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Non-relativistic and relativistic quantum kinetic equations in nuclear physics.

Wilhelmus Martinus Maria Botermans

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

In this thesis, we discussed the derivation of quantum kinetic equations appropriate for applications in nuclear physics, both in a non-relativistic and in a relativistic context. In each case we obtained a kinetic equation together with an equation for the effective interaction. The latter serves as the dynamical input of the former. In the kinetic equation both mean-field and collision effects are described in a self-consistent way because both are expressed in terms of the effective interaction. This self-consistency enforces us to solve a statistical problem (kinetic equation) and a dynamical problem (effective interaction) simultaneously. The interaction, that we obtained as a result of our considerations, is a generalized version of the usual Brueckner G-matrix in the sense that single-particle energies are corrected for collision effects by the so called rearrangement-term. 0nly in this way kinetic theory and interaction are consistent with one another and therefore constitute a conserving approximation.... Zie epilogue

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In this thesis, we discussed the derivation of quantum kinetic equations appropriate for applications in nuclear physics, both in a non-relativistic and in a relativistic context. In each case we obtained a kinetic equation together with an equation for the effective interaction. The latter serves as the dynamical input of the former. In the kinetic equation both mean-field and collision effects are described in a self-consistent way because both are expressed in terms of the effective interaction. This self-consistency enforces us to solve a statistical problem (kinetic equation) and a dynamical problem (effective interaction) simultaneously. The interaction, that we obtained as a result of our considerations, is a generalized version of the usual Brueckner G-matrix in the sense that single-particle energies are corrected for collision effects by the so called rearrangement-term. 0nly in this way kinetic theory and interaction are consistent with one another and therefore constitute a conserving approximation.... Zie epilogue
We generalize our previous work on the compatibilty of kinetic equations with second virial corrections to the inclusion of spin and particle indistinguishability; the system is supposed to be sufficiently dilute for higher order virial density corrections (interactions and statistics) to be negligible. We show that the general idea of the “free Winger transform” can be extended to this situation; the function which appears in the kinetic equation becomes here a matrix which acts in the space of spin states of the particles. Assuming that the collisions are described by a hamiltonian which does not act on the spins (a very good approximation for nuclear spins), we write explicitly a kinetic equation which is valid for this case. The right hand side of the equation is an 18 dimension integral, as for spinless distinguishable particles, but here it contains an additional term due to statistics, which introduces commutators and anticommutators. We discuss the local conservation laws in this formalism and find, as expected, a total number of 8 conserved quantities for spin 1/2 particles (including three components of the magnetization). When the gas is at equilibrium, we obtain a pressure dependence which is in agreement with known calculations on spin polarized gases. We finally study the gradient expansion of the collision integral, and show that the zero-order (local) part is identical with the 4 terms (including identical spin rotation terms) obtained previously by Lhuillier et al. The first order (non-local) part contains many terms, wich we compare with those obtained by Silin in a context more closely related to the Landau theory.
From the quantum kinetic equation of Bogoliubov, Gurov, Born, and Green, the equations of motion for macroscopic variables are examined in many different ways. First, in the configuration space a hierarchy of generalized fluid-dynamical equations can be obtained by taking the appropriate limits of the quantum kinetic equation. The resultant equations of continuity, of momentum, and of energy are similar in form to those one encounters in classical fluid dynamics with the exception of additional terms proportional to ${\ensuremath{\hbar}}^{2}$ Secondly, the quantum kinetic equation is examined in phase space. The same set of equations of continuity, of momentum, and of energy can be derived by taking the first three moments of the quantum kinetic equation. The exact results we obtained are utilized to form the starting point for many simplifying approximations for the investigation of the dynamics of quantum many-body systems such as the elastic response limit, the hydrodynamical limit, Landau's Fermi-liquid theory, and finally the time-dependent Hartree-Fock and the multideterminant time-dependent Hartree-Fock approximations. The fact that all these different dynamical descriptions can be traced to a common origin provides a unifying viewpoint to the present approach with the quantum kinetic equation.NUCLEAR STRUCTURE Dynamics of nuclear fluid, quantum kinetic equation, and generalized fluid-dynamical equation. Elastic response, hydrodynamical limit, Landau's Fermi-liquid theory, time-dependent Hartree-Fock approximation, and multideterminant time-dependent Hartree-Fock approximation.
A link between correlation contributions to kinetic equations for dense quantum systems and the energy conservation is considered. In order that the energy be conserved by an approximate collision integral, the one-particle density matrix and the mean interaction energy are treated as independent state parameters. It is shown how the density operator method can be used to derive non-Markovian kinetic equations including the correlation effect associated with the energy conservation. A quantum generalization of the Enskog theory is discussed.
The methods of quantum statistics previously developed by the author are applied to the determination of properties of the ground state of nuclear matter. An expansion in powers of the pair-function of quantum statistics is made and expressions are derived for the momentum distribution, pair-correlation function, binding energy, and effective single-particle energies. The leading terms of these expressions can be interpreted in terms of an effective two-body interaction, and a model of nuclear matter which consists of interacting quasi-particles whose energies are the effective single-particle energies is thereby suggested. The theory is compared with Brueckner's theory and also with Landau's phenomenological theory of the Fermi liquid.
We review the non-relativistic Green's-function approach to the kinetic equations for Fermi liquids far from equilibrium. The emphasis is on the consistent treatment of the off-shell motion between collisions and on the non-instant and non-local picture of binary collisions. The resulting kinetic equation is of the Boltzmann type, and it represents an interpolation between the theory of transport in metals and the theory of moderately dense gases. The free motion of particles is renormalised by various mean field and mass corrections in the spirit of Landau's quasiparticles in metals. The collisions are non-local in the spirit of Enskog's theory of non-ideal gases. The collisions are moreover non-instant, a feature which is absent in the theory of gases, but which is shown to be important for dense Fermi systems. In spite of its formal complexity, the presented theory has a simple implementation within the Monte-Carlo simulation schemes. Applications in nuclear physics are given for heavy-ion reactions and the results are compared with the former theory and recent experimental data. The effect of the off-shell motion and the non-local and non-instant collisions on the dynamics of the system can be characterised in terms of thermodynamic functions such as the energy density or the pressure tensor. Non-equilibrium counterparts of these functions and the corresponding balance equations are derived and discussed from two points of view. Firstly, they are used to prove the conservation laws. Secondly, the role of individual microscopic mechanisms in fluxes of particles and momenta and in transformations of the energy is clarified.
We discuss the description of a many-body nuclear system using Hamiltonians that contain the nucleon relativistic kinetic energy and potentials with relativistic corrections. Through the Foldy-Wouthuysen transformation, the field theoretical problem of interacting nucleons and mesons is mapped to an equivalent one in terms of relativistic potentials, which are then expanded at some order in 1mN\frac{1}{{m}_{N}}mN​1​. The formalism is applied to the Hartree problem in nuclear matter, showing how the results of the relativistic mean field theory can be recovered over a wide range of densities.