Dispersive Transport

8/20/2019 Dispersive Transport

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Why is dispersion anomalous?

This post touches on the nature of theoretical and experimental research and illustrates howa fundamental idea can take quite a circuitous route before it lodges in a remotely relatedapplication area. The acceptance of the original idea tends to create a momentum thatmakes it difficult to dislodge from the conventional wisdom and impenetrable to anyone butthe cognoscenti .

First off, prep yourself for some solid-state physics. But don't worry about the math as thecommentary makes up for the potential MEGO. The extended narrative traverses the scaleof applicability from statistical mechanics to environmental geology picking up arcs ofconnectivity along the way. I also buried a valuable nugget in here, presenting asurprisingly powerful analytical result that has laid dormant for over 30 years, perhaps even20 years prior to that, and has huge implications for the analysis of solar cells, MOStechnology, and quantum electronics. To put it another way, if I had to do another graduate

thesis, I could easily defend this argument, and on top of that, I would have fun doing it. Itall fits together tighter than a Peyton Manning spiral. The fact that it also connects acrossdisparate domains of science makes it frankly mind-blowing.

I call it mind-blowing from the fact that the actual argument derives from such a simplepremise, and I have to seriously wonder why no one has picked up on this before. I actuallyquestion some of the belief systems inherent in these fields of study and assert thelikelihood of a sunk cost effect getting in the way of a fundamental understanding. If thissounds familiar to those following our oil predicament, it should, as I have definitely seensuch oversight play out before.

Often in physics, experimental observations are termed "anomalous" before they areunderstood. Once theory succeeds in explaining and illuminating the observations, theyare no longer "anomalous" and instead come to be regarded as "obvious". A crucial paper

can trigger such an "anomalous => obvious" transition, and in the present case that keyrole was played by a 1975 paper by Scher and Montroll. That landmark paper hasbecome basic to our understanding of a striking characteristic of carrier motion (nowcalled dispersive transport) which is a common occurence in amorphous semiconductors,though foreign to our experience with crystals.

-- Richard Zallen, "The physics of amorphous solids", Wiley-VCH, 1998

Hmm ... Reserve growth also considered anomalous... 1

The term anomalous in scientific code-speak essentially means "dunno". We have to admitthat we don't understand lots of things, largely due to issues of complexity, observability, or

just too much noise. Yet that doesn't prevent us from trying to extract a fundamentalmeaning of some strange behavior that we observe.

The applied mathematicians Harvey Scher & Elliott Montroll originally tried to explain theconcept of anomalous behavior of photo-conductivity in amorphous semiconductors.

1Or "enigmatic" as some have referred to the oil situation, seehttp://mobjectivist.blogspot.com/2008/07/solving-enigma-of-reserve-growth.html

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Scientists had long understood the complementary non-anomalous behavior innon-amorphous materials. To set the stage see the schematic figure to the right.

In a crystalline semiconductorwith a contact electrode ateach end, a pulse of light

incident at one electrode will,upon the effect of an electricfield or potential dropbetween the electrodes,generate an almostimmediate flow of currentacross the load lasting as longas the transit time of thephoto-induced carriers. Thecarriers, either electrons or holes depending on the polarity of the electric field bias relativeto the absorbing electrode, will drift from the site of the photon-induced excitation, to theopposite contact. Experimentalists consider the behavior well-characterized; the mobility ofthe carriers at temperature, the strength of the electric field, and the contact separation, d ,

determine the transit time, t T , of the output current pulse. Because the carriers scatteragainst lattice imperfections, the speed does not continuously accelerate but insteadachieves a bounded drift velocity, v 0. This drift mobility has an intuitive real-world analog --think in terms of a drag coefficient for the analogous situation of a falling body undergravitational forces, which eventually achieve what we call terminal velocity. For all practicalpurposes, the equivalent terminal velocity occurs almost immediately in a semiconductor (ashort relaxation or quenching time) and sets the bound for the transit time duration.

Ideally, the pulse looks like a perfect square wave with temporal duration t T. In terms of amathematical expression, the current behaves as I(t ) = K *[u(t )-u(t-t T )] where u(t ) is theunit step operator with magnitude K . See the figure to the right.

