Characterizing N-dimensional anisotropic Brownian motion by the distribution ofdiffusivitiesMario Heidernätsch, Michael Bauer, and Günter Radons Citation: The Journal of Chemical Physics 139, 184105 (2013); doi: 10.1063/1.4828860 View online: http://dx.doi.org/10.1063/1.4828860 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/18?ver=pdfcov Published by the AIP Publishing

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THE JOURNAL OF CHEMICAL PHYSICS 139, 184105 (2013)

Characterizing N-dimensional anisotropic Brownian motion by thedistribution of diffusivities

Mario Heidernätsch, Michael Bauer, and Günter Radonsa)

Technische Universität Chemnitz, Faculty of Sciences, Institute of Physics, Complex Systems and NonlinearDynamics, D-09107 Chemnitz, Germany

(Received 29 August 2013; accepted 19 October 2013; published online 11 November 2013)

Anisotropic diffusion processes emerge in various fields such as transport in biological tissue anddiffusion in liquid crystals. In such systems, the motion is described by a diffusion tensor. For aproper characterization of processes with more than one diffusion coefficient, an average descriptionby the mean squared displacement is often not sufficient. Hence, in this paper, we use the distributionof diffusivities to study diffusion in a homogeneous anisotropic environment. We derive analyticalexpressions of the distribution and relate its properties to an anisotropy measure based on the meandiffusivity and the asymptotic decay of the distribution. Both quantities are easy to determine fromexperimental data and reveal the existence of more than one diffusion coefficient, which allows thedistinction between isotropic and anisotropic processes. We further discuss the influence on the anal-ysis of projected trajectories, which are typically accessible in experiments. For the experimentallymost relevant cases of two- and three-dimensional anisotropic diffusion, we derive specific expres-sions, determine the diffusion tensor, characterize the anisotropy, and demonstrate the applicabilityfor simulated trajectories. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828860]

I. INTRODUCTION

The random motion of suspended particles in a fluid,which is usually referred to as Brownian motion, is anold but still fascinating phenomenon. Especially, wheninhomogeneous1–3 or anisotropic media4–7 are involved,many questions are still open. From the theoretical pointof view, much work has been done8 to predict the statis-tical properties of the trajectories of such particles usingstochastic methods. On the other side, the development ofexperiments only recently allows obtaining the paths of indi-vidual molecules and particles. Especially the observation oftwo-dimensional trajectories using video-microscopic meth-ods, for instance, by single-particle tracking (SPT), is alreadysuccessfully applied to biological systems9, 10 or to under-stand the microrheological properties of complex liquids.11, 12

But also the observation of three-dimensional paths becomesfeasible.13–15 The statistical analysis of these trajectories isusually accomplished by measuring the mean square displace-ment (msd) in order to get the diffusion coefficients for thematching theoretical description. However, in the anisotropiccase the diffusive properties depend on the direction of mo-tion and are described by a diffusion tensor. In such sys-tems, the analysis of msd’s turned out to be not sufficientto determine the anisotropy and extract the values of the dif-fusion coefficients.16–18 For similar reasons, we already in-troduced the distribution of single-particle diffusivities as anadvanced method to analyze stochastic motion in heteroge-neous systems19 involving more than one diffusion coeffi-cient. Hence, we distinguish between diffusivities as fluc-tuating quantities and diffusion coefficients as mean values.

a)Electronic mail: radons@physik.tu-chemnitz.de

An integrated version of the distribution of diffusivities is al-ready successfully applied to analyze diffusion in inhomoge-neous media,20 spectral diffusion,21, 22 and even anisotropicmedia.4–6 It should be noted that this distribution is closelyrelated to the displacement distribution.23, 24 However, thedistribution of diffusivities is superior since it is station-ary for time-homogeneous diffusion processes. Thus, exper-iments conducted on different time scales can be comparedeasily. Furthermore, this new method was extended to thedistribution of generalized diffusivities to characterize datafrom anomalous diffusion processes, which offers, for in-stance, a deeper understanding of weak ergodicity breaking.25

For anomalous diffusion, our method is an efficient alterna-tive to other techniques26 for distinguishing between differ-ent anomalous diffusion processes. Here, we want to estab-lish an efficient method to distinguish between different nor-mal diffusion processes involving more than one diffusioncoefficient.

