A Brief Survey of Lévy Walks

Karlstads universitet 651 88 Karlstad Tfn 054-700 10 00 Fax 054-700 14 60

[email protected] www.kau.se

A Brief Survey of Lévy Walks with applications to probe diffusion

Lars Fredriksson

Diploma Thesis for Bachelor’s Degree in Chemistry Supervisor: Gunilla Carlsson Examiner: Jan van Stam

Investigating Lévy motions

Lars Fredriksson

Department of Chemistry, Karlstad University

I

A brief survey of Lévy walks with applications to probe diffusion

En översikt över Lévyprocesser applicerat på probdiffusion

Lars Fredriksson

Department of Chemistry

Bachelor Degree Project Bachelor of Science in Chemistry

Denna rapport är en deluppfyllelse av kraven för kandidatexamen. Allt arbete som inte är

mitt eget har identifierats och referenser upprättats.

A Brief Survey of Lévy Walks

Lars Fredriksson


II

Abstract Lévy flights and Lévy walks are two mathematical models used to describe anomalous diffusion (i.e. those having mean square displacements nonlinearly related to time (as opposed to Brownian motion)). Lévy flights follow probability distributions |)(|rp yielding infinite mean square dis-placements since some rare steps are very long. Lévy walks, however, have coupled space-time probability distributions penalising very long steps. Both Lévy flights and Lévy walks are domi-nated by a few long steps, but most steps are much, much smaller. The semi-experimental part of this work dealt with how fluorescent probes moved in systems of cationic starch and latex/ solutions of dodecyl trimethyl ammonium bromide, respectively. Visually, no Lévy walks could be detected. However, mathematical regression suggested enhanced diffusion and subdiffusion. Moreover, time-dependent diffusion coefficients were calculated. Also examined was how Micro-soft Excel could be used to generate normal diffusion as well as anomalous diffusion.

Sammanfattning Lévyflygningar och Lévypromenader är matematiska modeller som används för att beskriva anomal diffusion (i.e. dessa då medelvärdet av kvadratförflyttningarna är icke-linjärt relaterat till tiden (till skillnad från Brownsk rörelse)). Lévyflygningar följer sannolikhetsfördelningar |)(|rp som ger oändliga medelkvadratförflyttningar eftersom vissa steg är väldigt långa. Lévyprome-nader, å andra sidan, har kopplade rum-tid-sannolikhetsfördelningar som kraftigt reducerar de mycket långa stegen. Både Lévyflygningar och -promenader domineras av ett fåtal långa steg även om de flesta steg är mycket, mycket mindre. Den semiexperimentella delen av detta arbete studerade hur fluorescerande prober rör sig i katjonisk stärkelse respektive latex/lösningar av dodecyltrimetylammoniumbromid. Inga Lévypromenader kunde ses. Emellertid talade matematisk regression för att superdiffusion och subdiffusion förelåg. Tidsberoende diffusionskoefficienter beräknades också. I detta arbete undersöktes även hur Microsoft Excel kan användas för att generera både normal och anomal diffusion.


Lars Fredriksson


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Contents

1. INTRODUCTION...................................................................................................................................... 1

2. THEORY .................................................................................................................................................... 2 2.1 SOME MATHEMATICAL NOTES ..............................................................................................................................................2 2.2 DIFFERENT KINDS OF DIFFUSION .......................................................................................................................................4 2.3 LÉVY FLIGHTS..........................................................................................................................................................................6 2.4 THE CTRW APPROACH: LÉVY WALKS ................................................................................................................................7 2.5 RELATING MEAN SQUARE DISPLACEMENTS TO THE DIFFUSION COEFFICIENT D......................................................9 2.6 RICHARDSON’S LAW..............................................................................................................................................................11 2.7 THE LÉVY CONCEPT IN BIOLOGY ......................................................................................................................................12 2.8 THE LÉVY CONCEPT IN PHYSICAL CHEMISTRY................................................................................................................15 2.9 ANALYSING EXPERIMENTAL DATA....................................................................................................................................16

3 METHOD...................................................................................................................................................17 3.1 ANALYSING CARLSSON’S DATA ..........................................................................................................................................17 3.2 INVESTIGATION OF SELECTED CARLSSON’S FILMS.........................................................................................................18 3.3 INVESTIGATION OF SELECTED HEIDKAMP’S FILMS .......................................................................................................18 3.4 SIMULATIONS OF DIFFUSION IN MICROSOFT EXCEL ....................................................................................................19

4 RESULTS AND DISCUSSION..................................................................................................................21 4.1 ANALYSING CARLSSON’S DATA ..........................................................................................................................................21 4.2 THOROUGH INVESTIGATION OF SELECTED CARLSSON’S FILMS ..................................................................................26 4.3 ANALYSING HEIDKAMP’S FILMS ........................................................................................................................................33 4.4 SIMULATIONS OF DIFFUSION IN MICROSOFT EXCEL ....................................................................................................50

5 CONCLUSIONS............................................................................................................................................

6 ACKNOWLEDGEMENTS ....................................................................................................................... 56

REFERENCES............................................................................................................................................ 56

APPENDIX 1. CORE FORMULAE............................................................................................................. 59


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1. Introduction Diffusive processes are crucial in many applications, not only in chemistry. Theories of diffusion can also be used to construct models in geology, ecology and even in the social sciences. More-over, the concepts can be applied when it comes to networks, even though lengths in this case may have to be regarded as discrete.1 Diffusion is a kind of random walk. The history of describing such motions goes back at least three hundred years to works of two Bernoullis. In 1713, Jacob Bernoulli published Ars Conjectandi, in which he developed the concept of random walks concerning the game of chance. His nephew Daniel worked out a seed to the modern kinetic theory of gases. His work contained the concept of random flights.2 The most well known kind of diffusion is probably Brownian motion, a zig-zag, erratic motion always the case of all relatively small particles. This phenomenon was described by Robert Brown in the early 1800s. Motions of this kind were known earlier, but then thought to be caused by living organisms. What was revolutionary about Brown's work was that he demon-strated that this kind of motion occurred independently of the origin of the particles, i.e. no life spirit was called for – and he also showed that these motions were not caused by pressure or temperature differences.3 In this work, two different kinds of random walks known as Lévy walks and Lévy flights will be presented. Then, some of these theories will be applied to studies of diffusion of latex probes in some real systems (vide infra). Lévy walks and Lévy flights, in this paper collectively described as Lévy motion, can imply so-called superdiffusion. Besides, they are scale invariant and fractal processes, i.e. they look the same no matter if zooming in or not. 4 Thus, Lévy walks and Lévy flights can be said to be self-similar, i.e. no particular length scale is dominating. 5 As will be seen below, the overall motion in a Lévy flight is determined by a few rare steps. These patterns of motion are named after Paul Lévy, who in the 1920- and 1930-s developed the mathematics concerning the infinite mean-squared jump distances, which are characteristic of Lévy flights.6 Lévy flights were first used by physicists to describe photoconductivity in amorphous materials.7 Since then, these concepts have been applied in many areas. For example, Lévy motions have been used extensively in biological contexts to model the foraging patterns of a variety of organisms: dinoflagellates, fruit flies, bees, spider monkeys, sharks…8 Lévy statistics is also well suited to describe turbulent diffusion, slow relaxation in glassy materials and phases in certain kinds of electric junctions.9 In addition, the Lévy flight concept can be used to describe the motions of particles in crystals,10 turbulent diffusion, micelle dynamics, diffusion at liquid-solid boundaries, vortex dynamics and to improve image quality.11 Furthermore, it has also been used to model diffusion in rotating flows (vide infra).12

1 Viswanathan G.M., Raposo, E.P., da Luz M.G.E.. Lévy flights and superdiffusion in the context of biological encounters and random searches. (2008). Physics of Life Reviews, 5, 133-150 2 Shlesinger, Michael F., Klafter, Joseph. Random Walks in Liquids. (1989). Journal of Physical Chemistry, 93, 7023-7026 3 Lindley, David. Uncertainty. Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science. (2007). Anchor Books. 4 Viswanathan et al. 5 Shlesinger, Michael F., Zaslavsky, George M., Klafter, Joseph. Strange kinetics. (1993). Nature, 163, 31-37 6 Shlesinger et al (1993). 7 Weeks, Eric R., Swinney Harry L.. Random walks and Lévy flights observed in fluid flows. (1998). Springer-Verlag New York Inc. Available online: www.physics.emory.edu/%7Eweeks/papers/nst98.pdf 8 Viswanathan et al. 9 Shlesinger et al. (1993). 10 Greenenko, A.A., Chechkin, A.V., Shul’ga, N.F., Anomalous diffusion and Lévy flights in channeling. (2004). Physics Letters A, 324, 82-85. 11 Klafter, Joseph, Shlesinger, Michael F., Zumofen, Gert. Beyond Brownian Motion. (1996). Physics Today, 49, s.33-39. 12 Solomon, T.H., Weeks, Eric R., Swinney, Harry L.. Observation of Anomalous Diffusion and Lévy Flights in a Two-dimensional Rotating Flow. (1993). Physical Review Letters, 71, 3975-3978.


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2. Theory

2.1 Some mathematical notes In this section, some basic mathematical formulae and procedures will be reviewed in brief. In particular, two commonly encountered integral transforms will be included.

2.1.1 The mean square displacement Throughout this work, the quantity 2|| r is vital. This is termed the mean square displacement, i.e.

the mean of the square displacements 2||r of a probe. For two dimensions,

20

20 )()(|| yyxx r (1)

where ),( 00 yx is the initial position.

2.1.2 -functions The Dirac delta function can informally be defined as

ba if

if0

baba )( . (2)

The Dirac delta function is not a proper function in the common sense, but is called a generalised function i.e. while having an infinitely narrow breadth but an infinitely extended height. 13 It is

characterised by

1)( badx , i.e. the area of integration is exactly unity. Another characterist-

ic feature is that

)()()( 00 xfdxxxxf .14 So, the delta function here picks the very value

of x from where the function is peaked (the 'sifting' property of the Dirac delta function). 15 Another kind of delta-function is the conceptually related Kronecker delta defined as16

ba if0

if1)(

baba (3)

2.2.2 Fourier transforms Moreover, Fourier transforms are referred to. A well-known example from wave physics is the interference patterns caused by light passing through a pair of slits. Not surprisingly, they form the basis for X-ray crystallography. These transforms are often used to help solve differential equations. Carrying out a Fourier transformation is commonly thought of as changing the function from the time domain with the variable t to the frequency domain with frequency ν. The pair of variables t and ν is known as conjugate variables. Another pair of conjugate variables 13 Kreyszig, Erwin. Advanced Engineering Mathematics. 8th Edition. (1999). John Wiley & Sons, Inc. 14 Beddard, Godfrey. Applying Maths in the Chemical and Biomolecular Sciences – an example based approach. (2009). Oxford. 15 Sharma, Shama. Vishwamittar. Brownian Motion Problem: Random Walk and Beyond. 2005. Resonance. 10. 49-66. 16 Beddard.


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frequently encountered (especially in infrared spectroscopy) is distance (in cm) and wave-number (in cm-1). 17 In the following, t and ν will be used as an example of a conjugate pair. When having performed a Fourier transformation, the new function is known as the Fourier transformation, F, of the original function )(tf . The Fourier transformation can be defined in this way:18

)(e)(})(F{ 2- Fdttftf ti

(4)

The Fourier transform analyses the function )(tf into its constituent parts, i.e. revealing any repetitive parts of f. This feature is more easily seen by using Euler’s formula:19

dttittfdttftf ti

)2sin()2cos()(e)(})(F{ 2- (5)

Also noteworthy is that, in principle, one has to integrate from to , but in practice a much narrower range is usually possible since the function of interest may decrease fast outside a particular region.20 Since the Fourier transform has the property of being invertible, it is possible to recover the original function by taking the inverse Fourier transform -1F 21:

dFF tπi

21- e)(})({F (6)

2.1.3 Laplace transformation Laplace transformations are somewhat similar to the Fourier transformation. The ‘input’ variable is often t , often thought of as time. The output is a new function, F , with a new variable, commonly s , often thought of as frequency. The definition of the (unilateral) Laplace transfor-mation is as follows:

0

)(e)( sFdttff(t)L st (7)

where ste is called the kernel. The definition can thus be applied to calculate the transform. In order the get the original function back, i.e. )(sF )(tf , the inverse Laplace transform,

1L is applied.22

17 Beddard.

18 Note that i denotes the imaginary unit i.e. 1i . Some authors, like Beddard, scales this by 2

1 . Others, according to

Beddard, reverse the sign of the exponent. 19 Beddard. 20 Beddard.