Because of carrier diffusion, the actual drop-off in current has rounded leading and trailing

edges as the charged carrier pulse spreads out a bit into a Gaussian packet as itpropagates. This relatively innocuous but well-understood form of dispersion, known asdiffusion, occurs from random walk excursions as the carrier makes its way across thetransit width. In the ideal case, the diffusion constant varies linearly with the mobility (ordrag for particle systems) according to the Einstein relation. For high mobilities and smallcontact separations, the amount of diffusion that occurs does not appreciably round thepulse edges. The prized high mobility solid-state material allows device manufacturers tofabricate ultra-high speed photo-detectors as the sharp transition and short transit timegenerates an excellent and well-characterized frequency response.

This class of semiconductor has an ordered structure due to the crystalline lattice structureand it has properties such as carrier mobility which remain uniform through the sample.

Such behavior shows little dispersion, either through diffusion or disorder. The narrower thedistribution of velocities, the sharper the transition. Scientists have generally understoodthis for years and no one raises the spectre of anomalous behavior.

The truly anomalous behavior observed occurs in amorphous versions of certainsemiconductors. The narrow pulse of carriers seen in an ordered sample now shows a hugespread in its concentration profile as it makes its way between the contacts. Obviouslydispersion plays some role in this behavior, as it goes by the name "dispersive transport".Scientists had known about this "anomalous dispersion" since 1957 but it took nearly twodecades before Scher and Montroll presented the solution to the problem mathematically.


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Figure 1: (a) Normal transport shows a

drifting packet of carriers. As long as thepacket travels, we can detect a currentproportional to the amount of carriers active.As they reach the opposing contact, thecurrent rapidly declines to zero. TheGaussian-shaped packet widens slightly as ittravels due to diffusion about the mean. Afew extra fast carriers reach the far contactsooner than the bulk of the carriers, while afew stragglers take up the rear. (b) Indispersive transport, the velocity of thecarriers varies over a wide range so that theoriginal narrow impulse of carriers quickly

spreads out as it drifts and diffuses acrossthe width. This gives a long tail to thephoto-response profile.The stragglers keep arriving in this "fat tail"world, with progressively fewer in number asthough they had joined and tried tocomplete a long marathon race (seeMarathon Dispersion).

2

Figure 2: The typical measurement in thephoto-response current starts with a spikefollowed by a soft plateau, then a shoulder

or transition region, followed by theubiquitous long tail. Shapes of the currentprofile taken over a range of experimentalconditions show invariance in the generalshape with respect to the electric field andspecimen thickness. Importantly, this profiledoes not follow from the expected spread ofthe Gaussian packet. Scale invariance oruniversality manifests itself in statistics ifone can first transform the ordinates intodimensionless quantities while conservingthe moments. Transport of holes andelectrons near the absorption region

electrode contributes to the initial spike,which has no impact on the longer tail due tothe complementary carrier type (thistransient spike is also known as prompttransport)

567

2"The physics of amorphous solids" Richard Zallen, Wiley-VCH, 1998

http://mobjectivist.blogspot.com/2008/09/marathon-dispersion.html


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Figures 1 and 2 give a qualitative view of the dispersive transport that occurs in adisordered semiconductor. The first figure describes what we think happens internally andthe second figure provides a view of the observable result. Figures 3 and 4 illustrate acouple of experimental results of widely studied amorphous (a-As2

Se3) and organicmaterials (TNF-PVK).

Figure 3: Typical experimentalTime-Of-Flight curve shows a set ofsuperimposed measurements from an earlyScher-Montroll behavior which exhibited the"universality" property of the scaling acrossdifferent measurement conditions (theapplied voltage in this case).

Figure 4: The TOF curve of an organicsemiconductor illustrates the characteristicknee that Scher and Montroll had predicted;the two slopes differ but must sum to -2according to their theory.

The anomaly in the title of the Scher-Montroll paper referred to the fact that no one

previously could formally explain the long tails in the response (so called "time of flight")measurements. Other researchers clearly had an inkling that it had something to do withthe high amounts of disorder leading to greater amounts of diffusion and dispersive spreadthan in an ordered material. Amorphous materials naturally have many inhomogeneities,defects, and carrier traps that can lead to varying delays in transit time. Scher and Montrollderived a statistical formulation of random walk called the Continuous Time Random Walk(CTRW) that they then applied to the experimental results.

