Reduced density matrices characterized

Problem 10.9. Pi. system that comprises a two-level sub-system and a bath is found in a pure state, characterized by finite probabilities to exist in states 1 and 2 of the two-level subsystem where each of them is associated with a different bath state, b and b, respectively. Show that the corresponding reduced density matrix of the 2-level subsystem does not describe a pure state but a statistical mixture. 

Systems of many atoms are usually characterized in terms of ensemble-averaged quantities, such as correlation functions or the reduced density matrix. Consider, for concreteness, a correlation function of the type... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

Considerable effort would clearly be needed to characterize complex colloids in such a complete way. In many cases, it is likely that one would only need to focus on a certain limited region of the size-density matrix, thus considerably reducing the experimental labor. In addition, other techniques (such as chemical analysis) might be brought into play to simplify the experiments and, at the same time, extend the information base. We are also examining an approach to the two-dimensional (size-density) characterization of complex colloids without the requirement for fraction collection. 

When V 0 transitions between L and R can take place, and their populations evolve in time. Defining the total L and R populations by our goal is to characterize the kinetics of the L R process. This is a reduced description because we are not interested in the dynamics of individual level /) and r), only in the overall dynamics associated with transitions between the L and R species. Note that reduction can be done on different levels, and the present focus is on Pl and Pr and the transitions between them. This reduction is not done by limiting attention to a small physical subsystem, but by focusing on a subset of density-matrix elements or, rather, their combinations. 

The use of stearic acid as a modifier for silica and other fillers like CaCOs and Mg(OH)2 has been reported. The authors found that the presence of adsorbed stearic acid on the filler surface reduces the hydrophilicity of the silica surface and enhances the compatibility between filler and matrix, which may lead to an improvement in filler dispersion and the related mechanical performance of composites. Kosmalska et al also investigated the adsorption of DPG, ZnO and sulfur on the silica surface and reported that the bonding of DPG/ZnO and ZnO to silica causes a reduction in the surface energy of silica from 66 mN/m to 28.75 mN/m and 35.49 mN/m, respectively. A similar effect of ZnO on the surface tension of silica was also found by Laning et alP and Reuvekamp et al. The adsorption of that additive and its impact on the scorch time and reduction of the crosslink density in silica-filled rubber compounds have been frequently characterized. ... 

Positron annihilation lifetime spectroscopy (PALS) is a more recent tool used to probe free volume and free volume distribution in polymers (38, 59). PALS uses orthoPositronium (oPs) as a probe of free volume in the polymer matrix. oPs resides in regions of reduced electron density, such as free volume elements between and along chains and at chain ends (38). The lifetime of oPs in a polymer matrix reflects the mean size of free volume elements accessible to oPs. The intensity of oPs annihilations in a polymer sample reflects the concentration of accessible free volume elements. The oPs lifetime in a polymer sample is finite (on the order of several nanoseconds), so PALS probes the availability of free volume elements on nanosecond timescales (40). The minimum free volume cavity diameter required by oPs for localization is 3.SA (41), which is equal to the kinetic diameter of methane (42). Thus, PALS probes the dynamic availability of free volume elements similar in size to those important for gas separations applications. Several recent studies demonstrate the strong correlation of PALS parameters and transport properties in polymers (34, 38, 43-45). The chapter by Yampol skii and Shantarovich in this book describes the use of PALS to characterize free volume distribution in membrane polymers. 

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