Gaussian transport

Fig. 8.10 A Gaussian transport process. Above the distribution function C(x,t) of the charge carriers at various times t. C(x,t) is the probability that a charge carrier will be at the position x at time if it was at the position X = 0 at the time t = 0. xjd is the normalised distance of the charge carriers from X = 0, where they were generated, d is the sample thickness (see Fig. 8.9). In Gaussian transport, the transit time tx is defined by x tx)) = d. Below transients of the displacement current l(t)/l(tx). In Gaussian transport, the width of the decay of the transients is governed by diffusive broadening of the charge-carrier packet. For transient (1), the transit time tx is longer than for transient (2).

In many cases, one observes no sharp decay of the transient current and thus a fundamental deviation from Gaussian transport. The reason for this is that the charge carriers have spread out over the whole sample. This transport is termed dispersive transport. It is characteristic of strongly disturbed crystals, or especially of non-crystalline organic solids, and will be treated in Sect. 8.6. 

Nearly ideal TOF transients for Gaussian transport are, however, exhibited e.g. by high-purity sublimation-grown anthracene crystals at room temperature [21], Fig. 8.11a. The transit time for the electrons in this experiment is about 12.5 /xs. On cooling the anthracene crystal to 40 K, on observes for the same orientation of the electric field, (F//c), a nearly featureless decay of the current (Fig. 8.11b). 

Gaussian Plume Model. One of the most basic and widely used transport models based on equation 5 is the Gaussian plume model. 

Recently the effect of intrinsic traps on hopping transport in random organic systems was studied both in simulation and experiment [72]. In the computation it has been assumed that the eneigy distribution of the traps features the same Gaussian profile as that of bulk states. 

In this connection let us remark that in spite of several efforts, the relation between Lyapounov exponents, correlations decay, diffusive and transport properties is still not completely clear. For example a model has been presented (Casati Prosen, 2000) which has zero Lyapounov exponent and yet it exhibits unbounded Gaussian diffusive behavior. Since diffusive behavior is at the root of normal heat transport then the above result (Casati Prosen, 2000) constitutes a strong suggestion that normal heat conduction can take place even without the strong requirement of exponential instability. 

Starting from these initial conditions, the composition PDF will evolve in a non-trivial manner due to turbulent mixing and molecular diffusion.13 This process is illustrated in Fig. 3.10, where it can be seen that the shape of the composition PDF at early and intermediate times is far from Gaussian.14 As discussed in Chapter 6, one of the principal challenges in transported PDF methods is to develop mixing models that can successfully describe the change in shape of the composition PDF due to molecular diffusion. 

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