Most experimentalists around that time got good results using the CTRW formulation so thatit has become fairly well accepted in semiconductor circles for the last 30 years. The mathgets fairly hairy in spots, and soon experimentalists began to simply use the empiricalsloped lines to get at the Scher-Montroll disorder parameter (ɑ = alpha). High values ofalpha indicate more order and low values more disorder (0 < ɑ < 1).

Research continues in understanding dispersive transport as new electronic materials comeon line. Physicists have had a long-standing interest in disorder, as the finding of anorder/disorder transition easily classifies as a type of "holy grail" discovery, certain to elicitoohs and aahs from their colleagues. As the figure to the right shows, one can add acontrolled amount of disorder to a sample and observe the results of diffusive transport. Theupper TOF trace shows linear transport while the lower trace shows the effects of dispersivetransport. In the latter case the disorder comes in the way of the intentional introductionof impurities, apparently forming electronic traps which slow down the carrier motion as ittraverses the width of the sample.

Generally I have noticed that the basis for much of the current research has to do with

finding some novel aspect of dispersion relating somehow to material properties. In reality,the rather mundane effect of randomness due to heterogeneity likely plays a far moreimportant role. For many of these materials, we simply can't control the distribution ofdefects and traps and the disorder evolves into a garden variety randomness, with which wehave a single mean rate, say average drift velocity, to characterize the behavior.

If you look at the curve to the right, the red line shows my simple assumption for maximumdisorder. I had noticed this same shaped curve in my studies of dispersive discovery that Iposted to TOD [], and had a hunch that I could use the same formulation in the


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semiconductor case. After all, as I assert that dispersion is just dispersion, and I haveenough experience dealing with semiconductor physics that I didn't expect any gotchas. Asfor the ideas of Scher and Montroll, I turn their formulation upside down and don't evenconsider a random walk premise, as this leads to overly complex math.

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Breakthrough

I started to look at this problem because I had a nice intuitive way of modeling dispersivebehavior in oil production [google links] and figured that I could try applying my generaldispersive model to dispersive transport. And that I could do it much more simply than theapproach by Scher and Montroll. At certain places in their seminal papers and reviewarticles, I find passages that amount to "... and then a miracle occurs" and knew that thismeant some messy first-principles work had gone missing. The way I turned their model onits head basically amounted to working in the rate domain, corresponding to velocities,instead of the time domain The latter derives from the classical work used to describe

everything from Brownian motion to large scale diffusion. The former relates to a more orless pragmatic view of the world which relies on entropy considerations instead of thestatistics of hopping over energy barriers with small probabilities.

As a basic premise, I use the Maximum Entropy Model (MEM) to select a stochastic rateProbability Density Function (PDF or more precisely PMF for probability mass function) inwhich I can then derive dispersive transport.

that corresponded to random-walk hopping (see figure).1011 One way to choose the “right”distribution p is by using the principle of maximum entropy. This principle states that theleast biased probability assignment is that which maximizes the system entropy subjectto the constraints supplied by the available information.

For the constrained system of interest, all we really know is the mean carrier transport3

velocity. If we don't know the higher order moments, the MEM says to use a damped

exponential as the PDF to maximize entropy. In a general sense, this maximizes theamount of disorder that exists in this quasi-equilibrium system. It says that many slowcarriers exist, with an exponentially diminishing supply of fast carriers. For the fixedgeometry shown in the schematics at the top, the normalized expression for the timedependence of dispersed current reaching the far contact derives as follows:

I(t ) = I0

[1 - e-1/t (1 + 1/t )] (EQ 1)

This essentially describes the integral over a PDF of normalized velocities

p(v ) = (1/v 0)exp(-v/v 0 )(EQ 2)

for carriers that have not yet swept through the transport layer. We assign t T = d/v 0 = 1 toshow the scale invariance of the result of Equation 1. Crucially the formulation maintainsthe moments of the distribution. If the velocity distribution becomes dispersed as a dampedexponential then the cumulative position distribution of a particle/carrier also advances by a

