Spectral line shapes. II

A spectral line shape function theory is extended for gaseous systems where reactive collisions can occur. The theory of subdynamics and the Fock-Tani representation are used in order to attempt this goal. The dependence of the spectral shape on the pressure and the reactive transition probability is clarified. Finally^a discussion concerning the above aspects and their possible applications has been included as well.     

Subdynamics and nonintegrable systems

Work on the application of Poincaré's theorem to large classical or quantum systems with a continuous spectrum is continued. In situations where it is applicable, Poincaré's theorem prevents the construction of a complete set of eigenprojectors which would be hermitian as well as analytic in the coupling constant. In contrast, the theory of subdynamics as developed by the Brussels group permits the construction of a unique set of projectors , giving up the requirement of hermiticity which is replaced by “star-hermiticity”.

The theory of subdynamics is presented in a new self-contained way, starting from the commutation relation , where L_H is the Liouvillian. This presentation is far more direct, and avoids some of the lengthy discussions associated with previous presentations (based mainly on the resolvent of the Liouvillian).

Subdynamics appears to be of interest from many points of view. It generalizes the concept of spectral representation while permitting to retain all the degrees of freedom present in the unperturbed Hamiltonian. In contrast, degrees of freedom are lost when going to the spectral representation (e.g. in the Friedrichs model). Subdynamics permits us to solve the initial value problem associated with the Liouville equation retaining the “non-Markovian” contributions which appear in the standard presentation. Finally, it introduces a classification of large dynamical systems, classical or quantum, into integrable and nonintegrable ones. It is therefore of direct interest for a number of basic problems which belong to the class of nonintegrable dynamical systems, such as the interaction of matter with light. The applications of this technique to these problems will be worked out in subsequent papers.     

A generalization of the van Kampen-Case treatment of the Vlasov equation

In the inhomogeneous collisionless system the collective modes are constructed in the physical representation using the method of subdynamics. They correspond to the explicit solutions of the non-linear Vlasov equation and they appear as the generalization of the van Kampen-Case treatment. The absence of entropy production is shown.     

Analysis of Prigogine's theory of subdynamics

We discuss in full detail the basic results of the theory of subdynamics. We analyse the regularity requirements of the projective components of the resolvent in the theory and correct a recent mathematical result of M. Courbage regarding the impossibility of simultaneous analytic continuation of the collision and creation operators. We offer further some critiques on the interpretation of the various results of the subdynamics formalism.     

Macroscopic cell representation of density matrix equation and optimal choice of projectors by variational principle

We introduce a complete set of projectors on Liouvillespace and a “cell representation” of the equation of motion of the density matrixρ adjusted to the macroscopic observables of the system. Using a variational principle, the projector on the relevant density matrixρ _rel is determined by the postulate that the subdynamics ofρ _rel should include an optimal part of the Liouvillean?. The relation of the reduced subdynamics equation to wellknown master equations is investigated.     

On the derivation of linearized hydrodynamics from kinetic theory

The relation of the linearized hydrodynamic equations to kinetic theory is discussed. We show that the theory of subdynamics provides a general framework for their derivation. This theory and the related transformation theory permit the construction of a Lyapounov functional that for the hydrodynamical processes, considered here, becomes the macroscopic entropy as given by Gibbs relation.     

Vacuum subdynamics in large quantum systems

We present a constructive procedure to deal with large quantum systems in the thermodynamic limit. Starting with a discrete spectrum, we perform a complete decomposition of the evolution into one-dimensional subdynamics. We then go to the limit of a continuous spectrum after collecting them into global subdynamics of given degrees of correlation. Previously obtained results for the vacuum subdynamics are justified. The procedure is applied to the problem of potential scattering.It is a privilege for the authors to dedicate this paper to Prof. I. Prigogine. They are both genuine offspring of the Brussels school. As such, most of the ideas they profess owe their inspiration to I. P., their mentor, colleague, and friend through fascinating teaching, stimulating discussions, and illuminating speculations, but they take full responsability for their obvious mistakes, pernicious misconceptions, and malignant deviations from the orthodoxy they have modestly helped to create.     

