Expansion properties, momentum density

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

The definition and properties of B(r) may be summarized as follows. For simplicity, we treat the spinless one-electron wave function assuming the independent-particle model or the natural orbital expansion (Lowdin, 1955 Benesch and Smith, 1971). Based on the three-dimensional momentum density p(p),... 

The first complete calculation for the onset and the progress of spontaneous condensation was described by Oswatitsch for a one-dimensional expansion of a vapour with ideal gas properties [5]. By means of a computer such calculations can now be performed even for vapours at high density with a complex equation of state to describe the real gas behaviour. In addition to the equations for the conservation of mass, momentum and energy and to the equation of state, a further equation for the increase of the mole fraction of condensate... 

For analytical studies assuming single-phase fluid dynamics, mixing length models are employed for turbulence. Since this type of model requires the distribution of turbulent viscosity in advance, a special assumption is used to incorporate effects of excessive change of thermophysical properties. In this case, validity of the special assumption is somewhat contentious even if the calculation results agree with the experimental values. In addition, change of density is not considered in the continuity and momentum equations, which implies that buoyancy force and fluid expansion are not incorporated. Therefore, these studies are applicable only to limited flow conditions. 

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