21 Like the forward transform, this inverse transform can be scaled by 2

1 and/or having the reverse sign in the exponent.

22 James, Glyn. Modern Engineering Mathematics. Second Edition. (1996). Addison-Wesly Longman.


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2.1.4 Scale invariance When the whole looks the same as the constituent parts, the system is said to be scale invariant. This concept is widely known as fractals.23

2.1.5 Skewness coefficients Many statistical methods exist being able to tell how skewed (i.e. non-Gaussian) a distribution is. In its most elementary form a comparison between the mean and the median can be used. In this work the easily computed Pearson’s second skewness coefficient will be used24:

deviation standard

median)-(mean3skewness (8)

2.2 Different kinds of diffusion Simple diffusion is described by the simple linear relationship tt ~)(|| 2r (9) where )(|| t2r is the mean square displacement after time t.25 This motion is often described as Brownian.26 Equivalently, this means that the mean displacement |r| grows as the square root of t.27 These Brownian motions have a particle probability distribution (the ‘diffusion front’) that is Gaussian due to the Central Limit Theorem.28 If the linear equation (eq. 9) does not hold, anomalous diffusion is the case. This is char-acterised by a similar, though non-linear equation: tt ~)(|| 2r (10) If 1 the kind of diffusion is known as dispersive transport regime29 or subdiffusion30. The probability distribution ),( trP is non-Gaussian.31 If 1 the diffusion is known as enhanced transport regime32 or superdiffusion33. 1 is typical for so-called chaotic dynamics. 34 Motions with 2 are also known as ballistic35, which commonly have constant velocity36. For turbulent motion, 3 .37 As

23 Klafter et al. (1996). 24 Wolfram Mathworld. Pearson's Skewness coefficients. mathworld.wolfram.com/PearsonsSkewnessCoefficients.html (as by 21th Nov. 2009) 25 Klafter, J., Blumen, A., Zumofen, G., Shlesinger, M.F.. Lévy walk approach anomalous diffusion. (1990). Physica A, 168, 637-645 26 Shlesinger et al. (1993). 27 Bouchaud, J.P., Ott, A., Langevin, D., Urbach, W.. Anomalous Diffusion in Elongated Micelles and its Lévy Flight Interpretation. (1991). Journal de Physique II France, 1, 1465-1482. 28 Bouchaud. 29 Klafter et al. (1990). 30 Greenenko et al. 31 Klafter et al. (1990). 32 Klafter et al. (1990). 33 Greenenko et al. 34 Klafter, J., Blumen A., Shlesinger, M.F.. Stochastic pathway to anomalous diffusion. (1987). Physical review A, 35, 3081-3085. 35 Shlesinger et al (1993). 36 Weeks, Eric R., Swinney Harry L.. Random walks and Lévy flights observed in fluid flows. (1998). Springer-Verlag New York Inc. Available online: www.physics.emory.edu/%7Eweeks/papers/nst98.pdf 37 Klafter, J., Blumen A., Shlesinger, M.F.. Stochastic pathway to anomalous diffusion. (1987). Physical review A, 35, 3081-3085.


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opposed to normal diffusion, a few rare steps determine the general motion in the case of anomalous diffusion38 [at least in enhanced diffusion]. Most commonly, anomalous diffusion is subdiffusive. 39 However, most cases are readily described by Brownian diffusion and its Gaussian distribution. In order to replace Brownian diffusion with anomalous diffusion, there must be strong reasons, or ‘pathologies’ present.40 Diffusion is often described by the diffusion equation, i.e. Fick’s second law. This equa-tion allows jumps with infinite velocity since a jump with an arbitrary length has a probability greater than zero. The opposite is true for the telegrapher equation, derived by Goldman. This equation only deals with finite velocities.41 The diffusion equation as applied to probability densities (with the diffusion constant

1D ) is

),(),( ttt

rr

r 2

2

(11)

where 22 r / is the Laplace operator [commonly written 2 ]. This diffusion equation leads to a Gaussian distribution and a linear dependence: t~|| 2r .42 To Fourier transform equation (11), the following is done to convert from the ordinary displacement-time space to the Fourier-Laplace space: ),(),( ut kr . In the limit 0u ,0k , the Fourier transformation solution of equation (11) is:

)(

),( 2

1ku

u

k (12)

A modified diffusion equation is needed to allow anomalous diffusion. One such is:

),(),(),( ttktt

rr

rr

r

(13)

A Fourier transform solution to this equation is (using 2ttk ~),(r ):

4

21ukC

uu ~),(k . (14)

This solution corresponds to 32 tt ~)(||r . Generally, the following equation corresponds to

tt ~)(|| 2r in the Fourier-Laplace space:

1

21

ukC

uu ~),(k . (15)

38 Weeks et al. (1998). 39 Bouchaud et al. 40 Bouchaud et al. 41 Shlesinger et al. (1989). 42 Klafter et al (1987).


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‘[W]ell behaved functions’ have:

02

22

k

kk

rrr ),(),()(|| tdrtt 43.

2.3 Lévy flights As mentioned in the introduction, Paul Lévy worked on these special kinds of diffusion in the 1920- and 1930-s seeking scale invariant distributions. 44 Scale invariance is the case with normal, i.e. Gaussian, distributions since the sum of Gaussians is again a Gaussian. Lévy, however, show-ed that scale invariances could be obtained also by using other distributions: 45 He found scale in-variance when

bn kp |k|nconstant -e)( with ]2,0]b n denotes the number of steps. b is significant: it

reveals the dimension of the point set that is visited by the Lévy flight. It also reveals the kind of distribution and hence the kind of motion: 2b means Gaussian distribution, whereas

20 b indicates Lévy flights.46 Physically, a Lévy flight is not a smooth process like those described by the diffusion equa-tion. Rather, they have some very long displacements and also many smaller, thus acquiring a very special shape with some long jumps (‘flights’) linked to small patches made up of a large number of tiny steps.47 For Lévy flights, the exponent in tt ~)(|| 2r is larger than one but typically smaller than two.48, (49) Lévy flights have immediate displacements requiring no time, which means they possess an infinite velocity.50 They follow a probability distribution:

1|~|)( rrp . (16)

Steps of all sizes occur. 51 The Fourier transformation coupled to eq. (16) is ttkP

|k|)-(constante~),( . (17) Using the Fourier transformation, it is possible to show that the mean square displacement,

)(|| t2r , is divergent52, which may be written 2||r . The exponent in the Lévy distribution above, here denoted , is known as the Lévy index and ranges from 0 to 2.53 So, if 2 , then the mean square displacement is divergent.54 What is the meaning of infinite mean square displacement? Although most steps do have a finite size, not all have. This is what makes the mean square displacement, )(|| t2r , infinite.55

43 Klafter et al (1987). 44 Shlesinger et al. (1993). 45 Klafter et al. (1996). 46 Shlesinger et al. (1993). 47 Shlesinger, Michael F. Physics in the noise. (2001). Nature. 411, 641. 48 Weeks et al. (1998). 49 Viswathan et al. 50 Fowlkes, J.D., Hullander, E.D. Fletcher, B.L., Retterer S.T., Melechko, A.V., Hensley, D.K., Simpson, M.L., Doktycz, M.J. Molecular transport in a crowded volume created from vertically aligned carbon nanofibres: a fluorescence recovery after photobleching study. (2006). Nanotechnology, 17, 5659-5668. 51 Klafter et al. (1990). 52 Klafter et al . (1990). 53 Greenenko et al. 54 Weeks et al. (1998). 55 Weeks, Eric R., Urbach, Jeff, Swinney Harry. Random walks and Anomalous Diffusion. http://chaos.utexas.edu/research/annulus/rwalk.html (as by April 2nd 2009).


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The moment of l, nl , could be calculated from a probability distribution function (also known as PDF).

)(lP :

)(lPldll nn . (18)

If it decays according to a power law, i.e. µtlP ~)( , then the moment is

µnn ldll constant)( (19)

(n=1 describes the mean length while n=2 then describes the mean square lengths). When n=2 and 3µ , the [second] moment becomes infinite, as can be seen when evaluating the integral. When [,] 21µ both the first as well as the second moment is divergent, so also the mean flight length is infinite. With [,] 32µ the motion is intermediate between ballistic and normal diffusion.56 As can easily be deduced from the integral above, only the second moment is divergent this time, whilst the first moment converges, i.e. the mean length l is finite. These features are summarised in table 1. µ n n- µ First moment <l > Second moment <l2 > Type of motion

1 -1 to 0 - 1 to 2 2 0 to 1 - ballistic

2 -1 to 0 - 2 to 3 1 -1 to -2 converges - intermediate

Table 1 Connection between different values of µ and n, and the first and second moments described in equation (19).

2.4 The CTRW approach: Lévy walks As stated above, Lévy flights have immediate displacements requiring no time. 57 Therefore, they are not possible in Euclidean space. 58 Moreover, Lévy’s mathematics does not include the time needed for each jump per se.59 In order to apply the Lévy mathematics to real applications, time is needed as a variable. 60 Physically, this could be explained by placing one’s interest in the distance actually travelled after time t (i.e. vt ) rather than focusing at the total displacement.61 This average displacement after a time t is called a Lévy walk, which is a modification of the Lévy flight con-cept; for example, both motions may look the same62, 63. What is needed is to make the mean square displacement )(|| 2 tr finite.64 A method

to obtain this is by the CTRW (Continuous time random walk) approach 65. Moreover, the CTRW approach allows a particle to rest before instantaneously moving to a new position.66

56 Weeks et al. (1998). 57 Fowlkes et al. 58 Fowlkes et al. 59 Shlesinger et al. (1993). 60 Shlesinger et al. (1993). 61 Shlesinger (2001). 62 Shlesinger (2001). 63 Klafter et al. (1990). 64 Klafter et al. (1990).


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The CTRW approach relies on a probability distribution like the following to describe the probability density of displacement |r| in time t (this time also being the lifetime until being trapped next time):

))(

||()(),(

rrr

vtrpt (20)

)(rv denotes the length-dependent velocity. The first factor, )(rp , is the probability a trajectory,

between two halting points, has the distance r . The second factor, ))(

||(

rr

vt , can be

interpreted as )|( rtp , i.e. the conditional probability that given the displacement r the time for this displacement will be t . Since µp |~|)( rr , self-similarity is preserved (just as with Lévy flights).67 Another slightly different distribution describing Lévy walks is )|(|||),( vµ tAt rrr (21) where the function couples t and r. This distribution provides a superlinear, though finite, re-lationship between )(|| t2r and t. Any jump length is possible but, due to the coupling between t and r, longer lengths are less probable. 68 The coupling is necessary for enhanced transport ( )1 , whereas for dispersive transport ( 1 ), the components may be decoupled: i.e.

)()(),( tt rr .69 An attempt to visualise eq. 21 is shown in figure 1.

65 Shlesinger et al. (1993). 66 Shlesinger et al. (1989). 67 Shlesinger et al. (1993). 68 Klafter et al. (1990). 69 Klafter et al. (1987).


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Figure 2. α as a function of µ. In the interval µ ]2,3[, the motion changes from being

ballistic to normal diffusion.

0

0.5

1

1.5

2

2.5

1 1.5 2 2.5 3 3.5 4

µ

α

The Lévy walks are scale invariant. The value in tt ~)(|| 2r

takes different values depending on µ. For 1v (indicating motion of the ballistic

kind) the following holds: For [,] 21µ , 2 , for [,] 32µ , µ 4 and finally

for 3µ , 1 .70 Note the continuity in (figure 2). As stated above, Lévy flights have immediate displacements requiring no time. Therefore, they are not possible in Euclidean space. A Lévy walk, on the other hand, does have a finite, and actually often constant, velocity. The lengths of these displacements are chosen from a probability distribution.71

2.5 Relating mean square displacements to diffusion coefficients The connection between trajectories (i.e. motion) and the common diffusion coefficient, here denoted D’, can be made by relating the variance, )(t2 , which is given by 22 )()( txtx , with the diffusion coefficient and time t by:72 tDtxtx '2)()( 22 (22)

This applies in the one dimension case. In the case of a symmetric walk, the mean displacement,

)(tx is zero, and the diffusion coefficient is then directly proportional to the mean square

displacement: tDtx '2)(2 . For three dimensions, variance can be defined as73

tDtztytx '6)()()( 222 (23)

Generalising and using vector notation it is possible to write: tdD'2|| 2 r (24)

where d is the number of dimensions being investigated.74, 75 For make adequate comparisons possible, however, the time-independent diffusion coefficient D will, in the rest of this work, simply be defined as

t

D2||r

(25)

i.e. without any numerical coefficients (hence '2dDD ). 70 Klafter et al. (1990). 71 Fowlkes et al. 72 Weeks et al. (1998). 73 Sharma. 74 To, My. Interactions between cross-linked latexes and ionic surfactants. (2007). Karlstad University Studies. 75 Sharma.


