Spatially variable diffusion coefficient

If the diffusion coefficient is spatially variable, the one-dimensional diffusion equation has the form  

The correct way to differentiate this equation relies on the following centered-difference approximation of the spatial derivative at timestep tn  

The midpoints xi+i/2 = (xi +xi+i)/2 and the corresponding values of the diffusion coefficient Di+i/2 = D(Xj+i/2) have been introduced to ensure appropriate centering of the implied derivatives. A convenient approximation for Di+1/2 results from considering the average of the values at the neighboring gridpoints  

In the case of unequal spacing of the spatial gridpoints, Eq. (8-39) can be generalized to give  

Application of the Crank-Nicholson method based on the spatial difference scheme (5.39) results in the following discretized form of the diffusion equation  

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In a spatially extended system, fluctuations that are always present cause the variables to differ somewhat in space, inducing transport processes, the most common one being diffusion. In the case of constant diffusion coefficients /), the system s dynamics is then governed by reaction-diffusion equations ... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the time variable and numerical in the spatial dimension) for linear parabolic partial differential equations using Maple, the method of lines and the matrix exponential. 

On the other hand, the effectiveness of the signaling reactions also depends on the diffusion coefficient, as shown in Eq. (33.11). Although other parameters in Eq. (33.11) (rs, rA, and nT) are not variable as determined for each reaction (33.10), only the diffusion coefficients (Ds and DA) can be controlled by the existence of the surrounding media. Moreover, as mentioned in the previous section, the diffusion coefficient of anomalous diffusion depends on the diffusion time and the dimensions of the reaction space. In such a situation, the diffusion coefficient observed by one method (e.g., FCS, FRAP) is only a local value, depending on the time constant and the spatial size of a proper experiment. As mentioned for Figure 33.4 in the beginning of this section, the size of the reaction volume for signaling reaction is of the order of pL-fL and measurement of the diffusion coefficient in such a microspace is important. 

Note that in a general case the effects associated with diffusion in three spatial dimensions r, the dependence of diffusion coefficients on the r, x variables and the presence of (non-zero) off-diagonal diffusion coefficients should be accounted for in equations of reactions with diffusion. 

Here u(f) is the inhibitor and a x, t) is the activator variable. In the semiconductor context u t) denotes the voltage drop across the device and a(x, t) is the electron density in the quantum well. The nonlinear, nonmonotonic function /(a, u) describes the balance of the incoming and outgoing current densities of the quantum well, and D(a) is an effective, electron density dependent transverse diffusion coefficient. The local current density in the device is j a, u) = (/(a, u) + 2a), and J = j jdx is associated with the global current. Eq. (5.22) represents Kirchhoff s law of the circuit (5.3) in which the device is operated. The external bias voltage Uq, the dimensionless load resistance r R, and the time-scale ratio e = RhC/ra (where C is the capacitance of the circuit and Ta is the tunneling time) act as control parameters. The one-dimensional spatial coordinate x corresponds to the direction transverse to the current flow. We consider a system of... 

Eq. (5-28) is not suitable for describing spatially and temporally variable diffusion processes. Because of the principle of the conservation of mass, however, the divergence of the flux can always be set equal to the time derivative of the local concentration. This leads to Pick s second law which may be written as follows for the case of binary systems with constant diffusion coefficients ... 

The complete discussion of migration of the reagents through tubes is quite complex. We shall here give a sketchy, albeit representative, overview based on the notion of a diffusion coefficient D r) that depends on the spatial coordinates. We choose the variable r to be the distance between two reagents. The flux at a distance r is given by ... 

In this chapter we have provided evidence of nontrivial interference between macroscopic behavior and dynamics at the microscopic level. In the vicinity of the Hopf bifurcation this interference is essentially manifested in spatially extended systems where the presence of inhomogeneous modes can lead to the loss of coherence of the uniform limit cycle, even if the latter is macroscopically stable. On the other hand, in well-stirred (0-dimensional) systems (phase) coherence is recovered in the large size limit, since both the phase diffusion coefficient and the variance of fluctuations of the radial variable tend to zero. 

The general equation of convective diffusion in liquids, equation (15), is a second-order, partial differential equation with variable coefficients. Its solution yields the spatial distribution of c, as a function of time, namely its transient behaviour. On an analytical level, solution of equation (15) into the transient c(t) is possible only for a number of relatively simple systems with well-defined geometry and flow properties. The problem is greatly simplified if the concentration function Cj(x,y,z) is essentially independent of time t, i.e. in the steady-state. Then equation (15) reduces to ... 

The method of separation of variables can be applied in the same manner to other initial distributions of diffusant. The effort lies only in determining the Fourier coefficients, which, for many cases, can be looked up in tables. If the spatial dimension of the system is higher [e.g., c(x, y, z, f)], a separate Fourier series must be obtained for each of the three separate functions in the product X(x)Y(y)Z(z). 

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