3A MAXIMUM ENTROPY ANALYSIS OF SINGLE SERVER QUEUING SYSTEM WITHSELF-SIMILAR INPUT TRAFFICA. Asars, E. Petersons

http://mobjectivist.blogspot.com/2008/08/general-dispersive-discovery-laplace.htmlhttp://mobjectivist.blogspot.com/2008/08/general-dispersive-discovery-laplace.htmlhttp://www.google.com/search?as_q=Dispersive&btnG=Google+Search&as_sitesearch=mobjectivist.blogspot.com


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damped exponential. Nothing more to it than that!

I pulled out the fundamental transport coefficients such as mobility and the diffusionconstant for the time-being as this assumes that a uniform drift plays the prominent role. Inthe normalized case, I show the response profile below superimposed on the figure fromKao. At a subjective level, it follows the qualitative plateau/decline behavior quite well.

Figure 5 : The maximum entropy dispersion for time of flight according to the normalized(EQ 1)

Since we know that dispersion plays a role in the transport, we just have to figure out howto use the much simpler dispersion formulation instead of the hideous Scher-Montrollderivation. The key to understanding physics is to keep it simple, but not too simple (quoting Einstein I believe). In fact the maximum entropy formulation that I had usedpreviously in the dispersion analysis for oil field sizes, discovery, and reserve growth, Iretain in this analysis. Also known as the "method of least information", it essentially relieson using common sense in not trying to under- or over-estimate the variance of thedispersive spread. In one sense, the interpretation I make looks similar to the schematic atthe right. I assume the equivalence of multiple mobility pathways through the device.

For any one pathway, the advance in the particles motion has a diffusive component as wellas a drift component. This leads to an expression involving time as shown below

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= sqrt( Dt + (v 0 t )2 ) (EQ 3)

This essentially incorporates the concurrent diffusion component along with the driftcomponent of the velocity and we can make an implicit transform into the actual timeline.The drift velocity v 0 relates to the electric field by v 0 = uE , where u is the carrier mobility(pronounced "mu") and E is the electric field strength. I would consider this a routineparametrization into a Hilbertian space where we can maintain moments of the distributionsacross dimensions, /t T = /w .

http://www.ee.technion.ac.il/orgelect/Spatially_dispersive_transport-A_mesoscopic_phenomenon_in_disordered_organic_semiconductors.pdfhttp://www.theoildrum.com/node/4311http://www.theoildrum.com/node/2712http://mobjectivist.blogspot.com/2009/02/usa-field-size-distribution-update.html


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I(t ) = I0 [1 - exp(-w /sqrt(Dt + (vt )2

)) (1 + w /sqrt(Dt + (vt )2))](EQ 4)

Qualitatively the constant (drift) velocity drops as t -2 while diffusional velocity drops by t -1 . Iam not certain whether the formulation by Scher and Montroll take this into account. They

simply say that long-range correlations go as t-a-1

when they set up their CTRW model. Ibelieve this step links my exponentially damped rate dispersion to their long range timecorrelations.

Many experimental results show the knee in the curve of Scher and Montroll, but withusually not much dynamic range. I looked at a few material studies done fairly recently tosee how well the simple theory works.

Transport in SiO 2

For verifying any theoretical formulation, you usually want to match the behavior to as widea dynamic range as experimentally feasible. The larger the dynamic range in the measured

quantities, the more confidence that you have in its worth or value.

Figure 6: Dispersive transport via the MEM

model compared to SiO2 measurements.This shows mainly diffusion with the driftcatching up at longer times.

Figure 7 : The effect of changing the width

of the transport layer. The Montroll-Scherknee does not show up prominently.