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These time-independent diffusion coefficients determined experimentally using the mean square displacement, may be compared to the one obtained through the Stokes-Einstein equation76, 77, 78:

R

TkD B

6' (26)

where Bk is the Bolzmann constant ( )K J 1038.1 -123 , is the viscosity and R is the hydrodynamic radius (as if the particles were spheres). 79 The higher viscosity and/or radius, the lower rate of diffusion. On the other hand, if the thermal energy (~ TkB ) is increased, diffusion is enhanced. The diffusion coefficients may also be assumed to be time-dependent, as viewed by Weiss et Al. For example, they could follow 1)( ttD . (27) However, this kind of dependence is problematic, since, according the authors, D diverges as

0t [in the case of subdiffusion]. Using this time dependent diffusion coefficient, it is easily deduced that80 2| | ( ) ( )t D t t t r (28) For a Lévy flight, having an infinite second moment, i.e. 2||r , the diffusion

coefficient is infinite as well .81 Interestingly, in the limit of Brownian motion, i.e. 1 , then the time-dependent and time-independent diffusion coefficients merge: DtD constant)( and

hence Dtt )(|| 2r . The larger deviations from Brownian motion, the greater discrepancy be-

tween D(t) and D. Moreover, it is possible to use limiting values as the definition of the diffusion constant. Then, the normal diffusion coefficient could be defined as

t

xD

xt

2

0,0lim (29)

A generalised diffusion coefficient applicable in the anomalous case is:

||||

lim2

0,0 tx

Bxt

(30)

76 Carlsson, Gunilla. Latex Colloid Dynamics and Complex Dispersions. (2004). Karlstad University Studies. 77 To. 78 Heidkamp, Hannah. How interactions between anionic latex and the cationic surfactant DoTAB affect latex film formation. (2009). Karlstad University Studies. 79 Note, in order to compare with the values calculation using the Stokes-Einstein formula, one must include the numerical coefficients in the experimental determination of the diffusion coefficient, i.e. use D’ and not D.. 80 Weiss, Matthias, Elsner, Markus, Kartberg, Fredrik, Nilsson, Tommy. Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells. (2004). Biophysical Journal, 87, 3518-3524 81 Ott, A., Bouchaud, J.P., Langevin, D., Urbach, W. Anomalous Diffusion in “Living Polymers”: A Genuine Levy Flight? (1990). Physical Review Letters, 65, 2202-2204.


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Both D and B should be constant. The last equation produces the scaling /2 ~ tx . It is also possible to show that the in t~||r is then 2/ .82 The diffusion coefficient can also be linked to the probability distribution by the following equation (valid in one dimension for ordinary diffusion, i.e. 1 ):83

Dtx

DttxP 42

41 /),( e

(31)

2.6 Richardson’s law In 1926, L.F. Richardson empirically obtained the following connection between the diffusitivity K and pair separation R 84, 85: 3/4~),( RtRK (32) The accompanying probability distribution is

tCRttRp /3/22/9 e~),( (33) This function is of non-Gaussian type. Using this, it is possible to obtain a formula for the mean square displacement for pair separations: 3

22 tCtR )( (34)

where denotes the mean energy dissipation and 2C is the Richardson constant.86 Equation 33 is not the only one implying a 3t dependence. Actually, every space- and time-dependent diffusion coefficient ba tRtRK ~),( with 423 ba is ‘compatible’ with the 3t law. However, the distribution functions may differ.87 There has not been consensus for the value of 2C . Usually, boundary conditions affect the so-called prefactors of scaling laws, whilst the exponents appear stable. However, for the exponent to take the value of 3, the time t must be large enough, otherwise, the finite size makes the )(tR 2 flatten out. According to Gioia et al, the clue to the hard-determined 2C -values may

be due to the time-scales as well; one can not obtain an appropriate 2C using limited time-scales since the )(tR 2 is very sensitive to the starting-off pair-separations.88 Richardson’s law has been utilised to describe various trajectories. For example, it can describe particles moving in trajectories encountering vortices. 89 The Richardson law has, as well, been used with many simulations. Jullien et al used comprehensive data sets to solve the Lagrangian equation dx/dt=v . They found pair separations with alternating periods of rest and motion. They obtained close to a straight line when plotting )0,(/),( tRtR against t/ . The

82 Shlesinger et al. (1993). 83 Chaves, A.S.. A fractional diffusion equation to describe Lévy flights. (1998). Physics Letters A, 239, 13-16. 84 Klafter et al. (1987). 85 Gioia, G., Lacorata G., Marques Filho E.P., Mazzino A., Rizza. U.. Richardson’s law in large-eddy simulations of boundary layer flows. (2004). Boundary layer meteorology, 113, 187-199. 86 Gioia et al. 87 Boffetta, G., Celani A.. Pair dispersion in turbulence. (2000). Physica A, 288, 1-9. 88 Gioia et al. (2004). 89 Shlesinger et al. (1993).


Lars Fredriksson


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distribution was non-Gaussian. The authors explain this by a ‘memory effect’. Moreover, there was no need for a Lévy flight model to explain this.90 Boffetta et al. also used simulations to pursue the possibility of a 3t dependence. They simulated pair separations for 64 000 particles and found the Richardson law to hold for a limited time interval. The dependence depended on the initial conditions, i.e. the data with largest initial

)0(R best fitted the 3t -curve. The authors found this dependence on initial conditions remark-able. At 0.2 s, the distribution was of the peaked Richardson type, whereas for 5.0t s, the distribution was clearly Gaussian.91

2.7 The Lévy concept in biology As stated implicitly above, Lévy walks can, as opposed to Lévy flights, take place in Euclidean space. However, since they extend to infinity, the walks must be truncated to make the model viable to realistic situations (e.g. in biology).92

2.7.1 H2O molecules performing Lévy walks in lenses In the framework of traipsing water molecules, the concept of ‘strong adsorbers’ is essential. The time a hydrated molecule spends near the surface is significantly larger than the time of desorp-tion. Thus, the molecule may let go from the surface but without indefinitely going into the bulk solution. It is, in nearly all cases, readsorbed, possibly on another location. This is the foundation for discussing surface displacements. According Bodurka et al, this results in a non-Fickian [i.e. anomalous] type of movements, since the displacement [note: not the square displacement] with length scales that are less than or greater than Dt . The distribution connected to the first case ( Dt||r ) is the Cauchy distribution: 93

90 Jullien, Marie-Caroline, Paret, Jérôme, Tabeling, Patrick. Richardson Pair Dispersion in Two-Dimensional Turbulence. (1999). Physical Review Letters, 82 (14), 2872-2875. 91 Boffetta et al. 92 Viswanathan et al. (2008). 93 Bodurka, Jerzy, Seitter R.-O., Kimmich R., Gutsze A.. Field-cycling nuclear magnetic resonance relaxometry of moleculardynamics at biological interfaces in eye lenses: The Lévy walk mechanism. (1997). Journal of Chemical Physics, 107, 5621-5624.

05

1015202530354045

0 0.5 1 1.5 2| r|

Γ(r

,t)

t=1

t=2t=3t=40

0.20.40.60.8

11.21.41.61.8

0 0.5 1 1.5 2| r|

Γ(r

,t)

t=1

t=2t=3t=4

0

0.10.2

0.3

0.4

0.50.6

0.7

0 0.5 1 1.5 2 2.5t

Γ(r

,t)

| r|= 1

| r|= 2| r|= 3

| r|= 4 0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5t

Γ(r

,t)

| r|= 1

| r|= 2| r|= 3

| r|= 4

Figure 3. The behaviour of the Cauchy distribution. In a) and c) c=1 and in b) and d) c=0.2. As can be seen, with small relative speeds and small times the probability of reaching a given location is proportional to the time. As seen in the inflection, the distribution became more spread out with larger t-s. The authors assumed c=0.01 ms-1, which make the vs |r| graphs even steeper near the origin, though not diverging to infinity. The graphs in d) become virtually linear with c=0.01.

a) c)

b) d)

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1| r|

c=1

c=1 c=0.2

c=0.2


Lars Fredriksson


13

232221

/])[(),(

rctctt

r (35)

Here hDc / is the speed and HDth is the adsorption depth. The displacement grows linearly with time, i.e. ct||r .94 This distribution is investigated in figure 3. When Dt||r the following distribution is valid:

Dt

Dtttt

/Dtr

H

2

2

-2e

||),(

rr . (36)

This distribution had the form of a modified ‘Gaussian propagator’ and describes molecules per-forming lengthy journeys from the surface. 95 The behaviour of this distribution is analysed in figure 4. The authors used the Cauchy distribution when they set up a correlation function for the motions of water molecules in a rabbit lens using NMRD (Nuclear magnetic resonance dis-persion) data. Their conclusion is that the displacement of water molecules along inner surfaces follows the patterns of Lévy walk [but] with Dt||r . When exchanging positions, i.e. desorb-ing and adsorbing, on the surface, water molecules position themselves favourably. The longest correlations times 410 s. Moreover, the variations of relaxation times allow deductions of the surface topology of structures of the rabbit lens.96

2.7.2 Lévy walks to simulate searches Lévy walk theory has been used to simulate searching strategies in ecology. For example, pre-dation and pollination behaviours can be modelled this way. Thus, the ‘searcher’ can be a pre-dating animal or an insect seeking pollen. The ‘target’ is often identified as food. Since the loca-tion of a target is commonly not known a priori, these searching processes are largely stochastic processes. In other words, since the search problem is undetermined, statistical calculations are adequate.97 The Lévy flight foraging hypothesis states that the random walker, i.e. the searcher, makes increments of length lj every time step j. These increments are chosen from a probability distribution µllP ~)( with ],] 31µ . The values of µ strongly characterise the movements. Values close to unity reflect paths that are rather straight while values approaching 3 suggest more of Brownian behaviour. The authors showed that, in the absence of a priori knowledge and in case of randomly distributed targets, the optimum value of µ is 2. However, the value of the Lévy walk strategy decreases with increased food abundance; for 3µ a Brownian search would be comparable in efficiency.98 When searching, if the target is located within a radius of vision rv, the searcher moves to it straight away, otherwise the animal chooses a direction at random and while moving along the distance lj, keeps looking for the target. If not located, the searcher reiterates the process when having travelled a distance lj. Viswathan et al characterise three different kinds of searches: regenerative (where the target reappear after a while), destructive and non-destructive searches.99

94 Bodurka et al. 95 Bodurka et al. 96 Bodurka et al. 97 Viswanathan et al. 98 Viswanathan et al. 99 Viswanathan et al.


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Reynolds (2009) shows that so-called adaptive Lévy walks can outperform Brownian walks (as well as non-adaptive Lévy walks) when it comes to searching resources that are heavily patched in a non-destructive way. In this model, the starting point for the search is the origin. If a target is not located in a distance r, then the direction is chosen randomly and a walk with length l is then carried out. This follows the power-law distribution

rlrllrµ

lPµµ

if if

01 1)(

)( . (37)

After the first walk has been performed, there are two alternatives. If no target has been located, the process starts all over again with a new walk with length l. If, however, a target is located, there is a change from a Lévy walk to a more intensive Brownian movement. This change in strategy opposes classical Lévy walk theories. The strength of the Lévy processes is to search for patchily distributed resources in ‘unpredictable environments’. Levy walks characteristically have

Figure 4. The behaviour of the distribution in equation 36. All graphs show some kind of minimum. In a) and b) D=0.1. In c and d), D=0.5 and in d) and e) D=1. In b) d and e) it is clearly seen that longer times cause greater displacements. However, the function diverges to infinity. Physically, according to Boffetta, the reason is that some molecules perform very long excursions. This behaviour is mirrored in a), c) and d). Surprisingly, smaller diffusion coefficients seem to promote these longer excursions. In these graphs, tH=1.

