One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.
One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter. Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.
As introduced in Chapter 1, for many problems in statistical mechanics one is interested in the detailed behavior of only a part of the overall system (call it the system) interacting with its surrounding (call it the environment). In field theory one can accordingly decompose the field describing the overall system ϕ = ϕ S + ϕ E into a sum of the system field ϕ S and the environment field ϕ E . This decomposition is always possible formally but only when there is a clear physical discrepancy between the two sectors will it be physically meaningful and technically implementable. The division could be made between slow and fast variables, low and high frequencies or light and heavy mass sectors. Drawing examples from cosmology, in the stochastic inflation scenario one regards the system field as containing only the lower modes and the environmental field as containing the higher modes with the division provided by the event horizon in de Sitter spacetime. A similar problem in quark-gluon plasma is to ascertain the effect of the hard thermal loops on the soft gluon modes. Another is the effect of the atoms in the noncondensate on the Bose–Einstein condensate (BEC). These cases will be discussed in later chapters. Usually the reason for performing such a decomposition is because one is interested more in the details of the system (the “relevant” variables or the “distinguished” sector), and less in that of the environment (the “irrelevant” variables).
Recently, a number of authors have begun to study the evolution of quantum fields in the early Universe characterized by a time-dependent density matrix ${\ensuremath{\rho}}_{S}$(t). All of this work is predicated on the assumption that one's ``subsystem'' of interest is in some sense ``decoupled'' from the rest of the Universe, so that ${\ensuremath{\rho}}_{S}$ satisfies a Liouville--von Neumann equation which implies, e.g., an isentropic evolution. Starting from ``first principles,'' i.e., the Schr\"odinger equation for the totality of ``subsystem'' plus surroundings (``bath''), it is shown here how such a picture can be derived as the limiting case of a more complete statistical description. Quite generally, one finds that ${\ensuremath{\rho}}_{S}$ and the ``bath'' density matrix ${\ensuremath{\rho}}_{B}$ satisfy coupled nonlinear generalizations of the Liouville--von Neumann equation and evidence a nonisentropic evolution. However, in a Vlasov-type approximation, ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{B}$ satisfy instead much simpler bilinear equations which imply an isentropic evolution. And finally, in the limit that the ``back reaction'' of ${\ensuremath{\rho}}_{S}$ on ${\ensuremath{\rho}}_{B}$ can be neglected in computing the evolution of ${\ensuremath{\rho}}_{B}$, one recovers a true Liouville--von Neumann equation for the evolution of ${\ensuremath{\rho}}_{S}$ in an external field.
Dissipation and noise in mean field dynamicsIn Chapter 6 we presented the main computational schemes to derive the dynamical laws for the mean field, including the back-reaction from quantum fluctuations.These equations may be derived from the variation of the CTPEA.The result of this approach is a semiclassical theory of a c-number condensate interacting with a quantized fluctuation field.This approach developed at this level of sophistication is limited as it offers no description of the fluctuations themselves.In most applications the magnitude of the fluctuations can be comparable and at times dominates the effects of the mean field in the semiclassical description.One possible way to incorporate fluctuations is to use the 2PI formalism, where the propagators describing the fluctuations are considered as dynamical variables evolving along with the mean fields.In this chapter we shall explore a different strategy, which is to allow for a stochastic component in the mean field.This component arises from both the uncertainty of the initial configuration of the mean field, and from the fluctuations in the back-reaction from the quantized excitations.Both sources of randomness combine so that stochastic averages in the noisy theory reproduce suitable quantum averages in the underlying quantum field theory.Formally, this approach lifts the seemingly overladen CTPEA.So far in this generally complex object, only the real part is enlisted in the derivation of the relevant equations of motion of the mean field.By regarding the CTPEA as a kind of influence functional, we shall see that the imaginary part contains the information about the stochastic sources.The material in this chapter also clarifies the relationship between the CTP and the influence functional approaches.This issue has been addressed by Su et al. [SCYC88] and the authors [CalHu94].In Chapter 5, we derived a Langevin equation for the long-wavelength modes of a quantum field, viewed as an open system interacting with the environment made up of short-wavelength modes.The system-environment divide we shall assume in this chapter is more elusive, since it depends on the c vs. q-number nature rather than on the value of a "hard" observable such as wavelength.In the end, as we shall discuss in detail, the physics is very much the same in one or the other approach.The stochastic mean field approach we shall discuss in this chapter has the redeeming feature that it does not force us to choose an a priori separation between modes which go into the system and which are relegated to the environment.In this sense, it is more pliable to the demands of a particular application: for example, if
Perhaps due to its technical complexity, oftentimes one sees in research papers on nonequilibrium quantum field theory (NEqQFT) more emphasis placed on the field-theoretical formalisms than the ideas these sophisticated techniques attempt to capture, or the issues such problems embody. All the more so, we need some basic understanding of the important issues and concepts in nonequilibrium statistical mechanics (NEqSM), and how they are manifested in the context of quantum field theory. Many important advances in this field came from asking such questions and finding out how to answer them in the language of quantum field theory. Because of this somewhat skewed existing emphasis in NEqQFT, and since we do not assume the reader to have had a formal course on NEqSM before, we shall give a brief summary of the basic concepts of NEqSM relevant to the field-theoretical processes discussed in this book. Many fine monographs and reviews written on this subject take a more formal mathematical approach. approach. Since our purpose here is to familiarize readers with these issues and their subtleties, rather than training them to work in the rich field of NEqSM (which includes in addition to the traditional subject matter such as the projection operator formalism and open system concepts, also current topics at their foundation, such as dynamical systems and quantum chaos), we choose to approach these topics in a more intuitive and physical way, sacrificing by necessity rigor and completeness.
Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).
In the previous chapter we have seen that departures from equilibrium generate a plethora of new phenomena in statistical physics. While our discussion so far has been limited to classical phenomena, in this chapter we want to ask how quantum mechanics interferes with the conditions of nonequilibrium. Experience shows that, in some cases, nonequilibrium quantum systems can be understood in terms of reasonably straightforward quantization of their classical limits. In others, an interplay of quantum coherence and nonequilibrium driving leads to entirely new phenomena – such as lasing, a paradigm for a steady state nonequilibrium quantum system. At any rate, we have every reason to anticipate that quantum nonequilibrium physics will be as diverse and colourful as its classical counterpart. Even minor departures from equilibrium call for a theoretical description entirely different from the ‘Matsubara formalism’ developed in earlier chapters of this book. The applicability of the latter is rigidly tied to the existence of a quantum grand canonical density operator, the hallmark of a many particle equilibrium system. But how then, can an functional theory of nonequilibrium quantum systems be developed? An obvious idea would be to once more go through the different elements of classical nonequilibrium theory developed in the previous chapter, and subject every one of them to an individual quantization scheme. Luckily, there are more efficient ways to achieve our goal: some decades ago, a many particle formalism suitable to describe nonequilibrium systems under the most general conditions was introduced by Keldysh.