various metals dropped to zero when the temperature of the sample was lowered below a certain critical value Tc, the transition temperature, which depends on the specific material being dealt with.

Another equally important feature of this new phase was its perfect diamagnetism which was discovered by Meissner and Ochsenfeld (1933), the so-called Meissner effect. A metal in its superconducting phase completely expels the magnetic field from its interior (see Fig. 3.1). The very fact that many molecules and atoms are repelled by the presence of an external magnetic field is quite well known, as we have already seen in the preceding chapter. The difference here lies in the perfect diamagnetism, which means that it is the whole superconducting sample that behaves as a giant atom!

This effect persists (for certain kinds of metal) until we reach a critical value of the external magnetic field, H = Hc(T), above which the superconducting sample returns to its normal metallic state. Moreover, at fixed temperature, this effect is completely reversible, suggesting that the superconducting phase is an equilibrium state of the electronic system. The temperature dependence of the critical field is such that Hc(Tc) = O and Hc(0) attains its maximum value as shown in Fig. 3.2.

In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables.

In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems.

The immediate problem we have to face concerns the choice of dissipative system to be studied.

On deciding to write this book, I had two main worries: firstly, what audience it would reach and secondly, to avoid as far as possible overlaps with other excellent texts already existing in the literature.

Regarding the first issue I have noticed, when discussing with colleagues, super-vising students, or teaching courses on the subject, that there is a gap between the standard knowledge on the conventional areas of physics and the way macroscopic quantum phenomena and quantum dissipation are presented to the reader. Usually, they are introduced through phenomenological equations of motion for the appropriate dynamical variables involved in the problem which, if we neglect dissipative effects, are quantized by canonical methods. The resulting physics is then interpreted by borrowing concepts of the basic areas involved in the problem – which are not necessarily familiar to a general readership – and adapted to the particular situation being dealt with. The so-called macroscopic quantum effects arise when the dynamical variable of interest, which is to be treated as a genuine quantum variable, refers to the collective behavior of an enormous number of microscopic (atomic or molecular) constituents. Therefore, if we want it to be appreciated even by more experienced researchers, some general background on the basic physics involved in the problem must be provided.

In order to ill this gap, I decided to start the presentation of the book by introducing some very general background on subjects which are emblematic of macroscopic quantum phenomena: magnetism and superconductivity.

In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O(n)-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension dc = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η. The critical exponents ν, η, and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension dc = 4.

We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.

The emergence of generic scale invariance, i.e., algebraic behavior without tuning to special critical points, appears to be remarkably common in systems that are settled in a non-equilibrium steady state. Prototypical examples are simple non-linear Langevin equations that describe driven diffusive systems and driven interfaces or growth models far from thermal equilibrium, whose distinct phases are characterized by non-trivial RG fixed points and hence universal scaling exponents. We start with driven lattice gases with particle exclusion that are described by generalizations of the one-dimensional noisy Burgers equation for fluid hydrodynamics. Symmetries and conservation laws completely determine the ensuing stationary power laws, as well as the intermediate aging scaling regime and even the large-deviation function for the particle current fluctuations. Next we address the non-equilibrium critical point for driven Ising lattice gases, whose critical exponents can again be computed exactly. We then turn our attention to the prominent Kardar–Parisi–Zhang equation, originally formulated to describe growing crystalline surfaces and the dynamics of driven interfaces, but also closely related to the noisy Burgers equation and even to the equilibrium statistical mechanics of directed lines in disordered environments. After introducing the scaling theory for interface fluctuations, we proceed to a renormalization group analysis at fixed dimension d. For d > 2, a non-trivial unstable RG fixed point separates a phase with Gaussian or Edwards–Wilkinson scaling exponents from a strong-coupling rough phase that is inaccessible by perturbative methods.

Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n-component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = (d + 2 − η)/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O(n)-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets (n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d/2 for all slow modes.

In the second part of this book, we consider the dynamics of systems far away from thermal equilibrium. This departure from equilibrium may be caused by an external driving force, as is the case for driven diffusive systems or growing interfaces considered in Chapter 11; there also exist fundamentally open athermal systems which never reach equilibrium, as is true for some of the reaction-diffusion systems considered in Chapter 9. In both instances, the constraints imposed by detailed balance and the ensuing fluctuation-dissipation theorem on the form of phenomenological equations describing the temporal evolution of such systems are absent. Even at the fundamental level of quantum mechanics, the dynamical description of such open dissipative systems is still quite poorly understood. Generally, it may thus seem a hopeless task to derive coarse-grained equations of motion for only a few mesoscopic degrees of freedom from such an unsatisfactory foundation.

Fortunately, this conclusion is too pessimistic, at least if we are interested in systems whose non-equilibrium steady state is tuned close to a critical point, or displays generic scale invariance. For, in these situations, we may appeal to the concept of universality to allow us to constrain through basic symmetry and conservation arguments the terms which must be retained in an effective dynamical description. Similarly, we may hope that the universal properties of drastically simplified models which happen to be exactly solvable may extend to more realistic and technologically relevant systems.

In this chapter, we develop the basic tools for our study of dynamic critical phenomena. We introduce dynamic correlation, response, and relaxation functions, and explore their general features. In the linear response regime, these quantities can be expressed in terms of equilibrium properties. A fluctuation-dissipation theorem then relates dynamic response and correlation functions. Under more general non-equilibrium conditions, we must resort to the theory of stochastic processes. The probability P1(x, t) of finding a certain physical configuration x at time t is governed by a master equation. On the level of such a ‘microscopic’ description, we discuss the detailed-balance conditions which guarantee that P1(x, t) approaches the probability distribution of an equilibrium statistical ensemble as t → ∞. Taking the continuum limit for the variable(s) x, we are led to the Kramers–Moyal expansion, which often reduces to a Fokker–Planck equation. Three important examples elucidate these concepts further, and also serve to introduce some calculational methods; these are biased one-dimensional random walks, a simple population dynamics model, and kinetic Ising systems. We then venture towards a more ‘mesoscopic’ viewpoint which focuses on the long-time dynamics of certain characteristic, ‘relevant’ quantities. Assuming an appropriate separation of time scales, the remaining ‘fast’ degrees of freedom are treated as stochastic noise. As an introduction to these concepts, the Langevin–Einstein theory of free Brownian motion is reviewed, and the associated Fokker–Planck equation is solved explicitly.