Drift term

In the fluid model the momentum balance is replaced by the drift-diffusion approximation, where the particle flux F consists of a diffusion term (caused by density gradients) and a drift term (caused by the electric field ) ... 

This equation can be interpreted as the drift term of a collisionless Boltzmann equation for the one-particle Wigner distribution p(q,p). To see that, let us explore the physical meaning of p(q,p) in this context. First note that p(q, p ) is in principle a Lorentz scalar. Thus an invariant solution of Eq. (59) is... 

The diffusion term in this expression differs from the Fokker-Planck equation. This difference leads to a fT-dependent term in the drift term of the corresponding stochastic differential equation (6.177), p. 294. 

Note that the turbulent diffusivity Tt(x, t) must be provided by a turbulence model, and for inhomogeneous flows its spatial gradient appears in the drift term in (6.177). If this term is neglected, the notional-particle location PDF, fx>, will not remain uniform when VTt / 0, in which case the Eulerian PDFs will not agree, i.e., i=- f0. 

As a consequence, the drift term corresponding to the first order in Eq. (4.154) vanishes, and we only consider the second- order term as the lowest nonvanishing... 

The s terms in Eq. (80) contribute only the term E,2 in Eq. (97). Thus, the term represents the quantum diffusional. v-terms in the Fokker-Planck equation. The other terms in Eqs. (93)-(100) originate in the drift terms of the Fokker-Planck equation. The terms B12 and C in Eqs. (93)-(94) play the role of feedback terms that pump quantum fluctuations into the classical Bloembergen equations. If the s terms in Eq. (80) do not appear (the classical case), the term in Eq. (97) does not appear, either. In this case the subset (95)—(100) with zero initial conditions has zero solutions and in consequence leads to the first truncation [171]. 

We shall meet more general Fokker-Planck equations the special form (1.1) is also called Smoluchowski equation , generalized diffusion equation , or second Kolmogorov equation . The first term on the right-hand side has been called transport term , convection term , or drift term the second one diffusion term or fluctuation term . Of course, these names should not prejudge their physical interpretation. Some authors distinguish between Fokker-Planck equations and master equations, reserving the latter name to the jump processes considered hitherto. 

Equation (8.14) demonstrates once more that the cation flux caused by the oxygen potential gradient consists of two terms 1) the well known diffusional term, and 2) a drift term which is induced by the vacancy flux and weighted by the cation transference number. We note the equivalence of the formulations which led to Eqns. (8.2) and (8.14). Since vb = jv - Vm, we may express the drift term by the shift velocity vb of the crystal. Let us finally point out that this segregation and demixing effect is purely kinetic. Its magnitude depends on ft = bB/bA, the cation mobility ratio. It is in no way related to the thermodynamic stability (AC 0, AG go) of the component oxides AO and BO. This will become even clearer in the next section when we discuss the kinetic decomposition of stoichiometric compounds. 

These equations show that the A and B fluxes are composed of both a drift term and a reaction term. The drift term stems from the electric field. The reaction term was already deduced in the kinetics of heterogeneous reactions. From Eqns. (8.78) and (8.79), we obtain the reaction product s rate of thickness increase to be... 

The solution of the equation (4.2.26) cannot be found in an analytical form and thus some approximations have to be used, e.g., variational principle. Its formalism is described in detail [33, 57, 58] for both lower bound estimates and upper bound estimates. Note here only that there are two extreme cases when a(r)/D term is small compared to the drift term, reaction is controlled by defect interaction, in the opposite case it is controlled by tunnelling recombination. The first case takes place, e.g., at high temperatures (or small solution viscosities if solvated electron is considered). 

Before discussing mathematical formalism we should stress here that the Kirkwood approximation cannot be used for the modification of the drift terms in the kinetics equations, like it was done in Section 6.3 for elastic interaction of particles, since it is too rough for the Coulomb systems to allow us the correct treatment of the charge screening [75], Therefore, the cut-off of the hierarchy of equations in these terms requires the use of some principally new approach, keeping also in mind that it should be consistent with the level at which the fluctuation spectrum is treated. In the case of joint correlation functions we use here it means that the only acceptable for us is the Debye-Htickel approximation [75], equations (5.1.54), (5.1.55), (5.1.57). 

