Third-order transport coefficients are also required for the conversion of the hydrodynamic transport coefficients into transport data measured in steady state Townsend and arrival time spectra experiments 3 , 4. This could be particularly important for discharges where ions play an important role 5 , or in situations where the hydrodynamic approximation is at the limit of applicability, as in the presence of sources and sinks of particles or in the close vicinity of physical boundaries.

In this study, we are concerned with the form of the skewness tensor for charged-particle transport in the presence of trapped localised states. In particular, we are interested in the scenario where transport is dispersive. Dispersive transport is characterised by a mean squared displacement that increases sublinearly with time 6. Due to this non-integer power-law dependence, we refer to dispersive transport as fractional transport throughout this study.

Some examples of fractional transport include the trapping of charge carriers in local imperfections in semiconductors 7 , 8 , 9 , 10 , 11 and both electron 12 , 13 , 14 and positronium 15 , 16 , 17 trapping in bubble states within liquids. Third-order transport coefficients are expected to be more sensitive to the influence of non-conservative collisions than those of lower order, suggesting that the presence of such trapped states would significantly influence the skewness tensor.

Indeed, skewness and other higher order transport coefficients are used to characterise fractional transport in a variety of contexts, including transport in biological cells 18 , 19 , 20 , Consider also Fig. This solution exhibits a large skewness in comparison to the accompanying Gaussian solution of the corresponding classical advection-diffusion equation. Skewed solution of the Caputo fractional advection-diffusion equation alongside the corresponding Gaussian solution of the classical advection-diffusion equation.

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## Buckley-Leverett Theory for Immiscible Displacement | Fundamentals of Fluid Flow in Porous Media

Both pulses have evolved from an impulse initial condition. The cusp in the fractional solution denotes the location of this initial impulse. In the following, we describe charged particle transport using a full phase-space kinetic model as defined by a generalised Boltzmann equation with a corresponding trapping and detrapping operator. In our previous papers 22 , 23 , 24 , we introduced and studied such a generalised Boltzmann equation, deriving lower-order transport coefficients up to diffusion and generalisations of the Einstein relation.

We will extend the results of these papers to determine the skewness tensor. Calculations of the skewness tensor for the Boltzmann equation have been performed previously by a number of authors 2 , 25 , 26 , 27 , We will use these earlier studies to confirm the structure of the skewness tensor and to benchmark our results in the trap-free case.

In Sec. This model is capable of describing both normal and fractional transport. We follow in Sec. In this section, we also provide a physical interpretation of trap-induced skewness. Finally, Sec.

We previously reported 22 , 23 , 24 the development of a phase-space kinetic model wherein charged particles scatter due to collisions, enter and leave traps and undergo recombination. For collisions, the Bhatnagar—Gross—Krook BGK collision operator 29 has been used, which scatters particles isotropically according to a Maxwellian velocity distribution of background temperature T coll. We define the Maxwellian velocity distribution of temperature T as.

Similarly, we use the BGK-type operator introduced by Philippa et al. That is, trapping events are described mathematically as delayed scattering events. This distribution appears in Eq. To determine these flux transport coefficients it is simply a matter of writing the solution of the generalised Boltzmann equation 1 itself as a density gradient expansion.

IV of ref. As in ref. Applying this time average to unity results in an implicit definition for the initial coefficient R :. Some values are tabulated in Appendix A of ref. Proceeding to evaluate Eqs 8 — 10 for the transport coefficients, we find. If we align the basis vector e 3 parallel to the applied electric field E , the transport coefficients 19—21 take on the known tensor structure 2 , 25 , 28 , 30 , 31 :. Here, the drift velocity is defined by the speed. Although this is the case in general, there are situations where the skewness can be defined using fewer than three components.