The case of carrier transport across SiO2

insulating layers for MOS devices provides somecases of amazing dynamic range, up to 8 orders of magnitude in current. I took data fromthe text "Ionizing radiation effects in MOS devices and circuits" by Ma and Dressendorfer.The fundamental idea remains the same in this situation as the photo-response experiment,


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although a different form of ionizing radiation supplies the pulse of carriers -- in this caseholes become the charge carrier instead of electrons. Otherwise, the same diffusivetransport occurs, with the authors trying to explain the results by applying the sameunwieldy Scher-Montroll formulation. As a side note, these kinds of measurements need adelicate touch as the dose of the radiation can actually effect the field due to space chargeformation. I did some pioneering work on a similar experiment years ago where I tried to

force dopant concentrations via ion bombardment into a growing junction and the bias ofthe junction alone pulled the mobile dopants from one side of the junction to the other. Thekey is that even though you see weird stuff happen, you can always explain it via somerather elementary considerations.

In any case the fits to the data using the simple diffusive transport model works over alarge dynamic range in ordinates. The sharp bend near the top indicates the potential startto the plateauing, and one can observe that some of the pairs of data indeed do flatten out.The other gradual bend indicates the transition between diffusion transport and drifttransport. The universality of this bend does not scale perfectly as drift does depend on theelectric field whereas the diffusion doesn't. And as we will see in the next example, thetemperature may not play a big role in deviations from universality.

Transport in a-Si:H

Amorphous semiconductorshave a huge influence on thesolar cell and photovoltaicindustry. In general, it costsmuch less to manufactureamorphous materials as thefabrication facilities do nothave to follow as strict amaterial process.Unfortunately theperformance characteristicsof the amorphous silicon incomparison to its crystallinebrethren leaves lots of roomfor improvement. Althoughnot as important for solarcells, the photo-responsetime for an incident lightstimulus shows the long tailscharacteristic of diffusive

transport.

I culled the data from a 2005paper by Emelianova, et al studying the photo-response of amorphous hydrogenated Silicon (a-Si:H). This material wasundoped and the investigation looked at hole carriers. I found this study verycomprehensive and it leans toward questioning the applicability of Scher and Montroll'soriginal formulation in terms of an alternate model that they formulate.For the curves fit to the right, I used the simple expression in Equation 4 and plugged in the

http://joam.infim.ro/JOAM/pdf7_2/Emelianova.pdfhttp://joam.infim.ro/JOAM/pdf7_2/Emelianova.pdf


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coefficients as stated in the legend and the table below. I have never seen a spanking newmodel that popped out with such obvious agreement in my life. This basically should setthe hairs on end; I really believe that after 50 years of first discovering the anomalousdispersive transport that a simple equation, suitable for spreadsheet entry, would agree sowell.

Mobility (u) Width (w ) Diffusionconstant (D)

Temperature(T)

CurrentScaling (C )

Electric Field(E)

0.00193cm2 /V/s

2.4 microns 0.69 * u 264 1.2e-13 varies as infigure (V/cm)

t/t 0 = normalized time = sqrt (2Dt + (uEt)2)/w (EQ 5)

I(t ) = I0 [1 - exp (-t 0 /t )(1 + t 0 /t )] (EQ6)

where I0

= C*E*E

The idealness of the fit should preclude me from over-analyzing the results but a fewinteresting issues remain.

(1) For one, in this case the relation between mobility and diffusion constant does not obeythe Einstein relation but this rarely happens in non-ideal and disordered materials as theenergy states get sufficiently smeared across the bandgap. The general Einstein relationrelates the diffusion constant D to the energy distribution of the carrier states.

D = u/q [N(E c )/N'(E c )]4

Diffusion exists in the absence of an electric field and so thermal energy acts as the onlystimulus to allow a carrier to move to an adjacent site. For a narrow variation of E c around

the Boltzmann distribution, the relation D=u/q*kT holds as an invariant, but as E c spreadsout -- and in the maximum entropy case of a large variance knowing only the mean -- thediffusion constant tracks E c more than it does temperature, T . I worked it out andD=u/q*(kT+E c ) in that case. Since E c typically exceeds the statistical value of thermalenergy, kT , we will see a higher diffusivity than one would expect from an ordered solid(see Schiff).

(2) Also, the initial transient spike has to do with the collection of complementary carriers atthe near electrode, and has no influence on the results (undoped material generates equalnumber of oppositely charged carriers). As a probability exercise, the results also show thatthe integrated area under each curve is identical to within 0.2% for each voltage bias. Infact the cumulative charge collection based on Equation 1 becomes the following simpleformula:

Q(t ) = I0

t*(1 - e-1/ t ) (EQ 7)5

4Hydrogenated amorphous silicon, R. A. Street

5The Maxwell-Boltzmann is an approximation to the actual Fermi-Dirac distribution at highertemperatures.