用子动力学理论研究介观体系的输运——电导响应函数的计算

从子动力学理论出发,计算了介观体系的输运公式,得到了包含高阶修正的Kubo公式,这对于任意外场的作用下的介观体系具有重要的意义。     

On generalized Waldmann-Snider equations: II. Π̌-subdynamics

Three non-equilibrium equations have been derived describing the evolution of the matrix elements, which are highly off-diagonal on the internal energy of the singlet density operator, ?⁽¹⁾, of a quantum dilute gas of molecules with internal levels. The first equation has been obtained by assuming the requirement of sufficient spacing of the spectral transition lines, the second one being its extension for weakly inhomogeneous gases and the third one the equation generalized for spectral transition lines of arbitrary spacing. They have been obtained inside the framework of the subdynamics theory of the Brussels Group where the Π̌-subdynamics have been explicitly constructed.Their linearized version is presented permitting to obtain closed equations for the ?⁽¹⁾-operator, which can be shown useful in the correlation functions, being interesting in spectroscopic measurements. A discussion concerning the above aspects has been included as well.     

On the standard form of the Bloch equation

The requirement is often made in non-equilibrium statistical mechanics that a transport equation should be derived as that which governs the subdynamics relative to a (small) part of a (large) conservative dynamical system close to equilibrium. We show that such a requirement on the Markovian relaxation of a 1/2-spin imposes that this process be described by a Bloch equation of a very specific form, which we call standard. We show that this reduced dynamics is quasi-free if, and only if, the relaxation time is maximally anisotropic.Research supported in part by NSF grant MCS 76-07286     

Field theory reformulated without self-energy parts. Kinetic extension for the Lee model

An alternative formulation of field theory is possible such that self-energy contributions do no longer appear as dynamical processes. The dynamics is now described by kinetic-like equations of motion, involving more degrees of freedom than in the original formulation, plus a constraint propagating in time. Considering situations outside the equivalence conditions provides a natural irreversible extension of quantum field. The relevance of such extension relies on the positivity and normalization of the extension. The feasibility is examplified here on the Lee model without focusing on a particular sector. The tool used is the single subdynamics approach [Trends Stat. Phys. 3 (2000) 115]. A partial treatment of the Lee model along those lines have been considered for a particular sector [Prog. Theor. Phys. 109 (2003) 881]. We intend to consider here a set of sectors using a reduction formalism [Physica A 171 (1991) 159] which provides directly the required properties. Outside the equivalence conditions, our approach couples sectors and the resulting statistical mixture changes its composition with time.     

Some remarks on subdynamics for systems interacting with reservoirs

It is shown that collision operators defined by the projector of Argyres and Kelley may provide useful examples for study of the subdynamics of the Brussels group. An application of this formalism to the optical pumping model of Barrat and Cohen-Tannoudji is discussed.     

Subdynamics Theory in the Functional Approach to Quantum Mechanics

The formalism of subdynamics is extended to thefunctional approach to quantum systems and is used forthe Friedrichs model in which diagonal singularities instates and observables are included. In this approach we compute the generalizedeigenvectors and eigenvalues of the Liouville-vonNeumann operator, using an iterative scheme. As complexgeneralized eigenvalues are obtained, the decay rates ofunstable modes are included in the spectraldecomposition.     

Dual instantons

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both position and momentum are expressed in terms of the collision cross section.     

Food chain chaos with canard explosion

The "tea-cup" attractor of a classical prey-predator-superpredator food chain model is studied analytically. Under the assumption that each species has its own time scale, ranging from fast for the prey to intermediate for the predator and to slow for the superpredator, the model is transformed into a singular perturbed system. It is demonstrated that the singular limit of the attractor contains a canard singularity. Singular return maps are constructed for which some subdynamics are shown to be equivalent to chaotic shift maps. Parameter regions in which the described chaotic dynamics exist are explicitly given.     

Exact Dimensional Reduction of Linear Dynamics: Application to Confined Diffusion

In their stochastic versions, dynamical systems take the form of the linear dynamics of a probability distribution. We show that exact dimensional reduction of such systems can be carried out, and is physically relevant when the dimensions to be eliminated can be identified with those that represent transient behavior, disappearing under typical coarse graining. Application is made to non-uniform quasi-low dimensional diffusion, resulting in a systematic extension of the “classical” Fick-Jacobs approximate reduction to an exact subdynamics.     

Canonical nonequilibrium ensembles and subdynamics

A method for constructing a canonical nonequilibrium ensemble for systems in which correlations decay exponentially has recently been proposed by Coveney and Penrose. In this paper, we show that the method is equivalent to the subdynamics formalism, developed by Prigogine and others, when the dimension of the subdynamic kinetic subspace is finite. The comparison between the two approaches helps to clarify the nature of the various operators used in the Brussels formalism. We discuss further the relationship between these two approaches, with particular reference to a simple discrete-time dynamical system, based on the baker's transformation, which we call the baker's urn.