Local-equilibrium approximation

In perhaps one of the simplest microscopic cases, a system of interacting Brownian (i.e., diffusing) particles, and in a local equilibrium approximation, one can write the time evolution of the ensemble-averaged one-body density as a functional of the density [3, 4]... 

Dynamic Density Functional Theory (DDFT), Fig. 1 Illustration of the local equilibrium approximation involved in the development of the DDFT. The left-hand side illustrates the nonequilibrium evolution of the density p(r t) thin lines) up to time t thick line). For the time evolution, the equal-time correlation function g(r, r t) is... 

The local equilibrium approximation for the two-point correlation function involved in the development of the DDET has two issues. Eirst, it is not a priori clear when it is justifiable to approximate the nonequilibrium correlations by equilibrium correlations. It has been shown that there are cases in which this approximation breaks down, in particular in driven steady-state systems. Second, the equilibrium sum rule in... 

The local equilibrium approximation breaks down if the frequency of the external field becomes comparable with the characteristic time of the fluctuation Xe — e lD o. The motion of the polymer in such a short time-scale can be treated by eqn (9.66) or a rotation model limited in a cone-like region. Both treatments gives qualitatively similar results e.g., af (to) and Koc((o) approach zero as with the relaxation time... 

In order to solve the conservation or transport equations (mass, momentum, energy, and entropy) in terms of the dependent variables n, Vo,U, and , we must further resolve the expressions for the flux vectors— P, q, and s and entropy generation Sg. This resolution is the subject of closure, which will be treated in some detail in the next chapter. However, as a matter of illustration and for future reference, we can resolve the flux vector expression for what is called the local equilibrium approximation, i.e., we assume that the iV-molecule distribution function locally follows the equilibrium form developed in Chap. 4, i.e., we write [cf Eq. (4.34)]... 

This result is essentially equivalent to the Chapman-Enskog local equilibrium approximation, which has proven quite successful for the theoretical representation of irreversible transport processes for real gases. Seasoning by analogy, the physical basis for Eq. 5 involves the simple notion that translational relaxation occurs isotropically and much more rapidly than other relaxation modes, notably including nonthermal chemical reactions. 

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

Transition structures are more dihicult to describe than equilibrium geometries. As such, lower levels of theory such as semiempirical methods, DFT using a local density approximation (LDA), and ah initio methods with small basis sets do not generally describe transition structures as accurately as they describe equilibrium geometries. There are, of course, exceptions to this, but they must be identihed on a case-by-case basis. As a general rule of thumb, methods that are empirically dehned, such as semiempirical methods or the G1 and G2 methods, describe transition structures more poorly than completely ah initio methods do. 

The local equilibrium curve is in approximate agreement with the numerically calculated profiles except at very low concentrations when the isotherm becomes linear and near the peak apex. This occurs because band-spreading, in this case, is dominated by adsorption equilibrium, even if the number of transfer units is not very high. A similar treatment based on local eqnihbrinm for a two-component mixture is given by Golshau-Shirazi and Gniochou [J. Phys. Chem., 93, 4143(1989)]. 

For adsorbates out of local equilibrium, an analytic approach to the kinetic lattice gas model is a powerful theoretical tool by which, in addition to numerical results, explicit formulas can be obtained to elucidate the underlying physics. This allows one to extract simplified pictures of and approximations to complicated processes, as shown above with precursor-mediated adsorption as an example. This task of theory is increasingly overlooked with the trend to using cheaper computer power for numerical simulations. Unfortunately, many of the simulations of adsorbate kinetics are based on unnecessarily oversimplified assumptions (for example, constant sticking coefficients, constant prefactors etc.) which rarely are spelled out because the physics has been introduced in terms of a set of computational instructions rather than formulating the theory rigorously, e.g., based on a master equation. 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Substituting Eqs. (35) and (36) into Eq. (34), the electrochemical potential fluctuation of dissolved metal ions at OHP is deduced. Then, disregarding the fluctuation of the chemical potential due to surface deformation, the local equilibrium of reaction is expressed as fi% = 0. With the approximation cm x, y, 0, if cm(x, y, (a, tf, we can thus derive the following equation,... 

We then use a Feautrier scheme [4] to perform spectral line formation calculations in local thermodynamic equilibrium approximation (LTE) for the species indicated in table 1. At this stage we consider only rays in the vertical direction and a single snapshot per 3D simulation. Abundance corrections are computed differentially by comparing the predictions from 3D models with the ones from ID MARCS model stellar atmospheres ([2]) generated for the same stellar parameters (a microturbulence = 2.0 km s-1 is applied to calculations with ID models). 

The local density approximation (LDA) binding energy curve in Fig. 3.6, which accurately follows the exact curve around equilibrium, can be approximated by the sum of five terms, namely... 

Fig. 3.12 The binding energies, equilibrium internuclear separations and vibrational frequencies across the first-row diatomic molecules. Note the good agreement between the self-consistent local density approximation calculations and experiment for R and coe but the larger systematic error of up to 2 eV for the binding energy. (After Gunnarsson et aL (1977).)...

Fig. 3.13 Left-hand panel The electronic structure of an sp-valent diatomic molecule as a function of the internuclear separation. Labels Et and Ep mark the positions of the free-atomic valence s and p levels respectively and = ( s + p). The quantity Rx is the distance at which the and upper tr9 levels cross. The region between the upper and lower n levels has been shaded to emphasize the increase in their separation with decreasing distance that is responsible for this crossing. Right-hand panel The self-consistent local density approximation electronic structure for C2 and Si2 whose equilibrium internuclear separations are marked by RCz and RSa respectively. (After Harris 1984.)...

Equality (1.20) is of primary importance because of the following reason. It is customary in most ionic transport theories to use the local electroneutrality (LEN) approximation, that is, to set formally e = 0 in (1.9c). This reduces the order of the system (1.9), (l.lld) and makes overdetermined the boundary value problems (b.v.p.s) which were well posed for (1.9). In particular, in terms of LEN approximation, the continuity of Ci and ip is not preserved at the interfaces of discontinuity of N, such as those at the ion-exchange membrane/solution contact or at the contact of two ion-exchange membranes or ion-exchangers, etc. Physically this amounts to replacing the thin internal (boundary) layers, associated with N discontinuities, by jumps. On the other hand, according to (1-20) at local equilibrium the electrochemical potential of a species remains continuous across the interface. (Discontinuity of Cj, ip follows from continuity of p2 and preservation of the LEN condition (1.13) on both sides of the interface.)... 

A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential. By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation. 

Assuming that (13.11) makes sense in the context of the system under investigation (i.e., that physical relaxation times are in the appropriate range for the condition of local equilibrium to be satisfactorily approximated), we seek the field-type differential equation that describes asymptotic (-evolution of fields Rfx, y, z, t) toward the known metric geometrical limit. Solutions of this equation are expected to describe a wide variety of thermal, acoustic, and diffusion phenomena in nonequilibrium conditions where local thermodynamic variables retain experimental meaning. 

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