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The build-up of charge starts linearly and then converges asymptotically to a valueproportional to the total number of carriers generated during the pulse duration (exceptingrecombination and other losses).

The authors apply their own model to the results and suggest that the dispersion is wider

than gaussian as the figure to the right shows, yet they also curiously indicate that is agaussian non-dispersive transport. Much of the confusion arises from the originalScher-Montrose formulation which demarcates the curves into ordered or non-disorderedinstead of what I would like to see -- a dispersed diffusion-dominated regime versus adispersed drift-dominated regime.

The upshot of the good agreement of my fundamental model with the results means thatany smart electrical engineer can start using the simple formulation right now, and shouldthat engineer want to calculate frequency response or impulse response of an amorphousmaterial device, they just have to use Equation 6. They can do FFT or Laplace transforms oranything they want since they have an analytical result which they can plop into theirnotebook or spreadsheet or Matlab and work out. I guarantee no one would want to messwith the Montroll-Scher result as it gets way too unwieldy and I dare say that no one

actually understands it. I consider this simplicity a huge benefit.

The only caveat: you need a disordered material to apply this to .... but, of course, thatgoes with the premise.

Quantum Dots

Scientists have looked to unique materials including a variety of organic semiconductors inthe hope of creating structures suitable for quantum dot devices. This paper Charge CarrierTransport in Poly(N-vinylcarbazole):CdS Quantum Dot Hybrid Nanocomposite provides a fewtime-of-flight curves in terms of a completely different material system. These TOF'sappear to obey the same simple maximum entropy model for dispersive transport as youcan see in Figure 8 and Figure 9.

Figure 8: TOF traces taken at differentapplied electric fields. The original diagramdid not have dimensions on the axis so I

Figure 9: All curves plotted on a universalscale. The t-2 drift dependence extendsbeyond the range of the data.

http://laserspark.anu.edu.au/lpc/pdf/JPCB108-1556-04.pdfhttp://laserspark.anu.edu.au/lpc/pdf/JPCB108-1556-04.pdfhttp://mobjectivist.blogspot.com/2008/08/general-dispersive-discovery-laplace.html


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guessed on the scaling based on the inset. Ishow the simple dispersive transport modelas symbols with the electric field dependenceas in Equation 6

Of course good agreement means that the disorder in the systems has to agree with the

maximum entropy model. Nothing precludes different diffusion mechanisms or even furtherdisorder, implying even fatter tails than t-1 . Some systems likely exist with a mix of orderand disorder, such as crystalline semiconductors with many defects. In that case, one couldconceivably separate out the effects.

The Connection

"When the weird gets going, the weird turn pro" -- Hunter S. Thompson

I got sidetracked into the dispersive transport behavior of carriers in disordered solid-statematerials as I searched for ideas that might substantiate the oil depletion models that I had

worked on. I have long asserted that everything about the behavior of oil, from reservoirsizes, to oil discovery, and on to reserve growth has as a basis the effects of dispersion. Justas the disorder in amorphous semiconductors causes a dispersion in carrier velocities, so toodoes the randomness and disorder in aspects of the fossil fuel process. Whether therandomness has to do with varying velocities in the drift of oil over eons or the variance ofhuman search efforts (see figure at right), these all lead to the same fundamentalformulation for dispersive analysis. Moreover, any chaotic or complex behavior getssmoothed out by the filter of dispersion. I essentially derived a new math shorthand todescribe oil, and stumbled across the fact that this same derivation applies equally well to afield totally removed from the macroscopic. I essentially went from the macroscopic to themicroscopic, and then back again to substantiate what I had earlier conjectured.

Recall again the two scientists Scher and Montroll, who originally formulated the CTRW

theory to explain dispersive carrier transport. They essentially worked as appliedmathematicians and have gained quite a bit of recognition for their ideas. Montroll, arguablythe more well-known of the two has since died, but Harvey Scher has continued on applyingthe same formalism to other application areas.

