Position-dependent diffusion equation

The monovariate Fokker-Planck equation with a position dependent diffusion coefficient D x),... 

In this equation, is the total interaction energy between the two colliding particles defined in the previous section. The stability ratio, W, for the system gives the ratio of rapid coagulation, Jp, to slow coagulation, J[= J W], DQi) is the position-dependent diflusion equation. This diffusion coefficient ratio is a factor that decreases the collision rate because of the difficulty in draining the liquid between the two solid surfaces. This diffiision coefficient ratio is given by [60,61]... 

The diffusion equation given by Fick s second law where Cj x,t) is the position dependent diffusible hydrogen concentration, t is time, D is the diffusion coefficient, and x is distance through the foil has been solved for the initial conditions Cr[x, f = 0) = 0 and for the boundary conditions Cr(0, t = ) = C , Cr L, f = ) = 0. In other words, the conditions are that before the experiment is started, the foil contains no hydrogen and, at any time after the start of the experiment, the concentration, Cr, on one side (x = 0) is a constant, Q, and on the other side (x = L) it is equal to zero. The time f = 0, in this case, corresponds to the time at which the concentration is switched from 0 to Q. The solution of Eq 47 with the above initial and boundary conditions yields a series, which, to a good approximation, can be given by its first term ... 

When the motion of electrons and positive ions in a particular system may be described as ideal diffusion, the process of bulk recombination of these particles is described by the diffusion equation. The mathematical formalism of the bulk recombination theory is very similar to that used in the theory of geminate electron-ion recombination, which was described in Sec. 10.1.2 ( Diffusion-Controlled Geminate Ion Recombination ). Geminate recombination is described by the Smoluchowski equation for the probability density w(r,i) [cf. Eq. (2)], while the bulk recombination is described by the diffusion equation for the space and time-dependent concentration of electrons around a cation (or vice versa), c(r,i). 

A solution of the reaction-diffusion equation (9.14) subject to the boundary condition on the reactant A will have the form a = a(p,r), i.e. it will specify how the spatial dependence of the concentration (the concentration profile) will evolve in time. This differs in spirit from the solution of the same reaction behaviour in a CSTR only in the sense that we must consider position as well as time. In the analysis of the behaviour for a CSTR, the natural starting point was the identification of stationary states. For the reaction-diffusion cell, we can also examine the stationary-state behaviour by setting doi/dz equal to zero in (9.14). Thus we seek to find a concentration profile cuss = ass(p) which satisfies... 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

To show this connection, consider an ion-pair as above (Sect. 2.1). Not only may the ion-pair diffuse and drift in the presence of an electric field arising from the mutual coulomb interaction, but also charge-dipole, charge-induced dipole, potential of mean force and an external electric field may all be included in the potential energy term, U. Both the diffusion coefficient and drift mobility may be position-dependent and a long-range transfer process, Z(r), may lead to recombination of the ion-pair. Equation (141) for the ion-pair density distribution becomes... 

While eqn. (211) is a bit complex, the similarity to the more familiar diffusion equation [e.g. eqns. (43), (44), (158) or (197)] is apparent. The diffusion coefficient has to be replaced by a position-dependent tensor which couples (or connects) the motion of one of the particles, e.g. the particle k, with that of the other particles, e.g. a particle j. When these particles are a long way apart, the solvent between them can be squeezed out easily. As the particles approach, this is no longer true because the two particles block certain directions for escape of the solvent as the particles approach. Increasingly, the solvent has to be squeezed out of the way in a direction perpendicular to that of the approach of the particles and this causes the solvent to impede the particle approach more and more effectively. When j = k, the effect on the same particle of its own motion is negligible. Hence, the diffusion coefficient tensor elements, Tjj = kBT/%, are the same as the diffusion coefficient for the particle... 

In this Appendix, the equivalence of the diffusion equation treatment and the molecular pair analysis is proved (see Chap, 8, Sect. 3.2) for the situation where there is a potential energy E/(r) between the reactants and the diffusion coefficient is tensorial and position-dependent. This Appendix is effectively a generalisation of the analysis of Berg [278]. The diffusion equation has a Green s function G(r, f r0, t0) which satisfies eqn. (161)... 

Exercise. In the Kramers equation (VIII.7.4) allow y and T to depend on the position x. The expansion for large y is again possible and produces the diffusion equation for inhomogeneous temperature510... 

The role of thermal fluctuations for membranes interacting via arbitrary potentials, which constitutes a problem of general interest, is however still unsolved. Earlier treatments G-7 coupled the fluctuations and the interaction potential and revealed that the fluctuation pressure has a different functional dependence on the intermembrane separation than that predicted by Helfrich for rigid-wall interactions. The calculations were refined later by using variational methods.3 8 The first of them employed a symmetric functional form for the distribution of the membrane positions as the solution of a diffusion equation in an infinite well.3 However, recent Monte Carlo simulations of stacks of lipid bilayers interacting via realistic potentials indicated that the distribution of the intermembrane distances is asymmetric 9 the root-mean-square fluctuations obtained from experiment were also shown to be in disagreement with this theory.10... 

Equation (130), in turn, allows the determination of the effective temperature as a function of t — t and t — to- Since the time-dependent diffusion coefficient is, at any positive time, a monotonic increasing function of the temperature, Eq. (130) yields for Teff t t. t — to) a uniquely defined value. 

The permeate is continuously withdrawn through the membrane from the feed sueam. The fluid velocity, pressure and species concentrations on both sides of the membrane and permeate flux are made complex by the reaction and the suction of the permeate stream and all of them depend on the position, design configurations and operating conditions in the membrane reactor. In other words, the Navier-Stokes equations, the convective diffusion equations of species and the reaction kinetics equations are coupled. The transport equations are usually coupled through the concentration-dependent membrane flux and species concentration gradients at the membrane wall. As shown in Chapter 10, for all the available membrane reactor models, the hydrodynamics is assumed to follow prescribed velocity and sometimes pressure drop equations. This makes the species transport and kinetics equations decoupled and renders the solution of... 

Kx,i dimensionless position-dependent frequency associated with diffusion of species i, see equation (13.23)... 

In the small damping limit it is also possible to obtain an energy diffusion equation for the case where the friction kernel (and the associated random noise) are position dependent. A convenient model with such property is given by ... 

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