Therefore, the approximate treatment of the A+B — 0 reaction for charged particles inavoidably requires a combination of several approximations the Kirkwood superposition approximation for the reaction terms and the Debye-Hvickel approximation for modification of the drift terms with self-consistent potentials. Not discussing here the accuracy of the latter approximation, note... 

Both equations 14 and 15 contain a diffusion term, D, for diffusivity and a drift term, x, for mobility. For positive ions, the diffusion term is nearly always negligible compared with the drift term, but both terms are retained for reasons that have to do with numerical stability. 

Equations (6.305) and (6.306) are different because of the drift term V (PV D), which is sometimes called a spurious drift term. These diffusion equations have different equilibrium distributions and are two special cases of a more general diffusion equation. 

Equation (6.308) implies that the isotropic diffusive motion along the coordinate axes is independent. Here, V V/KT is the drift due to an external potential force field V, while Vcr/cr represents an internal drift caused by a concentration gradient of the traps. The term PV(e /a2) is the spurious drift term. Equation (6.308) allows spatial variations of all parameters T, V, T>. and a with inhomogeneous temperature. FromEq. (6.308), the diffusion coefficient becomes... 

This is the traditional diffusion model given in Eq. (6.305) with the diffusion coefficient D proportional to 1/er. For this case, the so-called spurious drift term vanishes because the effects of a and a cancel each other out in the stationary state. The stationary distribution is proportional to the Boltzmann distribution exp(-I 7/c7 ) and independent of D. 2. a = constant. Then, Eq. (6.308) becomes... 

The anisotropy introduces two new features (i) equations (6.305) and (6.306) cannot in general be transformed into each other, as the drift term V D may not be a gradient field. Equation (6.306) can describe systems where the directions of the principal axes depend on the spatial position, (ii) Detailed balance implies that the diffusion flow J vanishes everywhere in the stationary state. However, this is not automatically satisfied for anisotropic systems and one needs to exercise extra care in the modeling of such systems. Inhomogeneity does not affect the detailed balance, (iii) The diffusive part of the diffusion flow must be represented by J = VD73, while the drift is represented by (PV D). 

The nonlinear friction coefficient y(V) thereby takes on the role of a confining potential while for y0 = y(0) the drift term Fy0, as mentioned before, is just the restoring force exerted by the harmonic Omstein-Uhlenbeck potential, the next higher-order contribution y2V3 corresponds to a quartic potential, and so forth. The fractional operator 0a/0 V a in Eq. (125) for the velocity coordinate for 1 < a < 2 is explicitly given by [20,64]... 

As it is possible to see, the drift term has disappeared since the continuous growth of particle size does not change the total number concentration (if Gl > 0). However, N is influenced by the rate of formation of particles (e.g. nucleation), and the rates of aggregation and breakage, which cause appearance and disappearance of particles. These processes are all contained in the source term /tL.o The third-order moment mL,3 is related to the fraction of volume occupied by particles with respect to the suspending fluid and can be easily found fromEq. (2.18) ... 

It is interesting to note that, if the drift term is null (i.e. particles are not growing in size) and if there is no introduction of new particles into the system (i.e. the nucleation rate is null ), the right-hand side of Eq. (2.23) is null, resulting in a continuity equation for the particle phase of the form... 

It is interesting to highlight that since the numerical solution of the drift term is problematic, especially when phase space is discretized, sometimes this perspective is inverted. In recent works, in fact, some continuous processes are described as if they were actually discontinuous processes. As we will see, this strategy solves some of the issues (i.e. numerical diffusion in phase space) but typically makes the problem very stiff, due to the different time scales governing the process. Readers interested in the details are referred to the work of Kumar etal. (2008b). 

The drift term represents a continuous variation of the internal coordinate f due to a rate of change f ... 

The discretization in terms of particle mass, as described in Eq. (7.8), is particularly popular for the treatment of the drift term. In this case the rate of change of the mass of particles belonging to the f interval becomes... 

We form the average of Eqs. (5.20)-(5.22) noting that (L(r)) will vanish throughout because, in the inertia corrected Langevin equation, M is statistically independent of the white noise field h(r). This is not, however, true of the noninertial Langevin equation where the multiplicative noise term L(f) contributes a noise induced drift term to the average (see Section VI). The averages so formed are... 

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