Guess where he has applied it?

Answer : Transport of materials underground via porous structures ... as you may haveguessed, pretty much the same life-cycle that petroleum operates under. So Scheressentially transitioned from the microscopic world of semiconductors to the macroscopicworld of the earth. Currently Scher works as a consultant for a group of geologists andenvironmental scientists that use the CTRW theory to explain the way that contaminationand other solutes spread over time via diffusive transport.

I have problems with the CTRW theory in that at a certain step in the derivation, theauthors invoke the legendary "and then a miracle occurs" argument into the proof. Thisturns into the essential observation that long-range correlations go as 1/talpha . Well, I cangenerate that just by invoking the Maximum Entropy assumption on the variance ofvelocities. In that case, the inverse time power-law behavior naturally takes over and anintegral exponent depending on the mean velocity from the specific type of motion occurring-- either diffusion (t-1) or drift (t-2). If a combination of the behaviors occurs, just solve theclassical equations of motion assuming Fick's Law and calculus, and the characteristic

http://mobjectivist.blogspot.com/2008/11/comprehensive-oil-depletion-model-life.htmlhttp://mobjectivist.blogspot.com/2008/10/significant-no-hyperbole.htmlhttp://www.theoildrum.com/node/3287http://mobjectivist.blogspot.com/2008/10/dispersive-discovery-field-size.htmlhttp://mobjectivist.blogspot.com/2008/10/dispersive-discovery-field-size.htmlhttp://mobjectivist.blogspot.com/2008/11/comprehensive-oil-depletion-model-life.html


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dispersive formula appears, just as for dispersive transport in amorphous semiconductors.

17181920

Figure 10: Application of the dispersivetransport to the motion of solute. Thisexperiment showed a transition as the solutemigrated over time.

Analysis of Tracer Test Breakthrough Curvesin Heterogeneous

Figure 11: The illustration shows some ofthe causes of pore-scale dispersion. Solutetraveling more tortuous pathways betweensediment grains will move more slowly thanthat moving along more direct pathways.Diverging pathways will also cause thecontaminant to spread perpendicular to theaquifer flow direction.

Double Breakthrough

Figure 12: Uranine dye moving downstream inFisher Creek after injection to trace thedestination of the water as it disappears.

(Groundwater Tracing in the Woodville Karst Plain)

Most of the solute transport measurements use something called breakthrough curveanalysis. "Breakthrough curves" enable a researcher to estimate the amount of dispersionoccurring in a flow of solute (or contamination or whatever) in a media. For anon-dispersive flow, the breakthrough curve looks like a unit step where the tracer materialis detected abruptly at a specific time at a certain point downstream. This has an analog tothe Time-of-Flight measurements used in photo-response studies described earlier. But dueto randomness and variability in the media due to pore structures (for example), the

http://www.gue.com/?q=en/node/798http://wrhsrc.oregonstate.edu/briefs/brief_9.htmhttp://www.weizmann.ac.il/ESER/People/Brian/CTRW/ctrwold/docus/DetailedDescription/gw_web.htmlhttp://www.weizmann.ac.il/ESER/People/Brian/CTRW/ctrwold/docus/DetailedDescription/gw_web.html


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dispersion smears the breakthrough curve over a broad time window.

A very simple model for a breakthrough curve involves solving the equation for a maximumentropy spread in velocities (for a given mean velocity) at a specific distance L. Given theaverage time taken is T=L/v , and a random variate would take time = t , then thebreakthrough curve looks like exp(-L/vt ). If you plot this curve it looks like what some

people refer to as a "reciprocal exponential". It isn't the classic exponential because thetime parameter goes in the denominator. That happens because we are dealing with rates,and not time for the stochastic parameter.

Next, we must realize that an idealized breakthrough curve assumes a fixed separation, L,in a very controlled experimental environment. In reality, the distance L's become spreadout over space and for a maximum entropy PDF of L, an uncontrolled "breakthrough curve"will have a temporal behavior that looks like 1/(1+L/vt ), where L becomes the meanseparation. This looks exactly like the formula for enigmatic reserve growth in oil discoveriesthat I derived before [ref].

