Momentum density symmetry

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

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... 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

Taking both the symmetry of the nuclear configuration and the inversion symmetry described above into account, it follows that the symmetry of the momentum density II(p) is (g) If contains the inversion operation i, then (g) However, if does not contain i, then the symmetry of the... 

In the framework of irreversible thermodynamics (compare, for example, [31, 32]) the macroscopic variables of a system can be divided into those due to conservation laws (here mass density p, momentum density g = pv with the velocity field v and energy density e) and those reflecting a spontaneously broken continuous symmetry (here the layer displacement u characterizes the broken translational symmetry parallel to the layer normal). For a smectic A liquid crystal the director h of the underlying nematic order is assumed to be parallel to the layer normal p. So far, only in the vicinity of a nematic-smectic A phase transition has a finite angle between h and p been shown to be of physical interest [33],... 

Let us briefly review the essential ingredients to this procedure (for more details of the method see [30] and for our model [42]). For a given system the hydrodynamic variables can be split up into two categories variables reflecting conserved quantities (e.g., the linear momentum density, the mass density, etc.) and variables due to spontaneously broken continuous symmetries (e.g., the nematic director or the layer displacements of the smectic layers). Additionally, in some cases non-hydrodynamic variables (e.g., the strength of the order parameter [48]) can show slow dynamics which can be described within this framework (see, e.g., [30,47]). 

Special techniques are required to describe the symmetry of fields. Since fields are defined in terms of continuous variables it is desirable to formulate suitable transformations of dynamic variables pertaining to fields, in terms of continuous parameters. This is done by using Hamilton s principle and defining quantities such as momentum densities for any field. The most useful parameter to quantify the symmetry of a field is the Lagrangian density (T 3.3.1). 

The Fourier transform (Eq. (1)) preserves direction, in the sense that one can refer to components of the total momentum in any particular direction. For example, one can distinguish components along any Cartesian axis, or paral-lel/perpendicular to a bond or plane. A further consequence of Eq. (1) is that the momentum density p p) possesses the same symmetry elements as its r-space... 

Consequently the momentum density of any mol ule will have inversion symmetry, even if p(r) does not. It can be useful to take advantage of this inversion symmetry when integrating functions of p(p such as the generalised overlaps to be described in Sect. 3. 

The one-electron momentum density for bound states of atoms and molecules always has inversion symmetry as a consequence of time-reversal symmetry or the principle of microreversibility [42,43] ... 

When the geometrical and inversion symmetries are considered together, it follows that the symmetry of the momentum density IT(p) is If does not contain the... 

Typical electron momentum densities with (3, — 1) and (3, +1) saddle points at zero momentum are found in MgO and acetylene (HCCH), respectively. A momentum density with a zero momentum (3, — 1) critical point is shown for MgO in Fig. 19.6. In the vertical plane of symmetry /T(p, 0,pj has the structure of two hills separated by a ridge or col, and one sees two local (and global) maxima located symmetrically along the p axis. The plot in the horizontal symmetry plane has the structure of a hill. 

According to the arguments presented in Section 4 it follows that the relaxation equations for these four variables consist of two coupled equations for n (g) and g (q) and two totally uncoupled equations for (q) and gy (q). This breakup is a consequence of our choice of coordinate axes such that the wave vector q defines the z axis, g (q) is the momentum density parallel to q, namely the longitudinal momentum, and g iq) and gy(q) are the momentum densities perpendicular to q, namely the transverse components of the momentum density. By symmetry, gz(q),gx(q), and gy(q) are independent, and furthermore gxiq) and gy q) should be dynamically equivalent. Let us therefore only treat g ,(q). 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

In density functional theories the potential is determined by the density, and consequently its Fourier components are related to those of the density. One can therefore connect the symmetry properties of the momentum funetions, in other words the transformation... 

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