Flux-force laws

Flux as a function of chemical and electrical forces, 474 Flux-force laws, 495... 

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

The variational derivation of the integral atomic force law, eqn (8.175), is applicable only to a region of space bounded by a zero-flux surface in Vp(r), i.e. to an open system whose Lagrangian integral vanishes at the point of variation. Thus the variational derivation of the atomic force. 

One of the most familiar linear flux-force relations is Ohm s law of electrical... 

Eq. (A.15) is a basic equation of Onsager s theory. Notice that so far it rests solely on macroscopic phenomenology, since both the flux-force relations and the Second Law derive from purely macroscopic experiments. 

An ion of mass m and charge q travelling with a velocity v in an electric field with strength E and a magnetic field with flux density B experiences a force F defined by the Lorentz force law. This equation can be combined with Newton s laws to yield the acceleration a as... 

Some heat flows in connection with entropy production are associated with other thermodynamic variables. Typical single fluxes and forces are summarized above. It may be noted that steady fluxes are considered. Kinetic theory provides theoretical justification of some of these flux force relations (J = LX). Here, L is called phenomenological coefficient. But kinetic theory has limitation in the sense that first approximation to distribution function corresponds to local equilibrium hypothesis. It may be noted that non-equilibrium molecular dynamics (model and simulation) provides justification of these laws for a wide range. Nevertheless, justification has to be provided by experiments (Table 2.1). 

WesterhofT and Chen [77] were able to demonstrate that the rate of ATP synthesis (output flux) is not a unique function of the average concentration of the intermediate when the latter occurs in small number. As the result of numerical simulation, under certain conditions the system displays the behavior (positive flux Jp generated by the negative driving force AG, Fig. 3.18) that contradicts the Second Law of Thermodynamics. This is the case when the volume of the domain in which the intermediate appears, F, is so small that an average number of intermediate iV < 1. Of course, for the macroscopic system the flux-force relationship follows the Second Law. 

The entropy production rate is equal to a sum of products of generalized forces by generalized fluxes. The laws of thermodynamics of irreversible processes enable us to express these fluxes as functions of these forces. When we do not stray too far from the state of equilibrium, where the fluxes and forces are null, linear relations appear between these terms. The coefficients of these linear laws are the Onsager phenomenological coefficients they are combinations of the coefficients of diffusion, viscosity, heat conduction, etc. In conductive media, the electrical resistance also appears as an Onsager coefficient. 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

It is empirically known that a linear relation exists between a potential gradient or the force X and the conjugate flux J, and the laws of Ohm, Fourier, and Pick s first law for electrical conduction, thermal conduction, and diffusion, respectively, within a range of suitably small gradients ... 

A complete model for the description of plasma deposition of a-Si H should include the kinetic properties of ion, electron, and neutral fluxes towards the substrate and walls. The particle-in-cell/Monte Carlo (PIC/MC) model is known to provide a suitable way to study the electron and ion kinetics. Essentially, the method consists in the simulation of a (limited) number of computer particles, each of which represents a large number of physical particles (ions and electrons). The movement of the particles is simply calculated from Newton s laws of motion. Within the PIC method the movement of the particles and the evolution of the electric field are followed in finite time steps. In each calculation cycle, first the forces on each particle due to the electric field are determined. Then the... 

This result shows that the most likely rate of change of the moment due to internal processes is linearly proportional to the imposed temperature gradient. This is a particular form of the linear transport law, Eq. (54), with the imposed temperature gradient providing the thermodynamic driving force for the flux. Note that for driven transport x is taken to be positive because it is assumed that the system has been in a steady state for some time already (i.e., the system is not time reversible). 

The simplest case of a flux and a driving force is shown in the conduction of electric current This process is governed by Ohm s law for the current density ... 

Thus, in an isothermal system, the mass flow rate depends on the difference in pressures of the gas across the orifice and does not depend upon the thickness of the plate. One may define an area-normalized resistance, R, for mass transfer through the orifice using a generalization of Ohm s law, i.e., Resistance = force/ flux. For Knudsen flow, the force is the pressure difference (analogous to voltage difference in Ohm s law) and the flux is the mass flow per unit area of the hole (analogous to the electrical current density in Ohm s law). Thus, we have... 

The situation with regard to convective (turbulent) momentum transport is somewhat more complex because of the tensor (dyadic) character of momentum flux. As we have seen, Newton s second law provides a correspondence between a force in the x direction, Fx, and the rate of transport of x-momentum. For continuous steady flow in the x direction at a bulk... 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

It is often easy to measure the flux density, e.g., using a flowmeter, and then determine the hydraulic conductivity or diffusion coefficient by dividing the flux by the driving force. One of the most difficult problems is determining how to represent the driving force. The symbol V is called an operator, which signifies that some mathematical operation is to be performed upon whatever function follows. V means to take the gradient with respect to distance. For Darcy s law under saturated... 

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