I am satisfied with using a maximum entropy estimator for the dispersion because theeffects could be due to many different possibilities. So, in a sense, variability overrules

complexity and if we can concentrate on understanding the mean value, we have a verysimple way to characterize the system. That is my premise and I have to be able to defendit from many angles.

Take a look at the figure to the right from a hydrogeology experiment. Granted, I do notknow anything about the particulars of the particular experiment, yet I assert that I can doa better job of fitting to the results solely because I do not place a bias on my estimator. Isimply apply maximum entropy to randomize the effect, making the only assumption themean transport rate2122.

I see some indication that Scher and his colleagues have at least considered this simplepremise. From the following extract, note that they imply that some sort of ensembleaverage acts as a precondition to further analysis. In other words, they make the

presupposition that geology is random and uncontrollable, just like amorphoussemiconductors and human processes.

The point average of v and D can be very sensitive to small changes in the local volumeused to determine the average. Conversely, if one fixes the volume to a practical pixelsize (e.g., 10 m3) the use of a local average v and D in each volume can be quitelimited, i.e., the spreading effects of unresolved residual heterogeneities are suppressed[e.g., Dagan, 1997]. We will return to this issue in a broader context in section 4. Itessentially involves the degrees of uncertainty and its associated spatial scales. We start, at first, with an ensemble average of the entire medium and discuss therole of this approach in the broader context .

-- "Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, RandomWalk and Fractional Derivative Formulations." Brian Berkowitz, Joseph Klafter, Ralf Metzler,

and Harvey Scher

Back to Oil

As a very general technique we can apply the equivalent of breakthrough analysis acrossmany domains. The usual problem remains that different application domains use differentterminology. I never used breakthrough analysis terminology because no one doescontrolled experiments when they look for or extract oil. Oil exploration is a commercialenterprises after all and oil prospectors get what they can, while they can, and don't

http://www.ws.chemie.tu-muenchen.de/groups/hydrogeo/research/e18/


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necessarily ponder any deeper meaning. Yet, I view the over-riding dispersion analysis as avery general concept and I simply apply the same technique in oil depletion by making theanalogy to dispersion in human-aided discovery search rates. The fact that it also occurs forphysical processes such as contaminant flow in groundwater, carrier transport in amorphoussemiconductors, or TCP dispersion should not surprise anyone.

Over 50 years have lapsed since the day that Hubbert first sketched a Logistic curve tomodel oil depletion, and I think science has had a mental block on the dispersion problemall this time. We can easily and simply explain the dynamics of the oil production curve byusing these same ideas from dispersion analysis.

Getting meta for a moment, to you I exist only as a blogger. I don't know if this analysis willgo anywhere. As you may realize, I have some good ideas on the way we can analyze oildepletion. Yet, I have no credentials in that field. A pseudonymous writer can only waysway an argument based on the logic of his arguments. If you can follow the argument inthis post, and believe it applies, and that all experimental evidence backs up the theory, mycredibility builds. Someone will then say, "well, he got that part right, maybe this other partmakes some sense". As far as I can tell, no one has documented a similar simple approachto what I have formulated via my blog postings. It takes a bit of intuition to determine the

situations where disorder and diversity rules and where it does not. I say that where youcan appropriately apply these arguments you can start to understand the dynamics. I cancertainly understand the dynamics of the Hubbert curve via dispersion just as I canunderstand the transient of an amorphous semiconductor time-of-flight experiment byapplying dispersion. The fact that no one else sees it this way turns my task into one ofsalesmanship, unfortunate, since no one likes a salesman.

Putting that aside, I assert that bottom-line we really can use fundamental concepts tounderstand the dynamics of these behaviors. Absolutely nothing about any of theseempirical observations I would consider anomalous. No one resorts to calling the variabilityin rain "anomalous" and why we haven't universally figured out simple solutions asdescribed here remains the real mystery. Or that is indeed the real anomaly.

WHThttp://mobjectivist.blogspot.com

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FOOTNOTES
http://www.theoildrum.com/node/4171http://www.theoildrum.com/node/4171http://mobjectivist.blogspot.com/2007/06/finding-needles-in-haystack.html

Dispersive Transport

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