D=1

0102030405060708090

100

0 0.5 1 1.5 2| r|

Γ(r,

t)

t= 1

2 3 4

t= 5

D =0.5

0102030405060708090

100

0 0.5 1 1.5 2| r|

Γ(r,

t)

t= 1

2 3 4

t= 5

D =0.1

0102030405060708090

100

0 0.5 1 1.5 2| r|

Γ(r,

t)

t= 1 t= 2 t = 3 t= 4 t= 5

D=1

0102030405060708090

100

0 0.5 1 1.5 2 t

Γ(r,

t)

| r|= 1.4

1.2

|r|=0.61 |r|= 0.8

| r|= 0.40.2

D =0.5

0102030405060708090

100

0 0.5 1 1.5 2 t

Γ(r,

t)

| r|= 1.4

1

|r|=0.6

1.2

|r|= 0.8

| r|= 0.4

0.2

D =0.1

0

50

100

150

200

250

300

350

400

0 0.5 1 1.5 2 t

Γ(r,

t)

| r|= 1|r|=0.6

|r|= 0.8

0.4

0.2

| r|= 1.2

a)

c)

d)

b)

d)

e)


Lars Fredriksson


15

µ>1. The optimum for adaptive Lévy walks is about 2. Reynolds also refer to several known examples from biology with Lévy walks with .2µ 100

2.8 The Lévy concept in physical chemistry

2.8.1 Studies of micelles of CTAB Ott, Bouchaud and colleagues investigated surfactant systems of Cethyl Trimethyl Ammonium Bromide (CTAB, H3C-(CH2)15-N+(CH3)3 Br-) with different concentrations of KBr and also diffe-rent temperatures. They claim to be the first to find Lévy flights in CTAB, possibly they may as well be the first team to find Lévy flights in any real physio-chemical system. The self-diffusion coefficients were obtained by so-called FRAP – fluorescence recovery after photo bleaching. This technique uses small fluorescent probes that become permanently non-fluorescent after being ‘photo-bleached’ by a potent laser flash. This creates a hole in the fluorescent surroundings. The time it takes for other probes to occupy this area is used to calculate the diffusion coefficient. The chains of this surfactant can break and recombine in a time break .The signal

obtained for mono-disperse diffusion objects at the frequency 2 is proportional to t

I

e0 . The

diffusion coefficient is then obtained by 12 )]([ qqD (where iq /2 and )(t is the relaxation time). Normal diffusion is independent of the value of q. The authors used six fringe spacings where 100µm 2 i µm. When this fringe separation increases, the diffusion coefficient decreases. Practically, D was obtained as the slope a plot of 1)( t versus q2. Since 1 was found to be approximately equal to µq this gives superdiffusive motion because µ<2

vtr where ½/1 µv , i.e. enhanced diffusion. Ott et al also pinpoint two conditions necessary for enhanced diffusion to be possible:

EC LL : The length rep

breakCL

length) average( must be larger than the so-called

entanglement length LE. Salt diminishes this entanglement length Le. The measuring time must be adequately short. More specifically, anomalous behaviour

was only seen if the experimental time is shorter than the cross-over time ( breakN where N is the number of steps). Temperature effects could be behind these effects.

When the concentration of micelles was low, there was no anomalous diffusion. This could be explained by large entanglement lengths. Bouchaud et al also discussed the possible impact of the concentration of tracer particles. In their article, they assume that two probes move independently, each having a separate Lévy flight. This is justified for very small concentrations, but for larger concentrations, tracer particles are merely swopped around and all chains are visited by probes which makes the anomalous diffusion vanish. What, however was found to be of great importance is the temperature. b is significantly and Arrheniusly influenced by temperature. Diffusion coefficients were found to correlate with the concentration of CTAB (asymptotically, i.e. linearly in the range 5-60 mM). The authors have also earlier noted that for higher salinities, the diffusion coefficient is not scale-independent. 101, 102 100 Reynolds, A.M. Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios. (2009). Physica A, 388, 561-564 101 Ott.


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2.9 Analysing experimental data Equation 10, tt ~)(|| 2r , can be rewritten as

tt )(|| 2r (38) Upon taking logarithms one arrives at

tt lnlog)(||log 2 r (39)

which should give a straight line, yielding the characterising as slope. Solomon et al have made conceptually similar data treatment, but with rotating motion and hence they defined variance as 2)( where is the azimuthal angle103. When plotting the log against log t they obtained a line with very little curvature; their (although denoted by ) varies between 1.5 and 2. However, since they made real measurements, corrections for the finite duration of measured trajectories were needed. This was made by comparing data from numerical simulations. Thus, they created artificial trajectories, chopped them up to finiteness, and then obtained probability distributions for those as well as for those not chopped. By comparing these, the authors were able to make the necessary conditions for the experimental trajectories. They also found flight and sticking probabilities of the form µttP ~)( since )(log tP versus tlog gave straight lines. 104 Microsoft Excel provides a shortcut to the logarithm path outlined above (eq. 39). It is, if 0t is excluded, possible to straight away obtain by the means of potential regression, i.e. fitting the data to BAxy directly. However, this obviously gives the same as the log-log transformation method. Since it is considerably easier to inspect visually the appropriateness of curves fitting to the original data, this method will be used consequently throughout this work. However, it should be pointed out that this method is indeed mathematically equivalent to taking logarithms (that is why 0t must be excluded).

102 Bouchaud. 103 the angle between x-axis and the y-axis in the xy plane [i.e. as seen from above]. 104 Solomon.


Lars Fredriksson


17

3 Method The basis for all investigations in this work is video-based fluorescence microscopy. By using fluorescence, it is possible to resolve smaller particles than with an ordinary light microscope. This is because the radiation used for excitation of molecules consists of smaller wavelengths than those of visible light. Hence, the resolving abilities are significantly improved. Conceptually, video-based fluorescence microscopy can be divided into five essential parts:

The light source. The excitation wavelength is always smaller than the wavelength of the light emitted (the emission wavelength). Commonly, a UV source is used. Here, a mercury vapour lamp from Carl Zeiss LTD provides the necessary UV radiation that is then converted by phosphorescence to a green wavelength, which is used as excitation radiation.

A filter singling out the desired excitation wavelength. Fluorescent molecules. Since, in this case, diffusion is the subject of investigation, fluore-

scent probes (made of latex) are used. The radii of these are either 100 nm or 50 nm (vide infra).

A magnification device, i.e. a microscope. Here, Carl Zeiss LTD Axioskop 2 is used. A recording device, i.e. a camera. Here, Hamamatsu ORCA-ER. Software for viewing films: Aquacosmos (Hamamatsu, especially versions 1.3 and 2.6)

was used to view films and trajectories were noted by following particles manually in Microsoft Excel 97.

3.1 Analysing Carlsson’s data Carlsson et al carried out comprehensive studies on the diffusion of latex probes, some of these in starch solutions with different concentrations, others in suspensions of latex.105 In this work, some data, especially concerning different latex concentrations, is evaluated in order to character-ise them according to equation 38. In table 2, the different probes Carlsson used is presented. Carlsson’s Designation

Producer Producer’s designation

Radius r Miscellaneous

A Molecular Probes 1.0 µm orange fluorescent

0.5 µm F-8820 Lot: 6991-1 10 mL

B Molecular Probes 0.2 µm orange fluorescent

0.1 µm F-8809 Lot: 5491-4 10 mL

C Bangs Laboratories Inc.

Uniform Microspheres

0.265 µm Catalog code: FC03F Polymer: P(S/V-COOH)*(490,515); Fl Yellow Mean diamerer: 0.53 µm Bangs lot #: 4775 Inv#: LOO1012 BB

Table 2. The different probes Carlsson used.106 105 Carlsson. 106 Carlsson, Gunilla. Non-published data.


Lars Fredriksson


18

3.2 Investigation of selected Carlsson’s films In this work, Carlsson’s films of diffusion in Raifix 01035 using the probe with radius 0.05 µm107 were further investigated. Manual tracking of probes was performed for solutions of 0, 100, 200, 400, 600, 800, 1000 and 1500 ppm starch. The goal was to characterise motions according to equation 38. For all configurations, two chambers were examined. In order to classify the particles under investigation, five probes with particularly long displacements were investigated as well as five probes with particularly short total displacements. The foundation for this selection was the tracer pictures provided by Carlsson, in which the total displacements are pointed out. From these, five long and five short displacements were selected for each run. Then, the co-ordinates of each selected probe were recorded by hand in Excel. Sub-sequently, | |r (in µm) and | |r 2 (in m2) were calculated for each probe. For each quintet, the

mean square displacement 2| |r was calculated (in m2) and plotted against time, whereupon Excel could perform the regression analysis in order to find and . As a complement, but which is vital in order to detect Lévy walks and not only enhanced diffusion, the paths of all probes in a selection of chambers were plotted and shown in section 4.2. Furthermore, the mean step sizes were analysed. In order to test Richardson’s law the combination of all long probes were investigated according to eq. 34: The number of combinations of pairs is given by

10!3!2

!345!3!2

!525

.

The pair separation, R , was computed as 22 )()( ABAB yyxxR (40) where the individual coordinates have been normalised, i.e. all trajectories are defined to start in the origin.

3.3 Investigation of selected Heidkamp’s films Heidkamp carried out comprehensive investigations of the interactions between the surfactant Dodecyl trimethyl ammonium bromide (figure 5) and latex with low transition temperature. She investigated different concentrations of DoTAB (range 0-25 mM) always with 3 % latex in order to study film formation. As to determine the relevant time-independent diffusion coefficients, she used video-microscopy in the same way as Carlsson.108

CH3

CH3

CH3

N+

CH3 Br-

Figure 5. Dodecyl trimethyl ammonium bromide (DoTAB). In this work, most of Heidkamp’s configurations were re-examined in much the same way as was done with Carlsson’s films. Furthermore, time-dependent diffusion coefficients were 107 Orange flouroescent like the B probes above. 108 Heidkamp.


Lars Fredriksson


19

calculated according to eq. 27, both for the quintets of long and short displacements, respectively, and for all ten investigated probes studied together. Besides these studies, calculations were made to compare the time-independent diffusion coefficients determined by Heidkamp109 using

steps 360probes 180probesteps 2 with the time-independent diffusion coefficients interpreted as

the mean of the linear slopes110 of 2||r versus t for steps. 990probes 10probesteps

99 111

According to Carlsson112 either method should, theoretically, yield the relevant diffusion coefficients.

3.4 Simulations of Diffusion in Microsoft Excel Microsoft Excel can very easily be used to simulate normal diffusion as will be seen. 30 virtual particles (VPs) and 99 steps were used. In the uppermost section, distances are generated using C7=STEP*SLUMP() (in English versions: =STEP*RANDOM()) C8=STEP*SLUMP() were “STEP” is a cell containing an arbitrary size parameter. Directions are generated as well using D7=SLUMP*2*PI() (in English versions: =RANDOM()*2*PI()) D8= SLUMP*2*PI() … since a circle is 2 radians. Under these 99 rows, the very positions are computed arbitrarily starting off from (0,0). X Y C109=0 D110=0 C110=C109+C7*COS(D7) D111=D109+C7*SIN(D7) C111=C110+C8*COS(D8) D112=D110+C8*SIN(D8) … .. Thus, each step is independent of the previous step, but each position is dependent on the previous position, except for the first one which is set to (0,0). Under the positions, the total displacements are calculated using: X Y C211=(C109-C$109)*size D211=(D109-D$109)*size C212=(C110-C$109)*size D212=(D110-D$109)*size C213=(C111-C$109)*size D213=(D111-D$109)*size … … Where “size” is, here, an arbitrary scaling parameter. Then, the mean displacements, ||r , are calculated: 109 Appendix 1 in Heidkamp. 110 Intercept is set to 0. 111 To make these comparisons possible, Heidkamp’s diffusion coefficients were adjusted according to eq. 24, i.e. no numerical coefficients is included in the relation between <|r|2> and t. 112 Carlsson, oral information.


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20

D313=(C211^2+D211^2)^0.5 D314=(C212^2+D212^2)^0.5 … Under this part, the significant mean square displacements 2||r are calculated: D415=D313^2 D416=D314^2 … This arrangement (including the number of ‘probes’) is apart from scaling factors, times – and, naturally, the generating parts – exactly the same is the arrangement used in the experimental section in the upcoming master’s diploma work. In addition, other configurations, more precisely different exponents of the random function were used. The function =(RANDOM())n itself yields the mean step size, median step size and standard deviations shown in table 3.

n mean median std.dev. skewness0.5 0.68 0.72 0.23 -0.541.0 0.50 0.49 0.29 0.151.5 0.38 0.32 0.30 0.642.0 0.33 0.24 0.30 0.822.5 0.29 0.18 0.29 1.093.0 0.25 0.14 0.27 1.21

Table 3. Dependence of n on the mean, on the median, on the standard deviation and on the skewness.

