A renormalized kinetic theory is developed for the classical phase-space density correlation function by means of the projection-operator formalism. The technique consists of introducing a sequence of projection operators which project onto spaces involving successively larger numbers of particles. The first such operator is the Akcasu-Duderstadt projection operator, which yields a generalized Langevin equation (kinetic equation) for the correlation function. The structure of the memory function in this equation suggests a second projection operator, which is shown to lead to a two-body kinetic equation for one factor in the memory function. Further projection operators proceed similarly, and the result is a continued-fraction expansion of the correlation function. This expansion is renormalized in the sense that the interparticle potential always drops out in favor of static correlation functions. The expansion is shown to be equivalent to the result of the kinetic theory formulated by Gross. The analogous development for self-correlations is also given.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
Using the recently discussed quantum dynamics in phase space, we derive a master equation, starting from the phase-space equivalent to the Schr\"odinger equation of motion for the density operator. Use is made of Zwanzig's projection-operator techniques and some explicit realizations of the projection operators are given. The master equation is then applied to show that the time-correlation functions, as defined in the text, satisfy an integral equation of the Volterra type. Next, a master equation for a system interacting with a large system is derived. As an illustration, we determine the lowest-order Born approximation and carry out a short-memory-approximation calculation for an oscillator coupled to a reservoir and for a two-level system interacting with an oscillator heat bath; we obtain equations of the Fokker-Planck type. Some physical implications of these equations are also discussed.
The memory kernel of the kinetic equation for the time correlation function of a quantum fluid is determined both in third order of the interaction strength and in the low-density approximation. The results are obtained with the help of a diagram representation for the kernel. The connexion with the kinetic theory for classical fluids is established. The properties of a nonequilibrium fluid may be studied with the help of the kinetic theory of time correlation functions. These functions are determined by specifying the memory kernel in the kinetic equation that governs their time evolution. For classical fluids a suitable formalism that permits the approximate evaluation of this kernel has been developed several years ago (for a review see ref. 1). The generalization of these methods to the case of quantum fluids was considered in a few recent papers2f3). In particular, the memory kernel was established for a normal quantum fluid in the weakcoupling approximation. In the long-time and large-distance limit this kernel reduces to that of the linearized Uehling-Uhlenbeck equation4) in which the scattering amplitude is replaced by its first Born approximation. The purpose of the present paper is to extend the available results by employing higher-order perturbation theory for the evaluation of the memory kernel. An essential ingredient of the treatment will consist in the introduction of four-point diagrams that represent the various contributions to the kernel. With the use of this diagram representation we shall derive the memory kernel in third order of the interaction strength. In this order the kernel still has the same general form as the weak-coupling kernel, but with shifted energies and a generalized transition probability. Its relation to the kernel of the linearized Bogoliubov-Gurov’) equation will be established. An alternative approach to the approximate evaluation of the memory kernel makes use of an expansion in the density. The low-density limit of the
In this paper we derive master equations for two or more systems coupled to each other, perhaps strongly, by using a generalization of the usual projection-operator technique to include time-dependent projection operators. The coupled systems may be either similar or dissimilar and classical or quantum mechanical. Whereas the customary approaches to coupled systems are best able to treat situations in which some of the systems are "baths" with a specified density operator or phase-space probability density, our approach allows us to treat situations where it is necessary or convenient to treat the coupled systems on an equal footing. In our scheme the "relevant" part of the full density operator is considered to be the uncorrelated part of the full density operator and is a symmetric functional of the reduced density operators of each of the coupled subsystems. The "irrelevant" part of the density operator is then the part describing correlations between the coupled systems. Our formalism is particularly useful where systems are coupled to one another predominantly in a self-consistent fashion. First, we develop exact master equations for two coupled systems, taking as our prototype the dynamical problem of quantum optics, where a spatially extended collection of two-level atoms interact with a multimode optical field. We then generalize our results to
N coupled systems, taking as our prototype the kinetics of a classical nonideal gas interacting through two-body forces, and derive exact master equations for the system. We then consider as examples several approximate theories resulting from our exact equations. In the case of the imperfect gas we investigate the low-density limit and show how Bogoliubov's form of the Boltzmann equation emerges from our formalism, as well as corrections due to Klimontovich. We consider as special cases of our exact quantum-optical equations the equations in the first Born approximation, with and without memory, and show how several existing quantum-optical master equations are contained in our general results. As a second example in quantum optics, we consider the case where the predominant behavior of the system is described by the self-consistent-field or coupled Bloch and Maxwell equations and derive a first-order perturbation description for deviations from self-consistent-field behavior.
A time-dependent projection-operator formalism is used to derive a kinetic equation for the single-spin density operator in an Ising system of
N two-state spins in weak interaction with a heat bath. This equation contains a contribution at the self-consistent-field level and also higher-order contributions due to dynamically generated correlations. A detailed evaluation of the self-consistent-field terms is made. Possibilities for generalizing the method to obtain equations for multispin density operators are discussed.