Local density fields

The diffusion equation describes the evolution of the mean test particle density h(r,t) at point r in the fluid at time t. Denoting the Fourier transform of the local density field by hk(t), in Fourier space the diffusion equation takes the form... 

When describing transient effects in these systems, it often proves convenient to deal with the Fourier transform of the local density field. 

Local density fields can also be introduced easily. The number of molecules of species a at point r in the fluid is given by... 

To make connection with the macroscopic law, we must analyze the structure of the damping matrix. The local density fields can change by both motion of the center of mass of the molecules and by reactive events that change the number of molecules of a given species. Thus we have... 

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

Berces, A., Ziegler, T., 1992, The Harmonic Force Field of Benzene. A Local Density Functional Study , J. Chem. Phys., 98, 4793. 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

Some of the major areas of activity in this field have been the application of the method to more complex materials, molecular dynamics, [28] and the treatment of excited states. [29] We will deal with some of the new materials in the next section. Two major goals of the molecular dynamics calculations are to determine crystal structures from first principles and to include finite temperature effects. By combining molecular dynamics techniques and ah initio pseudopotentials within the local density approximation, it becomes possible to consider complex, large, and disordered solids. 

Field emission is the emission of electrons from a solid under an intense electric field, usually at ambient temperatures. It occurs by the quantum mechanical tunneling of electrons through a potential barrier (Fig. 13.1). This leads to an exponential dependence of emission current density J on the local electric field, as given by the Fowler Nordheim equation,... 

With TMS, a brief but powerful electric current is passed through a small coil held against the scalp of a conscious patient. This generates a powerful local magnetic field which passes unimpeded through the skull and induces a weaker, less focused electric current within the brain. Due to the non-invasive nature of this method, the important physiological effects of TMS are likely to be a consequence of the density of the electric current and the electric field which is induced in the cortex. It is believed that the induced electrical fields cause neuronal depolarization which changes the neurotransmitter release mechanisms. 

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