In table 3, the skewness coefficient (eq. 8) is computed. As seen in the table, the mean step size will decrease when raising the random exponent n. The average step size, i.e. the median, how-ever, diminishes more quickly while the standard deviation is nearly unaffected. This means that the skewness is far greater with higher exponents, n. This could be valuable in order to pursue Lévy flights that indeed are skewed.


Lars Fredriksson


21

4 Results and discussion

4.1 Analysing Carlsson’s data

4.1.1 Initial considerations113 In figure 6, the changes for one of the probes in each co-ordinate with respect to time are plotted against time. As can easily be seen, the pattern of motion is clearly haphazard. However, this diagram gives no clear information of what kind of chaotic motion is being performed. In figure 7, data for four different probes (and hence different trajectories) is shown. For two of the probes, the time-scale is significantly shorter than for the other two. In order to calculate the mean square displacement of all probes, only data for the first 83 s was used. Thus, data in series 2 and 4 were interpolated to fit the t-s in series 1 and 3.114 As can be seen in figure 8, three of the four square displacement are very similar to each other, while number two (the one given in fig. 7) behaves differently. The mean square displace-ment reflects all four probes.

113 In this section data from Carlsson from experiments performed 3-5 May 2002 is used. 114 Interpolation was made by linearly calculating an approximate 2|| ir value using

211

11

21

212 ||

||||||

iiiii

iii tt

ttrrrr .

Figure 6. The changes in each coordinate with respect to time.

Figure 8. Square displacements and mean square displacement for all probes in the time interval 0-81 s.

0.0E+00

5.0E-11

1.0E-10

1.5E-10

2.0E-10

2.5E-10

0 20 40 60 80 100

t (s)

squa

re d

isplac

emen

ts /

m

1 |r|² /m²

2 |r|² /m²

3 |r|² /m²

4 |r|² /m²

<|r|²> /m²

Figure 7. Square displacement for four different probes.

0

2E-10

4E-10

6E-10

8E-10

1E-09

1.2E-09

1.4E-09

0 50 100 150 200 250 300

t (s)

R² /

m²

1 R² /m²

2 R² /m²3 R² /m²

4 R² /m²

-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

0 20 40 60 80 100

t /s

chan

ge in

pos

ition

/(µ

m/s

)

Δy/ΔtΔy/Δt


Lars Fredriksson


22

When 2m/||log 2r is plotted against s)/log(t , this should, according to

equation 39, result in a straight line with the slope . The slope found here was 0.85, which according to section 2.1, corresponds to subdiffusion. When a similar analysis is carried out separately on series 1 and 3 and on 2 and 4, respectively (figure 10), the result is clearly altered. For probes 2 and 4 (with time measurement ranging 0-83 s), is found to be 0.63, thus still indicating subdiffusion. However, for probes 1 and 3, 451. , now suggesting superdiffu-sion. Thus, the pattern of diffusion seems to change with increasing measuring time.

4.1.2 Results from experiments with different latex concentrations Figure 11 shows the mean square dis-placement from different latex con-centrations (0%, 10%, 30%, and 50%). For comparison purposes, the same in-formation is found in the log-log graphs in figure 12. The first three concentra-tions result in a similar pattern of motion while the highest concentration results in a distinctly different pattern. For the first three concentrations, the mean square displacement is a mean of three trajectories; for the last concentration, it is a mean of two trajectories. The expon-ents of the 10%, 30%, and 50% are all about 1.3, indicating a mild kind of super-diffusion. For 0%, 90. indicating slight subdiffusion. But, this difference is hardly significant. However, the deviation of the solution with 50% latex is profound. Here, the mean square displacement is much lower, but still there is some kind of mild enhanced diffusion.

y = 0.8525x - 11.72R2 = 0.8986

-11.6-11.4-11.2

-11-10.8-10.6-10.4-10.2

-10-9.8

0 0.5 1 1.5 2 2.5log (t/ s)

log

(<|r

|²>

/m

²)

Figure 9. Log-log graph of the mean square displacement.

y = 0.6326x - 11.436R2 = 0.7038

y = 1.446x - 12.627R2 = 0.956

-14

-12

-10

-8

-6

-4

-2

0-0.5 0 0.5 1 1.5 2 2.5 3

log (t/ s)

log

(<|r

|²>

/m

²)

Figure 10. Log-log graph of the mean square displacement for probes 1 and 3 and for probes 2 and 4.

50% latexy = 1.3357x - 12.93

R2 = 0.908

30% latexy = 1.2878x - 11.809

R2 = 0.9385

10% latexy = 1.3669x - 11.677

R2 = 0.94370% Latexy = 0.8809x - 11.525

R2 = 0.8576

-14

-13

-12

-11

-10

-9

-8

-7-1 -0.5 0 0.5 1 1.5

log (t /s)

log

(<|r

|²>

/m²)

Figure 12. Log-log graph of the mean square displacements.

Figure 11. The mean square displacements of different tracer concentrations.

y = 2E-12x1.3669

R2 = 0.9437

y = 2E-12x1.2878

R2 = 0.9385

y = 3E-12x0.8809

R2 = 0.8576

y = 1E-13x1.3357

R2 = 0.908

0

5E-12

1E-11

1.5E-11

2E-11

2.5E-11

3E-11

3.5E-11

0 2 4 6 8 10

t /s

<|r

|²>

/m

²

0% latex10% latex30% latex50% latex


Lars Fredriksson


23

Latex probe A In figure 13, some of the displacements in the Latex probe A series (r=0.5 µm) are shown. An at-tempt to regress the data accord-ing to eq. 38 was made (figure 14). Since only one probe at a time is used in the re-gression, the re-sulting equation cannot be taken as particularly accurate, especi-ally not for the short time scale. For the longer scale, they might possibly suggest some kind of enhanced diffusion. In figure 15, the results of other latex A experiments are shown. In analogy with above,

|r²| = 3E-13t1.4542

|r|² = 7E-14t1.6264

0.E+00

2.E-10

4.E-10

6.E-10

8.E-10

1.E-09

1.E-09

1.E-09

0 50 100 150 200 250 300 350

t/s

|r|² A1 1/к=1 0s< t<259s

A1 1/к=100 0s< t<259s

|r|² = 2E-12t0.8402

|r|² = 5E-12t0.2916

0.E+002.E-114.E-116.E-118.E-111.E-101.E-101.E-102.E-102.E-10

0 20 40 60 80 100 120

t /s

|r|² A1 1/к=1 0s< t<82s

A1 1/к=100 0s< t<82s

Figure 14. The square displacement as a function of time for the A1 series.

-20

-10

0

10

20

30

40

-40 -35 -30 -25 -20 -15 -10 -5 0 5 10

x /µm

y /µ

m

(A1) 1/κ=1, 0- 82 s

(A1) 1/κ=100, 0- 82 s

(A1) 1/κ=100, 0- 259 s

(A1) 1/κ=1, 0- 25 9 s

Figure 13. The displacements in the A1 series.


Lars Fredriksson


24

the square displacements are plotted against time in figure 16. Since no means are used, real con-clusions are elusive. For the bare eye, however, both displacements in figure 15 seem to have about the same step length. Possibly, though, the series with 11 shows some Lévy character since, sometimes, the displacements are confined in a small area and sometimes make larger excursions. Exactly this is revealed when |r|2 is plotted against time: The series with 11 has 1.4 while the one with

1001 has a very small exponent ( 4.0 ) indicating subdiffusion.115 The real reason for the difference, however, is the direction of the trajectories; trajectories confined to the area around the origin regularly attain low exponents .

115 1 can loosely be thought of as the thickness of the electric double layer surrounding a particle.

Figure 16. The square displacement as a function of time for the A2 series.

|r|² = 2E-13t 1.3596

|r|² = 3E-12t 0.3843

0.E+005.E-111.E-102.E-102.E-103.E-103.E-104.E-104.E-105.E-105.E-10

0 20 40 60 80 100 120 140 160

t /s

|r|²

A2 1/к = 1 0 -109 sA2 1/к = 100 0- 109 s

-10

-5

0

5

10

15

20

-8 -6 -4 -2 0 2 4 6 8 10 12 14

x /µm

y /µ

m

(A2) 1/κ=1, 0- 10 9 s(A2) 1/κ=100, 0- 109 s

Figure 15. The A2 series.


Lars Fredriksson


25

Latex probe B In the latex probe B (r=0.1 µm) series, the long 0-82 s movements (at least the one with 11 ) clearly shows Lévy-like behaviour. Meanwhile, it is hard to classify the shorter 0-31 s movements (figure 17). However, this is not equally obvious from the mathematical analysis (figure 18) where one of the 0-82 s displace-ments has an close to unity. This can probably be attributed to the crossing of the x axis. Evidently, it is hard to make the regression analysis on a single displacement. The shorter time series shows normal or subdiffusion according to the exponent.

|r|² = 2E-11t 0.7405

|r|² = 2E-12t0.9707

0.E+00

5.E-11

1.E-10

2.E-10

2.E-10

3.E-10

3.E-10

4.E-10

4.E-10

0 10 20 30 40 50

t /s

|r|² B 1/к = 1 0- 31 s

B 1/к = 100 0- 31 s

|r|² = 1E-11t 0.9872

|r|² = 1E-12t 1.3792

0.E+00

2.E-10

4.E-10

6.E-10

8.E-10

1.E-09

1.E-09

0 20 40 60 80 100 120

t /s

|r|² B 1/к = 1 0- 82 s

B 1/к = 100 0- 82 s

-15

-10

-5

0

5

10

15

20

25

-30 -20 -10 0 10 20 30 40

x /µm

y /µ

m

(B) 1/κ=1, 0-8 2 s

(B) 1/κ=100, 0 - 82 s

(B) 1/κ=1, 0-3 1 s

(B) 1/κ=100, 0 - 31 s

Figure 17. The displacements in the B series.

Figure 18. The square displacement against time for the B series.


Lars Fredriksson


26

Latex probe C In the latex probe C series (r=0.265 µm), both the long and the short displacements are mostly Brownian (figure 19). This is apparently contradicted by the exponents revealed in figure 20. Here, the longer time series have

1 while the shorter have 2/3 . This contradiction is clearly the result of the probes in the long time series returning to the region close to the origin.

Figure 19. The displacements in the C series.

-15

-10

-5

0

5

10

15

20

-35 -30 -25 -20 -15 -10 -5 0 5 10 15

x /µm

y /µ

m

(C) 1/κ= 1, 0- 164 s

(C) 1/κ= 1, 0- 31 s

(C) 1/κ= 100, 0- 165 s

(C) 1/κ= 100, 0- 31 s

|r|² = 5E-13t 1.5971

|r|² = 4E-13t 1.453

0.E+002.E-114.E-116.E-118.E-11

1.E-101.E-101.E-102.E-102.E-10

0 10 20 30 40 50

t /s

|r|² C 1/к = 1 0- 31 s

C 1/к = 100 0- 31 s

y|r|²= 2E-12t 0.676

|r|² = 1E-10t0.1848

0.E+00

2.E-10

4.E-10

6.E-10

8.E-10

1.E-09

1.E-09

0 50 100 150 200

t /s

|r|² C 1/к = 1 0s< t <1 64s

C 1/к = 100 0s< t <1 65s

Figure 20. The square displacement against time for the C series.


Lars Fredriksson


27

4.2 Thorough investigation of selected Carlsson’s films In this section, the results of 0 ppm and of 1000 ppm will be reviewed comprehensively whereas all exponents , coefficients and mean step lengths will be found in the summary in section 4.2.3.

4.2.1 In detail: Starch content 0 ppm The regression analysis proposes that the long displacements are superdiffusive or Brownian (α=1.24 and 1.05 respectively). Short displacements are clearly subdiffusive (α=0.81 and 0.26) (figures 21-22). As figures 23-26 suggest, there are no significant traces of any Lévy walk (step lengths are of similar magnitudes). Besides, the long and short displacements look relatively similar.

y = 1.49E-12x8.05E-01

y = 3.19E-12x1.24E+00

0

5E-12

1E-11

1.5E-11

2E-11

2.5E-11

3E-11

3.5E-11

4E-11

0 2 4 6 8 10

t /s

<|r

|²>

/m

²

longshort

Figure 21. Regression analysis in the 0 ppm-12 run.116

y = 1.28E-12x6.44E-01

y = 1.90E-12x1.51E+00

0

5E-12

1E-11

1.5E-11

2E-11

2.5E-11

3E-11

3.5E-11

4E-11

0 2 4 6 8 10

t /s

<|r

|²>

/m

²

longshort

Figure 22. Regression analysis in the 0 ppm-13 run.