The component Q 2 vanishes in this case due to the simple Maxwellian source term used to describe scattered particles. In Cartesian coordinates x , y , z with the electric field E aligned in the z -direction, the transport coefficients take the form of Eqs 27 — 31 and the advection-diffusion-skewness equation becomes. An alternative form of the skewness tensor that makes use of these components explicitly is.

This form was used by Robson 26 when expressing the BGK model skewness 26 and is valid only when the skewness is triple-contracted with a symmetric tensor, as occurs in the advection-diffusion-skewness equation In both cases, it can be seen that skewness introduces asymmetry in the pulse in the direction of the field. In general, positive skewness can be seen to reduce the spread of particles behind the pulse, while enhancing the spread toward the front of the pulse. In Fig. Each profile has evolved from an impulse initial condition.

As the skewness tensor is odd under parity transformation, Eq. In our previous paper 23 , we interpreted the trap-induced anisotropic diffusion present in Eq.

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In a similar fashion, we can interpret the trap-induced skewness present in the perpendicular and parallel skewness coefficients 47 and To achieve this, we plot the skewness against the detrapping temperature T detrap for various mean trapping times in Fig. The resulting plots are linear with gradients that characterise of the type of skewness caused by traps.

That is, positive or negative gradients correspond respectively to positive or negative trap-based skewness. The gradients in b are of smaller magnitude than a due to the greater dependence of the parallel skewness 48 on the drift speed W as compared to the perpendicular skewness Thus, as the drift speed decreases, the plots in b coincide with those in a. This observation coincides with the illustration of skewness in Fig. When the mean trapping time is zero, the gradients in Fig.

This is to be expected as, in this case, trapping and detrapping simply act as an elastic scattering process with a positive skewness akin to Eq. As the mean trapping time increases, the nature of the skewness caused by traps changes, ultimately becoming negative for the parameters considered in Fig.

As illustrated in Fig. We interpret the increased spread here as being due to particles returning from traps. This interpretation implies that the skewness coefficients could become overall negative if particles remain trapped for a sufficient length of time before returning with a sufficiently large temperature. Indeed, these are the conditions for which the skewness coefficients become negative in Fig.

Only collisions were considered in this study and so trapping is evidently not a necessary condition for negative skewness to occur. However, it should be emphasised that the skewness is strictly positive when collisions are described by the simple BGK collision operator, as is seen in Eq.

The classical Einstein relation between diffusion, mobility and temperature is As seen by Eq. This enhancement manifests as the following generalised Einstein relation By relating the skewness to the temperature tensor though this diffusion coefficient, we find a skewness analogue to the Einstein relation:. Koutselos 34 has presented a similar relationship between the skewness tensor and lower-order transport coefficients for the case of the classical Boltzmann equation. For the phase-space kinetic model described by Eq. Analytical solution of eq. The resulted solution for the concentration profile shows that the profile is independent of the diffusion coefficient.

If you have any questions at all, please feel free to ask PERM! We are here to help the community. We respect your privacy! Carbon dioxide flux is defined as follows: Figure Simple Diffusion Experiment We can assume that the flux is proportional to the gas concentration difference and we can recognize that increasing the capillary tube length will decrease the flux, so: The new proportionality constant D is the diffusion coefficient. Example Find the diffusion flux and concentration profile in a steady diffusion across a thin film.

Solution The objective is to determine how much solvent moves across the film and how the solvent concentration changes within the film. As the first step mass balance on a thin layer located at some arbitrary position x within the film is written: Figure Diffusion Across a Thin Film Because the process is in steady state, the accumulation is zero.

There are two boundary conditions for this differential equation: Analytical solution of eq. Custler Questions? Integrating over the cytoplasm and integrating by parts we obtain where denotes the boundary of. Taking into account the rescaling Eq. This is the concentration to be used in Eq.

With this definition, the mass balance reads for. Note that, for , it holds such that the boundary term vanishes.

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The boundary term in this equation corresponds to mass inflow and outflow while the reaction term corresponds to sources or sinks depending on the sign.