Distribution functions in physics: Fundamentals

Theory and application of the quantum phase-space distribution functions

A time‐dependent picture of vibrational Raman scattering in the weak field limit is presented. From this viewpoint we can separate the static effects, due to the coordinate dependence of the electronic transition dipole, from the dynamic effects that arise from wave packet propagation on the Born–Oppenheimer surfaces. Away from resonance, the energy uncertainty relation gives the propagation time necessary to obtain the cross section as being inversely proportional to the mismatch of the excitation frequency with the excited surface. The wave packet, given by the initial vibrational wave function times the transition dipole, hardly moves on the excited surface when the excitation frequency is far from resonance. As the excitation frequency is tuned closer to resonance, the propagated wave packet samples a larger portion of the surface. Using the short time approximation to the propagator, we obtain formulas for the cross section that are applicable for Raman scattering by polyatomics. The short time approximation is expected to be good away from resonance independent of the nature of the surface, and also on resonance with a repulsive surface. For an attractive surface, the approximation gives the average resonant cross section useful in the case when the vibrational structures cannot be observed.

Time‐dependent theory of Raman scattering

Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.

Mixed quantum-classical dynamics

An unconventional time dependent formula for total photodissociation cross sections shows the importance of short time dynamics in direct photofragmentation. This is exploited to provide a systematic expansion in powers of h/ for the cross section. The lowest order term is a classical cross section which is shown to be an improvement upon the venerable reflection approximation. Terms to higher order in h/ lead to even greater improvements in accuracy as shown by simple numerical examples. Our formulas are directly applicable to polyatomic photofragmentation, and as a spinoff we derive the polyatomic generalization of the (diatomic) reflection method.

Quantum corrections to classical photodissociation models

The Wigner method of transforming quantum‐mechanical operators into their phase‐space analogs is reviewed with applications to scattering theory, as well as to descriptions of the equilibrium and dynamical states of many‐particle systems. Inclusion of exchange effects is discussed.

Wigner Method in Quantum Statistical Mechanics

The exact solution of the transmission and reflection problem for transverse electromagnetic waves incident on a bounded plasma has been discussed to some extent by several authors. Shure considered the special cases of perpendicular incidence on nonrelativistic half‐space and slab plasmas and made use of van Kampen‐Case modes to construct the solution. For both half‐space and slab plasmas, we generalize his results to (i) arbitrary temperatures (relativistic) and (ii) arbitrary angles of incidence. For simplicity, we consider the special case where the incident electric field is perpendicular to the plane of incidence and assume that particles are reflected specularly at the interface. We proceed somewhat differently from Shure, and use a Laplace transformation in obtaining our solution. We also show that present solutions can be expressed as a superposition of van Kampen‐Case modes appropriate to a relativistic plasma.

Transmission and Reflection of Electromagnetic Waves at the Boundary of a Relativistic Collisionless Plasma

The kinetic stage of an interacting, multi-component system is studied using Bogoliubov's formalism including terms up to the second orde formalism including terms up to the second order in interaction strength λ, but to all orders in nλ, n being the density. It is shown that the second order s-particle distribution function can be expressed in terms of zero and first order binaiy and zero order ternary correlation functionals. This functional form is also shown to be deducible in a prescribed manner from the so-called cluster expansion series of the equilibrium theory. The latter observation provides a means for guessing the structures of the higher order distribution functionals. The equations satisfied by the first order binary and zero order ternary correlation functionals are derived. They constitute a coupled set of equations determining the second order contribution to the kinetic equation. Explicit expressions are found for the correlation functions when the one-particle distribution function is Maxwellian. The results, in this case, are in agreement with those obtained from the equilibrium theory.

http://iopscience.iop.org/0029-5515/4/1/001/pdf/0029-5515_4_1_001.pdf

Correlations in plasmas: I. Ternary correlations

The exact solution of the linearized transmission and reflection problem for electromagnetic waves obliquely incident on a half‐space relativistic Vlasov plasma is obtained in the case when the electric field is polarized in the plane of incidence. It is assumed that electrons are reflected specularly at the boundary, and ion dynamics is ignored. Coupling of transversal and longitudinal components at the boundary and their propagation characteristics are studied in detail. Numerical results for the calculation of the skin depth and the phase velocity of the discrete modes in the first Riemann sheet are included.

Mode Coupling at the Boundary of a Relativistic Collisionless Plasma

The structure of cluster expansion which is widely used in statistical mechanics is studied from an algebraic point of view. In doing this, a commutative algebra is constructed which is generated by partitions of a finite set by regarding them as operators which divide the set into disjoint parts. Physically, these operators correspond to operations which remove interaction among certain clusters of particles. It is shown that the cluster expansion stems from the relation between two basis sets of this algebra; the first set is the set of all partitions and the second is the set of pairwise orthogonal minimal idempotents. This property enables one to demonstrate the equivalence of the product versus cluster properties of the distribution and correlation functions, respectively, in general terms. This is done by constructing a simple representation space for the partition algebra corresponding to the distribution functions. A second application of the partition algebra is considered in the case when correlations are well‐ordered with respect to the interaction strength λ, so that to a given order in λ the distribution functions are not all independent and can be expressed in terms of a finite irreducible set of functions involving smaller numbers of particles. The combinatorial problem of calculating the expansion coefficients is carried out explicitly using a graded representation space with respect to the order in λ. It is concluded that the partition algebra can be used as a mathematical tool in handling problems involving cluster expansion.

Ercüment Özizmir

Papers

Wigner Method in Quantum Statistical Mechanics

Transmission and Reflection of Electromagnetic Waves at the Boundary of a Relativistic Collisionless Plasma

Correlations in plasmas: I. Ternary correlations

Mode Coupling at the Boundary of a Relativistic Collisionless Plasma

On the Algebraic Structure of the Cluster Expansion in Statistical Mechanics