-6

-4

-2

0

2

4

6

8

10

-5 -4 -3 -2 -1 0 1 2 3

x /µm

y /µ

m

29235144

Figure 23. Long displacements in the 0 ppm-12 run.

-3

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6 8 10

x /µm

y /µ

m

24

27

10

7

1


-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2

x /µm

y /µ

m

3013121625

Figure 25. Short displacements in the 0 ppm-12 run.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-5 -4 -3 -2 -1 0 1 2 3 4

x /µm

y /µ

m

5

8

9

12

17


116 Note that x and y in this and the following graphs refers to time and mean square displacement respectively.


Lars Fredriksson


28

4.2.2 In detail: Starch concent 1000 ppm In this case, the regression analysis seems to suggest superdiffusion for one of the long cases ( 22.1 ) and Brownian motion for the other long one ( 06.1 ). As for the short displace-ments, the results are highly diverging (0.71 and 1.18 respectively) (Figures 27-28) Scrutinising the displacements, it is obvious that the short displacements are indeed short and often seem to be merely 'vibrating' around a certain point. This was not the case with 0 ppm (Figures 29-32). Also noticeable is that the resolution obtained by the by-hand method put a clear limit on what information that is possible to extract from short trajectories.

y = 1.22E-12x7.11E-01

y = 8.44E-12x1.06E+00

0

2E-11

4E-11

6E-11

8E-11

1E-10

0 2 4 6 8 10

t /s

<|r

|²>

/m

²

longshort


y = 8.70E-13x1.18E+00

y = 1.32E-11x1.22E+00

0

2E-11

4E-116E-11

8E-11

1E-101.2E-10

1.4E-10

0 2 4 6 8 10

t /s

<|r

|²>

/m

²

longshort


-10

-5

0

5

10

15

-10 -5 0 5 10 15

x /µm

y /µ

m

56101623


-20

-15

-10

-5

0

5

-15 -10 -5 0 5 10

x /µm

y /µ

m

911131421


-6

-4

-2

0

2

4

6

-4 -2 0 2 4 6

x /µm

y /µ

m

811131922


-4

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3 4

x /µm

y /µ

m

134629

Figure 32. Short displacements in the 1000ppm-13 run.


Lars Fredriksson


29

4.2.3 Summary of all investigated films In this section, all regression analyses are summarised. In table 4, the exponents are found. Two points are to be highlighted: The long displacements have values of ranging from about 1.0 to about 1.5 (curiously

often around 1.2) whereas the short displacements have values of between 0.5 and 1.2, most often significantly less than unity.

No obvious correlation exists between the content of starch and the exponent.

13: long 12: long mean 13: short 12: short mean0 1.51 1.24 1.38 0.64 0.81 0.73

100 1.25 1.18 1.22 1.10 0.76 0.93200 1.05 1.06 1.06 0.26 0.96 0.61400 1.35 1.27 1.31 0.54 0.45 0.50600 0.99 1.33 1.16 0.56 0.55 0.56800 1.22 1.29 1.26 1.01 0.68 0.851000 1.22 1.06 1.14 1.18 0.71 0.951500 1.55 1.16 1.35 0.61 0.85 0.73mean 1.27 1.20 1.23 0.74 0.72 0.73

starch content /ppm

exponent α

In table 5, the coefficients are found. Two points are to be highlighted: There is no obvious correlation between the concentration of starch and the values of for

the long displacements. For the short displacements, there is an obvious correlation, where higher concentrations make dwindle. This relationship is visualised in figure 33.

Not surprisingly, the shorter displacements have much smaller values of than the long displacements.


100 1.42 1.17 1.30 0.75 0.75 0.75200 0.79 0.78 0.79 0.59 0.37 0.48400 0.71 0.85 0.78 0.41 0.28 0.34600 1.90 0.99 1.44 0.50 0.21 0.36800 1.68 1.20 1.44 0.38 0.09 0.231000 1.32 0.84 1.08 0.09 0.12 0.101500 0.81 0.70 0.75 0.04 0.09 0.06

starch content /ppm

koefficienten Γ /10 -11 m² s-1

Table 5. The dependence of concentrations on the coefficients .

Table 4. The exponents from the regression analysis.


Lars Fredriksson


30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 200 400 600 800 1000 1200 1400 1600

concentration /ppm

coef

ficien

t Γ

longshort

Most conspicuous is the complex binding of probes occurring, with the highest concentrations of starch. In these cases, as were seen in detail for 1000 ppm, some probes do not accomplish much displacement at all, rather, they linger about a certain point, where they presumably have been complex-bound by the starch. In table 6, the mean step lengths are shown. While the long displacements seem to have about the same mean step length, there is a pronounced decrease for the small displacements. This is in analogy with what was pointed out concerning .


100 0.87 0.84 0.86 0.83 0.68 0.75200 0.85 0.83 0.84 0.57 0.65 0.61400 0.80 0.88 0.84 0.76 0.62 0.69600 1.00 0.87 0.93 0.72 0.70 0.71800 0.95 0.81 0.88 0.67 0.27 0.471000 0.83 0.86 0.85 0.23 0.31 0.271500 0.88 0.79 0.84 0.17 0.10 0.14

starch content /ppm

mean step lengths /µm

Table 6. The mean step lengths. In table 7, the standard deviations of the step lengths are shown. Note the striking similarities between all concentrations and also between long and short trajectories.

Figure 33. The correlation between starch concentration and the coefficient /10-12 m2 s-1.


Lars Fredriksson


31


100 0.53 0.49 0.51 0.45 0.40 0.42200 0.54 0.48 0.51 0.40 0.44 0.42400 0.44 0.49 0.47 0.47 0.55 0.51600 0.63 0.53 0.58 0.54 0.47 0.50800 0.60 0.47 0.54 0.52 0.40 0.461000 0.50 0.54 0.52 0.25 0.43 0.341500 0.60 0.53 0.57 0.24 0.13 0.19

starch content /ppm

step standard deviations /µm

Table 7. The mean step lengths. In table 8, the relative standard deviations are shown. There is a clear difference between the long and short displacements; the short ones have much larger values. Furthermore, this difference is greatly exaggerated for the higher concentrations.


100 0.61 0.58 0.60 0.54 0.59 0.56200 0.63 0.58 0.61 0.69 0.68 0.69400 0.56 0.56 0.56 0.63 0.89 0.75600 0.64 0.61 0.63 0.75 0.67 0.71800 0.64 0.59 0.61 0.78 1.51 0.981000 0.60 0.62 0.61 1.11 1.40 1.281500 0.69 0.67 0.68 1.41 1.28 1.36

starch content /ppm

step relative standard deviations

Table 8. The relative standard deviations of the step lengths. In figure 34, many of these features are visualised.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 200 400 600 800 1000 1200 1400 1600

concentration /ppm

long mean /µmshort mean /µmlong std /µmshort std /µmlong RSDshort RSD

Figure 34. The means, standard deviations and relative standard deviations of the step lengths.


Lars Fredriksson


32

4.2.4 Investigating Richardson’s law

When computing the mean square displacements for the 10 different pairs for the long displacements and using exponential regression, one gets the exponents shown in table 9. Clearly, there is apparently no connection between the concentration of starch and the exponents, no more here than when the means of individual cases were investigated (4.2.1-3). What is more striking, is that the exponents are very far off from the value in Richardson’s law (eq. 34): 3

22 tCtR )(

Since this law is apparently not applic-able, the constant 2C and the mean energy dissipation are not investigat-ed. What does a pair separation look like? Figure 35 shows two pair separations (in 400 ppm starch). Cutting time to less than one second (thus mimicking Bofetta et al, vide supra) did not increase the exponent; rather, the mean square separations became ‘subdiffusive’. However, this was only evaluated for a small number of configurations.

-5

0

5

10

15

20

25

-5 0 5 10 15

x /µm

y /µ

m

Figure 35. Two pair separations.


Lars Fredriksson


33

4.3 Analysing Heidkamp’s films

4.3.1 0 mM DoTAB With 0 mM DoTAB, the long probes seem to be diffusive according to Brownian motion (figures 36-37); no steps are substantially larger than others are. However, the exponent seems to suggest subdiffusion, but the discrepancy between 0.9 and 1.0 is probably simply caused by the highly limited number of probes investigated.

y = 1E-11x0.1786

y = 2E-11x0.9061

0

2E-11

4E-11

6E-11

8E-11

1E-10

1.2E-10

0 2 4 6

t /s

<|r

|²>

/m

²

longshort

Figure 36. The mean square displacements plotted against time with 0 mM DoTAB.

-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15

x /µm

y /µ

m

14142030

Figure 37. The long trajectories with 0 mM DoTAB. Inspecting trajectories, the short displacements in figure 38 do not look very different from their long counterparts. However, they are more confined to their start-area (hence being classified as ‘short’). This also causes to low value of .


Lars Fredriksson


34

-6

-4

-2

0

2

4

6

8

10

12

-6 -4 -2 0 2 4 6

x /µm

y /µ

m

36251828

Figure 38. The short trajectories with 0 mM DoTAB.

4.3.2 6.10 mM DoTAB There are no pronounced differences between 0 and 6.10 mM DoTAB, except some higher values of the exponent (figures 39-41). Again, the cause is most probably the limited number of probes being investigated.

y = 8E-12x0.4274

y = 2E-11x1.297

0

5E-11

1E-10

1.5E-10

2E-10

2.5E-10

0 2 4 6

t /s

<|r

|²>

/m

²

longshort

Figure 39. The mean square displacements plotted against time with 6.10 mM DoTAB.


Lars Fredriksson


35

-20

-15

-10

-5

0

5

10

15

-30 -20 -10 0 10 20 30

x /µm

y /µ

m

12

14

15

21

25

Figure 40. The long trajectories with 6.10 mM DoTAB. Note that one probe (29) in figure 41 have some particularly long steps. However, since the net displacement is short, the probe is included here as a ‘short’ displacement.

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8

x /µm

y /µ

m

4

8

20

22

29

Figure 41. The short trajectories with 6.10 mM DoTAB.


Lars Fredriksson


36

4.3.3 6.98 mM DoTAB Still, the long displacements have exponents close to unity while the small ones appear to be subdiffusive (figure 42).

y = 7E-12x0.6164

y = 2E-11x1.1236

0

5E-11

1E-10

1.5E-10

2E-10

2.5E-10

0 2 4 6t /s

<|r

|²>

/m

²

longshort

Figure 42. The mean square displacements plotted against time with 6.98 mM DoTAB. Considering the actual trajectories, however, both the long and the short trajectories appear Brownian, possibly suggesting that the low value of would increase with more probes studied (figures 43-44).

-20

-15

-10

-5

0

5

10

15

20

-15 -10 -5 0 5 10 15 20

x /µm

y /µ

m

4

5

17

28

30

Figure 43. The long trajectories with 6.98 mM DoTAB.


Lars Fredriksson


37

-8

-6

-4

-2

0

2

4

6

8

-10 -8 -6 -4 -2 0 2 4 6

x /µm

y /µ

m

8

14

16

27

29


4.3.4 7.62 mM DoTAB The trajectories seem much the same with 7.62 mM DoTAB (figures 45-47). Both the long and the short displacements look similar although the shorter ones are, naturally, more confined to the area around the origin (since all trajectories have been normalised).


y = 6E-12x0.465

y = 2E-11x1.3456

05E-111E-10

1.5E-102E-10

2.5E-10

0 2 4 6

t /s

<|r

|²>

/m

²

longshort


Lars Fredriksson


38

-20

-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15 20 25

x /µm

y /µ

m

4

7

20

21

28


-6

-4

-2

0

2

4

6

8

-6 -4 -2 0 2 4 6 8

x /µm

y /µ

m

1352430


4.3.5 8.68 mM DoTAB For 8.68 mM DoTAB, the long displacements behave just as before. However, the short ones behave very differently; so differently that they cannot be regressed (since some mean square displacements (other than for t=0) are zero) (figures 48-50). Note the length scale in these graphs; they are much shorter than in the previous cases.


