Momentum density tensor

We begin by describing the HPP model, which satisfies all of the above requirements except for the isotropy of the momentum flux density tensor. As we shall, however, this early model nonetheless has some very interesting and suggestive properties, despite not being able to reproduce Navier-Stokes-like behavior exactly. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

We make two additional comments. First, notice that when u 0, the momentum flux density tensor reduces to the diagonal term p5ij, where the pressure p = Cgp and Cg is the speed of sound. We thus conclude that the speed of sound in the FHP-I LG is given by... 

Isotropy of the Momentum Flux Density Tensor If we trace back our derivation of the macroscopic LG Euler s and Navier-Stokes equations, we see that the only place where the geometry of the underlying lattice really enters is through the form for the momentum flux density tensor, fwhere cp = x ) + y ), k = 1,..., V... 

Now, in order for us to recover standard hydrodynamical behavior, we require that the momentum flux density tensor be isotropic i.e. invariant under rotations and reflections. In particular, from the above expansion we see that must be isotropic up to order... 

The methods developed in the theory of liquids (Rice and Gray 1965, Gray 1968) was used by Pokrovskii and Volkov (1978a) to determine the stress tensor for the set of Brownian particles in this case. One can start with the definition of the momentum density, given by (6.3), which is valid for an arbitrary set of Brownian particles. Differentiating (6.3) with respect to time, one finds... 

Equation (2.16) consists of two contributions the molecular momentum flow tensor, it, and the convective momentum flow tensor, pvv. The term p8 represents the pressure effect, while the contribution t, for a Newtonian fluid, is related to the velocity gradient linearly through the viscosity. The convective momentum flow tensor pw contains the density and the products of the velocity components. A component of the combined momentum flow tensor of x-momentum across a surface normal to the x-direction is... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

In order to connect the above expressions to Onsager s theory, it is necessary to extend equations such as (6.3.1) to three dimensions. This equation shows that a force in one direction leads to velocity changes in the other two spatial directions. A general treatment of fluid flow requires that one identify the components of the pressure tensor associated with the force which leads to fluid flow. The force vector F has, in general, a component in each of the three Cartesian directions. The component in the x-direction, F, gives rise to three pressure components, one in the same direction, P x, and two shear components, Pxy and Pxz- Six more pressure components are obtained from the force components in the y- and z-directions, Fy and F. As a result, there is a second-rank pressure tensor Py with nine components. Analysis on the basis of classical dynamics leads to the conclusion that the off-diagonal elements, Py and Py,-, are equal. As a result, six distinct elements of this tensor must be determined to define it. The second important step is to write an equation of continuity in terms of momentum. Defining the momentum density If as... 

It should be noted that if y is a vector-valued function, as is the case for the momentum density, then the above equations are formally the same, but with appropriate vector or tensor interpretations for the other variables. 

Time constant for Hookean dumbbell model Time constants for Rouse chain model Solvent contnbution to thermal conductivity Tensor virial multiplied by 2 Momentum space distribution function Integration variable in Taylor series Stress tensor (momentum flux tensor) External force contribution to stress tensor Kinetic contribution to stress tensor Intramolecular contribution to stress tensor Intermolecular contribution to stress tensor Fluid density... 

For viscous liquids the law for the mass conservation remains unchanged. As to the momentum density conservation, it keeps the same form (9.8) but tensor 11 should be changed to take the dissipation into account. Now we write... 

The microscopic density of (4.177) becomes the smoodi function p of 4.S on averaging.) The change of momentum density with time is die gradient of die instantaneous value of die stress tensor. The pressure tensor of 4.3 is the ensemble average of the negative of this stress tensor in a system at equilibrium in the absence of an external field. Hence, cf. (2.82) and (4.94),... 

Stress enters in a development of hydrodynamics when one considers the equation of conservation of momentum. The rate of change of momentum in some volume element at point r is written as the acceleration produced by external forces on that element and a (negative) flux of momentum across the surface. The flux of momentum has two parts. The first is the momentum associated with the average velocity, u(r), of the fluid at r. Thus momentum density in the a direction (with a x,y, or z) is p(r)t/ (r), where p(r) is the mass density at r. This momentum is transported in the direction at a rate u ir). Therefore this contribution to the flux of a momentum in the /S direction is p(r)M (r)M (r). Additional observed momentum transfer is called minus the stress tensor. The stress tensor can be separated into contributions from two molecular sources. One is also kinetic, and arises from the fact that the particles have a distribution of velocities about the average fluid flow velocity. We can write this term as a statistical average... 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

While the general form of the generalized Euler s equation (equation 9.9) allows for dissipation (through the term Hifc) expression for the momentum flux density as yet contains no explicit terms describing dissipation. Viscous stress forces may be added to our system of equations by appending to a (momentarily unspecified) tensor [Pg.467]

In the case of the flux of mass, the result is the normal component of pua. But for the flux of momentum and energy, in general the flux density is not the normal component of a vector or tensor function of (t, x), since it will depend on the extended shapes of if and Y. But in the case of short-range forces and slowly varying p, ua, E, it can be shown to have this form with sufficient approximation. Thus one is led to the familiar pressure tensor and heat flow vector Qa, both as functions of (t, x). It is to be emphasized that the general expression of these quantities involves not only expected values of products of momenta (or velocities), but the effect of intermolecular forces. 

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