Phase-space density formalism

This formalism is the quantum-mechanical analog of the Klimontovich phase-space density formalism, and is particularly effective in the theory of long-range correlation and fluctuation in the kinetic theory of plasmas. 

In Sections 2 and 3, we set up a formalism for dealing with the dynamics of dense fluids at the molecular level. We begin in Section 2 by focusing attention on the phase space density correlation function from which the space-time correlation functions of interest in scattering experiments and computer simulations can be obtained. The phase space correlation function obeys a kinetic equation that is characterized by a memory function, or generalized collision kernel, that describes all the effects of particle interactions. The memory function plays the role of an effective one-body potential and one can regard its presence as a renormalization of the motions of the particles. 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

Remark. We assumed that Y(t) is a Markov process. Usually, however, one is interested in materials in which a memory effect is present, because that provides more information about the microscopic magnetic moments and their interaction. In that case the above results are still formally correct, but the following qualification must be borne in mind. It is still true that p y0) is the distribution of Y at the time t0, at which the small field B is switched off. However, it is no longer true that this p(y0) uniquely specifies a subensemble and thereby the future of Y(t). It is now essential to know that the system has aged in the presence of B + AB, so that its density in phase space is canonical, not only with respect to Y, but also with respect to all other quantities that determine the future. Hence the formulas cannot be applied to time-dependent fields B(t) unless the variation is so slow that the system is able to maintain at all times the equilibrium distribution corresponding to the instantaneous B(t). 

The term density matrix arises by analogy to classical statistical mechanics, where the state of a system consisting of N molecules moving in a real three-dimensional space is described by the density of points in a 6N-dimensional phase space, which includes three orthogonal spatial coordinates and three conjugate momenta for each of the N particles, thus giving a complete description of the system at a particular time. In principle, the density matrix for a spin system includes all the spins, as we have seen, and all the spatial coordinates as well. However, as we discuss subsequently we limit our treatment to spins. For simplicity we deal only with application to systems of spin % nuclei, but the formalism also applies to nuclei of higher spin. 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

The procedure for constructing kinetic equations using the generalized Langevin equation is well known - one uses as variables in this description the p/ioje-space density fields. We could of course simply use the solute phase-space fields, (7.1), and follow the methods of Section V to obtain a formal kinetic equation for their time evolution. This procedure... 

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix formulation yields a consistent treatment of electronic populations and coherences, and the momentum changes associated with an electronic transition can be directly derived from the formalism without the need of ad hoc assumptions. Employing a Monte-Carlo sampling scheme of local classical trajectories, however, we have to face two major complications, that is, the representation of nonlocal phase-space operators and the sampling problem caused by rapidly varying phases. At the present time, the... 

Note that the Liouville equation, formally, is identical with the first conservation equation, the so-called continuity equation of hydrodynamics, equation (la). The change of the mass density and the change of the phase-space-distribution can be derived based on the conservation of the total mass and the total number of systems, respectively.) The last step of equation (7) is a definition of the term A(/ ) called the phase-space compression factor. In the case of conservative systems (the most common example of which is Hamilton s equations), the Liouville equation describes an incompressible flow and the right-hand side of equation (7) is zero. (In many statistical mechanical texts, only this incompressible form is referred to as the Liouville equation.)... 

All results presented so far can be put in one overarching formalism, that of the Fokker-Planck equation. This equation governs the probability of finding a system in a particular area of phase space. The phase space probability density is denoted by p( r), [p], t), which is a function of the positions (r) and momenta (p) of the particles (or modes) involved in the barrier transition, and of the particles (modes) coupled to this barrier transition process. The equation describes classical Newtonian dynamics, but also incorporates damped, or decaying, motion to a final equilibrium state. It is based on a mesoscopic picture of the environment... 

There is a formal similarity between this equation and Eq. (9.32) the Poisson bracket in the latter is replaced by the a commutator in the former. Poisson brackets and commutators have a number of properties in common. They are Lie brackets, which means they are linear, are antisymmetric, and satisfy the Jacobi identity. Both 7f, p) and —j[H, p] can be viewed as an operations on the probability density. But here the similarity ends. The probability densities themselves hve in completely different spaces. The first is a function in classical phase space, and the second is an operator on a Hilbert state space, ft has up to now not been possible to unify those two spaces, so that, for instance, a correct classical limit can be taken. 

A fundamental hypothesis in statistical thermodynamics is that the two averages are equal. This is the ergodic hypothesis. The physical implication of the ergodic hypothesis is that any system afforded with infinite time will visit all the points of phase space with a frequency proportional to their probability density. In other words, in a single trajectory the system spends an amount of time at each microstate that is proportional to its probability. The ergodic hypothesis is still a hypothesis because there has been no formal proof of its truth. 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

Adsorption itself is, according to Gibbs formalism, an excess quantity. The situation at the gas/solid interface is schematically represented in Fig. 1. As consequence of the adsorption potential, the average number of molecules in an element voliune near the surface is larger than in an element volume of equal size in the bulk gas. A density profile is thus estabUshed from the solid surface to the ambient gas phase. Both molecular simulation of adsorption and experimental proof reveal that this density profile vanishes over quite a short range. Therefore, the layer on the solid surface, where adsorbate molecules are concentrated, is referred to as the adsorption space or adsorbed phase. It thus assumes a definite thickness (t) for the adsorbed phase. 

Contrary to the bulk liquid phase which is homogeneous in three directions in space, has a characteristic composition, and is also autonomous (i.e., its extensive properties depend only on the intensive variables characterising this phase such as the temperature T, the pressure P, and the chemical potentials of the solvent /ri and the solute /u-2), the formal thermodynamic description of a Solid-Liquid interface presents a serious difficulty. In the interfacial region, the density a> of any extensive quantity changes continuously throughout the thickness (Fig. 6.1a). 

The difference between the apparent densities in nonpenetrating liquids and in water can be formally attributed to the existence of an apparent void space, inaccessible to hydrocarbon liquids but accessible to water, and definining that this selectively accessible void space is completely filled when assigning normal density to the water phase. Since this selectively accessible apparent void space appears to be identical with the void space inaccessible to hydrocarbons, it may be reasonably assumed that both relate to the same real void space within particle aggregates. Indeed, it is likely that water would be able to reach this void space... 

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