Lars Fredriksson


39

y = 8E-12x1.176

0

2E-11

4E-11

6E-11

8E-11

0 2 4 6

t /s

<|r

|²>

/m

²

longshort


-12

-10

-8

-6

-4

-2

0

2

4

-8 -6 -4 -2 0 2 4 6

x /µm

y /µ

m

12

15

16

21

30



Lars Fredriksson


40

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

-3 -2 -1 0 1 2 3 4

x /µm

y /µ

m

1

7

19

24

25


4.3.6 11.7 mM DoTAB Now, diffusion has, in principle, totally ceased. However, the same analysis was carried out (figure 51). With the resolution obtained (assuming each probe is placed on a single pixel) some 'movements' are duly recorded. However, none of this could, even theoretically, be fitted into a potential regression diagram. The 'movements' could possibly simply be noise, or possibly the whole complex-bounded system being 'pushed' by water molecules. With a higher resolution, more motions – or noise – could be recorded. Without a higher resolution, it is better to regard all probes as fixed.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-0.5 -0.4 -0.3 -0.2 -0.1 0

x /µm

y /µ

m

No disignation

Figure 51. The short trajectories with 11.7 mM DoTAB. Note that Heidkamp did not assign numbers to the probes.


Lars Fredriksson


41

4.3.7 16.8 mM DoTAB The situation here bears high similarities with 11.7 mM DoTAB (figure 52). No regression is possible.

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

00 0.1 0.2 0.3 0.4 0.5

x /µm

y /µ

m

no designation


4.3.8 22.6 mM DoTAB Now, diffusion is back on track. In figure 53, it is clear that the mean square displacements are similar to the range 0-8 mM DoTAB, with the long probes apparently displaying enhanced diffu-sion and the short displacements seemingly somewhat subdiffusive (although with a higher value of than above).

y = 3E-12x0.8023

y = 7E-12x1.2995

0

2E-11

4E-11

6E-11

8E-11

1E-10

0 2 4 6

t /s

<|r|

²> /

m²

longshort

Figure 53. The mean square displacements plotted against time with 22.6 mM DoTAB. Scrutinising the trajectories, it is clear that the short displacements are very similar to the 8.68 mM case (figures 54-55).


Lars Fredriksson


42

-15

-10

-5

0

5

10

-15 -10 -5 0 5 10 15

x /µm

y /µ

m

1223616


-5

-4

-3

-2

-1

0

1

2

3

4

5

-4 -2 0 2 4 6 8

x /µm

y /µ

m

12

17

18

20

30


4.3.9 24.2 mM DoTAB For the highest concentration investigated by Heidkamp, 24.2 mM, the situation is similar to the previous case. However, the short displacements (figure 58) look less discrete. The mean square displacements, on the other hand, look much the same as for 22.6 mM DoTAB suggesting enhanced diffusion and subdiffusion, respectively. All in all, the situation now is similar to the lowest DoTAB concentrations.


Lars Fredriksson


43

y = 7E-12x0.7594

y = 2E-11x1.4276

0

5E-11

1E-10

1.5E-10

2E-10

2.5E-10

3E-10

0 2 4 6t /s

<|r

|²>

/m

²

longshort


-20

-15

-10

-5

0

5

10

-10 -5 0 5 10 15 20 25

x /µm

y /µ

m

1

3

5

19

24



Lars Fredriksson


44

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

-14 -12 -10 -8 -6 -4 -2 0 2 4

x /µm

y /µ

m

8

17

21

25

28


4.3.10 Summary of exponents and coefficients In table 10, all exponents and coefficients are reviewed with two decimal places. Naturally, the coefficients for the small displacements are smaller than the coefficients for the long displace-ments. More interestingly, the same pattern regarding the exponents is the case here, as was the case with Carlsson’s films: The long displacements tend to have 2.02.1 and the short ones tend to have much smaller values. This indicates enhanced diffusion and subdiffusion, respective-ly. However, these conclusions are ambiguous due to the highly limited number of probes exam-ined. In figures 59 and 60, the exponents and coefficients , respectively, are visualised.

long short long short0 0.91 0.18 1.79 1.00

6.10 1.30 0.43 1.57 0.856.98 1.12 0.62 2.31 0.717.62 1.35 0.46 1.79 0.638.68 1.18 * 0.81 *11.7 none ** none **16.8 none ** none **22.6 1.30 0.80 0.71 0.3324.2 1.43 0.76 1.63 0.66

[DoTAB] /mMexponent α koefficienten Γ /10-11 m² s-1

Table 10. The exponents and coefficients in summary.


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45

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 5 10 15 20 25DoTAB concentration /mM

expo

nent

s

longshort

Figure 59. The exponents .

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25 30DoTAB concentration /mM

coef

ficie

nts

long

short

Figure 60. The coefficients (1011 /m2 s-1). Heidkamp explained why diffusion ceases after about 7 mM DoTAB: At this stage, enough DoTAB has been added to first cover the negatively charged latex particles, then form a layer on the surface and also hydrophobically interact with the DoTAB attached to the latex particles. This third stage triggers flocculation of particles and hence causes diffusion to cease. With even more DoTAB, the structures responsible for flocculation break down and diffusion resumes.117

4.3.11 Mean step sizes and standard deviations In table 11, the mean step lengths, step standard deviations, relative standard deviations, medians, and skewness coefficients are shown. These features are visualised in figures 61-62. Note that a bit too many medians have the same numerical value (to more decimal places than is shown). This is due to the limited resolution obtained by using the by-hand method. Therefore, the calculated skewness coefficients are not very reliable. Another complication is that the exposure time was as long (0.06 s) compared to the interval time (also 0.06 s) which caused particles relatively frequently to appear twice. What was then recorded was the average position. Hence, some jumps are seemingly split into two. To some extent, this deteriorated the accuracy of the values in table 11.

117 Heidkamp.


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long short long short long short long short long short0 0.75 0.70 0.43 0.47 0.58 0.68 0.58 0.58 1.17 0.76

6.10 0.75 0.75 0.44 0.46 0.58 0.62 0.58 0.58 1.16 1.096.98 0.73 0.65 0.43 0.40 0.59 0.62 0.58 0.58 1.07 0.527.62 0.73 0.55 0.44 0.42 0.60 0.76 0.58 1.04 0.41 1.028.68 0.42 0.17 0.31 0.23 0.75 1.30 0.41 0.00 0.11 2.3111.7 none 0.01 none 0.07 none 5.66 none 0.00 none 0.5316.8 none 0.03 none 0.13 none 3.96 none 0.00 none 0.7622.6 0.49 0.42 0.38 0.27 0.77 0.65 0.41 0.41 0.64 0.1224.2 0.63 0.59 0.42 0.37 0.66 0.63 0.58 0.58 0.40 0.06

[DoTAB] /mM

median /µmskewness

coefficientstep rel. standard

deviationsstep standard

deviations /µmmean step lengths

/µm

Table 11. The mean step lengths, standard deviations, relative standard deviations, medians, and skewness coefficients.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25

concentration /ppm

long mean /µmshort mean /µm

Figure 61. The mean step sizes. Figure 61 reveals nothing surprising; there is a striking correspondence between the mean step size and the coefficients (compare figure 60). However, due to the much larger number of steps (as compared to the number of trajectories), the graphs look ‘neater’ without the peak for 6.98 mM observed in figure 60. The odd peak in relative standard deviations seen for 11.7 mM in figure 62 is simply a product of the highly discretised (i.e. rough) by-hand recording of positions.


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0

1

2

3

4

5

6

0 5 10 15 20 25

concentration /ppm

long std /µmshort std /µmlong RSDshort RSD

Figure 62. The standard deviations and relative standard deviations.

4.3.12 Time-dependent diffusion coefficients In table 12, all time-dependent diffusion coefficients are presented. These are calculated accor-ding to eq. 27, 1)( ttD . It is easily seen that superdiffusion, i.e. >1, yields a positive ex-ponent in this equation, hence resulting in an increasing diffusion coefficient function, with

0)(lim0

tDt

. With subdiffusion, on the other hand, the diffusion coefficient function is decreas-

ing (i.e. having a negative exponent) and diverging at the start, i.e.

)(lim0

tDt

(compare section

2.5). As an example, the diffusion coefficient functions for 6.98 mM DoTAB is plotted in figure 63.

06.106.987.628.6811.716.822.624.2 16.3∙t 0.428 3.3∙t -0.222

*8.1∙t 0.176

* ** *

7.1∙t 0.300

17.9∙t 0.346 1.4∙t -0.498

1.8∙t -0.178

D (t )/10-12 m2 s-1

long displacements

15.7∙t 0.297

23.1∙t 0.1241.6∙t -0.516

2.3∙t -0.353

D (t )/10-12 m2 s-1

short displacements[DoTAB]

/mM17.9∙t -0.094 1.0∙t -0.804

Table 12. The time-dependent diffusion coefficients. * no determination possible (since at least one position is zero (other than t=0)).


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y = 2.32E-12x-3.53E-01

y = 2.31E-11x1.24E-01

0

5E-12

1E-11

1.5E-11

2E-11

2.5E-11

3E-11

3.5E-11

0 1 2 3 4 5 6 7

t /s

D(t

) /m

²/s

long

short

Figure 63. The time-dependent diffusion coefficients for 6.98 mM.

4.3.13 Time-independent diffusion coefficients In table 13, the time-independent coefficients D are presented, both the ones obtained in this work (studying 10 probes concentration-1, 99 positions probe-1) as well as those obtained by Heidkamp (180 probes concentration-1, but only two positions probe-1). For comparison, time-dependent coefficients are presented as well.118 [DoTAB] /mM

0 9.4 ± 5.5 15.3 ± 2.16.10 14.4 ± 8.4 13.8 ± 2.06.98 16.5 ± 10.7 15.6 ± 2.17.62 16.7 ± 12.1 11.2 ± 1.88.68 5.6 ± 4.3 5.9 ± 0.911.7 0.005 ± 0.007 016.8 0.003 ± 0.005 022.6 7.2 ± 5.0 6.9 ± 0.824.2 19.6 ± 13.9 11.2 ± 1.59

D Fredriksson/10-12 m2 s-1 D Heidkamp/10-12 m2 s-1 D (t )/10-12 m2 s-1

14.4∙t -0.267

13.2∙t 0.055

15.4∙t 0.028

12.1∙t 0.2625.4∙t 0.169

4.3∙t 0.17713.2∙t 0.143

**

Table 13. The time-independent (with 95% confidence intervals) and time-dependent diffusion coefficients. See the main text for further explanations. Heidkamp’s data (here rescaled to fit eq. 25) used with permission. * no determination possible (since at least one position is zero (other than t=0)). In figures 64-65, the time-independent diffusion coefficients are visualised along with their confi-dence intervals. The major reason why the confidence intervals in first diagram are so large is that only ten probes were analysed. Anyway, it is clear that both methods, in most cases, give similar diffusion coefficients. The discrepancies that sometimes are present are caused by the fact that, although 99 steps were recorded for each probe, only 10 probes were analysed for each concentration; this is echoed in

118 These were calculated using all probe data in each experiment and then regressing according to eq. 25. Possible confidence intervals are beyond the scope of this work.


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the confidence intervals (compare those obtained by Heidkamp using 180 probes but just two steps concentration-1). Better agreement is, actually, obtained by comparing (i.e. the coefficient in the right column in table 13) with HeidkampD . This agreement probably arises since 01 , yielding

HeidkampDtD )( . Why, then, is the agreement better between and HeidkampD than between

nFredrikssoD and HeidkampD ? This could possibly be because the two-variable potential regression (yielding and ) carried out on the 990 points are better to use all data than the one-parameter linear regression yielding only nFredrikssoD – despite the fact that the same data (with only 10 probes concentration-1) was used in both cases.

0.05.0

10.015.020.025.030.035.040.0

0 5 10 15 20 25 30

[DoTAB]/mM

D

Figure 64. The time-independent diffusion coefficients /10-12 m2 s-1 determined using all data from 10 probes.

0.0

5.0

10.0

15.0

20.0

0 5 10 15 20 25 30

[DoTAB]/mM

D

Figure 65. The time-independent diffusion coefficients /10-12 m2 s-1 determined by Heidkamp using two-point data from 180 probes.


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4.4 Simulations of Diffusion in Microsoft Excel Very natural-looking trajectories were easily achieved using Microsoft Excel.

4.4.1 Using random exponent of one In figure 66, the trajectories of the first 10 (of totally 30) VPs are shown.

Figure 66. The first five VPs. Note that units are arbitrary. As is easily seen in figure 66, the trajectories are very similar to Brownian motion: Some steps are longer, others are shorter but they still have about the same magnitude; there are no real Lévy motions.

