Many-body problem statistical formulation

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

The path integral approach was introduced by Feynman in a seminal paper published in 1948. It provides an alternative formulation of time-dependent quantum mechanics, equivalent to that of Schrodinger. Since its inception, the path integral has found innumerable applications in many areas of physics and chemistry. Its main attractions can be summarized as follows the path integral formulation offers an ideal way of obtaining the classical limit of quantum mechanics it provides a unified description of quantum dynamics and equilibrium quantum statistical mechanics it avoids the use of wavefunctions and thus is often the only viable approach to many-body problems and it leads to powerful influence functional methods for studying the dynamics of a low-dimensional system coupled to a harmonic bath. 

The study of molecular scale devices has created the need for new theoretical tools which could be used for predictions of their structures and properties and to probe their new designs. Electronic devices are open systems with respect to electron flow, and a theoretical description of such devices should be done in terms of statistically mixed states which cast the problem in terms of quantum kinetic theory [100]. The only completely adequate theory that could currently address this task is the non-equilibrium Grin s function formulation of many-body theory. 

An accurate representation of the electronic structure of atoms and molecules requires the incorporation of the effects of electron correlation, and this process imposes severe computational difficulties. It is, therefore, only natural to investigate the use of new and alternative formulations of the problem. Many-body theory methods offer a wide variety of attractive approaches to the treatment of electron correlation, in part because of their great successes in treating problems in quantum field theory, the statistical mechanics of many-body systems, and the electronic properties of solids. 

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