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6 8

x

y

1

2

3

4

5


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In figure 67, the mean square displacement is shown. As is easily seen, points come very close to a straight line, thus indicating normal diffusion. Also, is close to unity having the value of 0.94. In figure 68, the time-dependent diffusion coefficient (see eq. 27) is visualised. Since was slightly below unity,

)(tD is decreasing slightly. In figure 69, the distribution of step sizes is shown. The distribution seems quite normal but is not perfectly Gaussian.

Figure 69. The step size distribution. The distribution is

relatively normal.

00.5

11.5

22.5

33.5

44.5

5

0 1 2 3 4 5 6 7t

D(t

)

Figure 68. Time-dependent diffusion coefficient. Note that the units are arbitrary. However, D(t) typically have m2 s-1.

Figure 67. The first 10 VPs. Note that units are arbitrary.

y = 3.9762x0.9402

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12

t

<|r

|²>

0

50

100

150

200

250

300

350

400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Step |r|

Coun

t


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When 300 particles were analysed, the exponent came, as expected, very close to one (1.028) In figure 71, the step size distribution for all the 300 VP trajectories is shown. The distribution no longer seems properly normal, since no steps have a zero displacement – and since a bit too many have steps around the mean.

y = 3.3713x1.0284

05

10152025303540

0 2 4 6 8 10 12

t

<|r

|²>

Figure 70. Mean square displacement for 300 VP trajectories. Note that the exponent comes very close to unity.

0

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Step |r|

Coun

t

Figure 71. Step size distribution for 2637099300 steps


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4.3.2 Using different random exponents in Excel When using the random exponent n=2 things change. Motions seem a bit more Lévy-like (figure 72). The mean square displacement versus fic-tive time is shown in figure 73. The step size distribution is shown in 74. Clearly, the higher value of n does not affect . However, the step size distribution is strikingly different.

Figure 72. The first five trajectories with n=2 of totally 30 Virtual probes. Figure 73. The mean square displacement with n=2. 30 VPs. Figure 74. The step size distribution with n=2. 30 VPs.

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Step |r|

Coun

t

y = 1.6941x0.9482

0

5

10

15

20

0 2 4 6 8 10 12

t

<|r

|²>

-7

-6

-5

-4

-3

-2

-1

0

1

-6 -4 -2 0 2 4 6

x

y

1

2

3

4

5


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When using n=3 these effects are exaggerated (figures 74-76). Note particularly the Lévy like trajectories in figure 75. Also noteworthy, is the step size distribution bearing high similarities with the Lévy distribution (eq. 16).

Figure 75. The first five trajectories with n=3. 30 VPs.

Figure 76. The mean square displacement with n=3. 30 VPs.

Figure 77. The step size distribution with n=3. 30 VPs.

y = 1.663x0.9255

0

5

10

15

0 2 4 6 8 10 12

t

<|r

|²>

-6

-5

-4

-3

-2

-1

0

1

2

3

-5 -4 -3 -2 -1 0 1 2 3 4

x

y

1

2

3

4

5

0

200

400

600

800

1000

1200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Step |r|

Cou

nt


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5 Conclusions The main conclusion is that it is possible to classify diffusion based on the total displacement. This was revealed when studying Carlsson's as well as Heidkamp's films. Short displacements have lower values of the coefficient and also tend to have lower values of the exponents, . Long displacements often have exponents larger than one, typically 2.02.1 , thus indicating some kind of enhanced diffusion. Often though, the shorter displacements were seemingly sub-diffusive having < 1. However, due to the limited number of probes traced, these conclusions are not very safe. As Weeks (1998) stated, the signature of anomalous diffusion is the predominance of a few rare steps (compare 119). All diffusion (from real experiments) encountered in this work is properly defined as Brownian, since there are no steps way longer than others, even though the step size, naturally, varies. Unfortunately, it has not been possible to use step size distributions since the by-hand method does not offer the required resolution. However, in the upcoming Master's thesis work, these highly valuable tools are used. These can provide excellent clues to tell anomalous diffusion from normal. How, then could the 'enhanced' values of of the long displacement be explained and, likewise, how could the small values of for the small displacements be accounted for? A very good guess is that by investigating a greater number of probes, the exponent would tend to go to unity, also in accordance with Bouchaud et al. However, for the short displacement where probes, more or less, vibrate around a spot, real subdiffusion could indeed be the case; at least this possibility cannot be rejected at this stage. Richardson’s law was indeed not obeyed for the systems studied. Hence, this concept will not be investigated further. Interestingly, the different methods of calculating diffusion coefficients yield similar values. However, it seems that the method using a large number of probes but only two positions probe-1 is far more efficient in the way that confidence intervals are much smaller. Hence, it is re-commended the path used by Carlsson, To, Heidkamp and others to be used in the future. How-ever, in order to obtain time-dependent diffusion coefficients the many-position method is, ob-viously, called for. Curiously, a striking correspondence was found between the time-independent diffusion coefficients calculated by Heidkamp, and the values of . This was due to the trajectories being quite Brownian hence hence having DtD )( . The mathematical tool skewness coefficient could be valuable in pursuing anomalous diffusion. However, the values of the median step sizes (which are used to calculate the skew-ness) do only have a very limited number of values (0, 0.41, 0.58, and 1.04 µm with Heidkamp’s data). This abnormality is generated by using the by-hand method assuming a probe being placed on one integer pixel only. In the upcoming Master’s thesis work, using automatic high-resolution probe tracking, these methods will be readily exploited. It is possible to generate neat trajectories using Microsoft Excel, both ‘Brownian’ trajec-tories were all steps have similar lengths and more Lévy like trajectories with some very long and many small steps. The distributions in these cases bear some resemblance to real Brownian (i.e. Gaussian) and Lévy systems, respectively, but the shapes of these histograms seem a bit odd. Furthermore, one must bear in mind that simulations of this kind always come more or less far off from reality. Hence, the practical value of such simulations is always debatable.

119 figures A-D p. 4 in Weeks (1998) .


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6 Acknowledgements The author is grateful for the support of his supervisor, Gunilla Carlsson and his examiner Jan van Stam. He is also grateful for the great deal of independence in this work. He also wants to thank the present graduate student Hannah Heidkamp for excellent cooperation regarding the diploma work exhibition in May 2009. Moreover, thanks to Gunilla and Hannah, this time for providing the comprehensive libraries of video films and tracer shots forming the basis for the semi-experimental part of this work.


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References Beddard, Godfrey. Applying Maths in the Chemical and Biomolecular Sciences – an example based approach. (2009). Oxford. Boffetta, G., Celani A.. Pair dispersion in turbulence. (2000). Physica A, 288, 1-9. Bodurka, Jerzy, Seitter R.-O., Kimmich R., Gutsze A.. Field-cycling nuclear magnetic resonance relaxometry of moleculardynamics at biological interfaces in eye lenses: The Lévy walk mechanism. (1997). Journal of Chemical Physics, 107, 5621-5624. Bouchaud, J.P., Ott, A., Langevin, D., Urbach, W.. Anomalous Diffusion in Elongated Micelles and its Lévy Flight Interpretation. (1991). Journal de Physique II France, 1, 1465-1482. Carlsson, Gunilla. Latex Colloid Dynamics and Complex Dispersions. (2004). Karlstad University Studies. Chaves, A.S.. A fractional diffusion equation to describe Lévy flights. (1998). Physics Letters A, 239, 13-16. Fowlkes, J.D., Hullander, E.D., Fletcher B.L., Retterer S.T., Melechko A.V., Hensley, D.K., Simpson, M.L., Doktycz M.J.. Molecular transport in a crowded volume created from vertically aligned carbon nanofibres: a fluorescence recovery after photobleching study. (2006). Nanotechnology, 17, 5659-5668. Gioia, G., Lacorata G., Marques Filho E.P., Mazzino A., Rizza. U.. Richardson’s law in large-eddy simulations of boundary layer flows. (2004). Boundary layer meteorology, 113, 187-199. Greenenko, A.A., Chechkin, A.V., Shul’ga, N.F., Anomalous diffusion and Lévy flights in channeling. (2004). Physics Letters A, 324, 82-85. Heidkamp, Hannah. How interactions between anionic latex and the cationic surfactant DoTAB affect latex film formation. (2009). Karlstad University Studies. James, Glyn. Modern Engineering Mathematics. Second Edition. (1996). Addison-Wesly Longman. Jullien, Marie-Caroline, Paret, Jérôme, Tabeling, Patrick. Richardson Pair Dispersion in Two-Dimensional Turbulence. (1999). Physical Review Letters, 82 (14), 2872-2875. Klafter, J., Blumen A., Shlesinger, M.F.. Stochastic pathway to anomalous diffusion. (1987). Physical review A, 35, 3081-3085. Klafter, J., Blumen, A., Zumofen, G., Shlesinger, M.F.. Lévy walk approach anomalous diffusion. (1990). Physica A, 168, 637-645. Klafter, Joseph, Shlesinger, Michael F., Zumofen, Gert. Beyond Brownian Motion. (1996). Physics Today, 49, s.33-39. Kreyszig, Erwin. Advanced Engineering Mathematics. 8th Edition. (1999). John Wiley & Sons, Inc.


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Lindley, David. Uncertainty. Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science. (2007). Anchor Books. Ott, A., Bouchaud, J.P., Langevin, D., Urbach, W. Anomalous Diffusion in “Living Polymers”: A Genuine Levy Flight? (1990). Physical Review Letters, 65, 2202-2204. Reynolds, A.M. Adaptive Lévy walks can outperform composite Brownian walks in non-destructive random searching scenarios. (2009). Physica A, 388, 561-564. Shlesinger, Michael F., Klafter, Joseph. Random Walks in Liquids. (1989). Journal of Physical Chemistry, 93, 7023-7026. Shlesinger, Michael F, Zaslavsky, George M., Klafter, Joseph. Strange kinetics. (1993). Nature, 163, 31-37. Shlesinger, Michael F. Physics in the noise. (2001). Nature. 411, 641. Sharma, Shama. Vishwamittar. Brownian Motion Problem: Random Walk and Beyond. 2005. Resonance. 10. 49-66. Solomon, T.H., Weeks, Eric R., Swinney, Harry L.. Observation of Anomalous Diffusion and Lévy Flights in a Two-dimensional Rotating Flow. (1993). Physical Review Letters, 71, 3975-3978. To, My. Interactions between cross-linked latexes and ionic surfactants. (2007). Karlstad University Studies. Viswanathan G.M., Raposo, E.P., da Luz M.G.E.. Lévy flights and superdiffusion in the context of biological encounters and random searches. (2008). Physics of Life Reviews, 5, 133-150. Weeks, Eric R., Swinney Harry L.. Random walks and Lévy flights observed in fluid flows. (1998). Springer-Verlag New York Inc. Available online: www.physics.emory.edu/%7Eweeks/papers/nst98.pdf Weeks, Eric R., Urbach, Jeff, Swinney Harry. Random walks and Anomalous Diffusion. http://chaos.utexas.edu/research/annulus/rwalk.html (as by April 2nd 2009). Weiss, Matthias, Elsner, Markus, Kartberg, Fredrik, Nilsson, Tommy. Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells. (2004). Biophysical Journal, 87, 3518-3524. Wolfram Mathworld. Pearson's Skewness coefficients. mathworld.wolfram.com/PearsonsSkewnessCoefficients.html (as by 21th Nov. 2009).


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Appendix 1. Core formulae 1. 2

02

0 )()(|| yyxx r Displacement

2.

ba if

if0

baba )( Dirac delta

3.

ba if0

if1)(

baba Kronecker delta

4. )(e)(})(F{ 2- Fdttftf ti

Fourier transform

6. dsFF tπi

21- e)(})({F Inverse Fourier transform

7.

0

)(e)( sFdttff(t)L st Laplace transform

8. deviation standard

median)-(mean3skewness [Pearson’s second] skewness coefficient

9. tt ~)(|| 2r Normal diffusion

10. tt ~)(|| 2r Anomalous diffusion

16. 1|~|)( rrp Lévy flight 21. )|(|||),( vµ tAt rrr Example of Lévy walk

25. t

D2||r

Time-independent diffusion coefficient

27. 1)( ttD Time-dependent diffusion coefficient

29. t

xD

xt

2

0,0lim Diffusion coefficient using limiting values

32. tCRttp // /

~),(3229 eR

34. 32

2 tCtR )( Richardson

38. tt )(|| 2r Regression formula, anomalous diffusion

39. tt loglog)(||log 2 r Logarithmic version of eq. 38

40. 22 )()( ABAB yyxxR Pair separation

A Brief Survey of Lévy Walks

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