In the current article, we show the applicability of thedistribution of diffusivities to analyze trajectories of homo-geneous anisotropic Brownian motion. Anisotropic processesare relevant in many different applications such as diffu-sion in porous media,27, 28 in liquid crystals,5, 29 or in biolog-ical tissues,30–32 where the anisotropy originates, e.g., fromaligned filaments in cells.33, 34 We present the properties ofthe distribution of diffusivities as well as their relations to es-tablished quantities. In order to assess the parameters of theprocess, we calculate the characteristic function, cumulants,and moments of the distribution. From these moments, thediffusion coefficients can be determined, which characterizethe mobility of the diffusing molecules or the viscosity of thesurrounding medium. For the asymptotic decay of the distri-bution of diffusivities, we derive a general expression, which

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184105-2 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

involves the largest diffusion coefficient of the system. In con-junction with the mean diffusion coefficient of the system, theasymptotic decay enables a data-based distinction betweenisotropic and anisotropic processes. Based on these quanti-ties, we provide a measure to characterize the anisotropy ofthe process from the analysis of SPT data. Such a quantitativemeasure provides valuable information about the dimension-ality of the process and characterizes the aspect ratio of thediffusing molecules or of the molecules in the surroundingmedium. Since in experiments the reconstruction of the com-plete diffusion tensor is of great interest, we extend our con-cept to tensorial diffusivities, which offer a simple method todetermine the entries of the tensor.

Due to restrictions in SPT experiments, the complete tra-jectory is often not accessible.4, 23 Hence, we investigate theinfluence on the distribution of diffusivities and the detectionof the anisotropy if only projections of the actual trajectoryare observed. Even in such cases it is possible to estimatebounds of the diffusion coefficients from the given projectionsof the diffusion tensor. Since especially two-dimensional andthree-dimensional diffusion processes have a high relevancein experiments, we apply our considerations to these systems.For homogeneous anisotropic diffusion in two dimensions, ananalytical expression of the distribution of diffusivities existsand its moments can be related to the diffusion coefficients,which enter the anisotropy measure. Moreover, we explain thedetails of reconstructing the diffusion tensor from the tenso-rial diffusivities as well as from projections of the trajectory.Three-dimensional processes are investigated analogously al-though a closed-form expression of the distribution of diffu-sivities does not exist. Additionally, we deal with anisotropicprocesses where one diffusion coefficient is degenerated cor-responding to diffusion of uniaxial molecules typical for liq-uid crystalline systems.35

The paper is organized as follows. In Sec. II, we brieflyrecall the theoretical principles of anisotropic Brownian mo-tion based on the diffusion tensor and introduce the distri-bution of single-particle diffusivities, its properties and rela-tions to established quantities. To apply our new concepts toN-dimensional homogeneous anisotropic diffusion processes,we provide in Sec. III a general expression for the distri-bution of diffusivities. We demonstrate how to distinguishbetween isotropic and anisotropic processes and explain thereconstruction of the diffusion tensor. Since in experimentstypically a projection of the motion is observed, we charac-terize the distribution of diffusivities of the projected trajec-tories. To illustrate our presented theoretical considerations,in Sec. V, we apply our results comprehensively to specificsystems of anisotropic diffusion which are typical for exper-imental setups. We substantiate the applicability of our find-ings by analyzing data from simulated anisotropic diffusionprocesses.

II. DEFINITIONSA. Anisotropic diffusion

A N-dimensional anisotropic Brownian motion is com-pletely defined by its propagator36

p(x, t |x′, t ′) = (2π )−N2√

[2(t − t ′)]N det D

× exp

[−1

2

1

2(t − t ′)(x − x′)TD−1(x − x′)

],

(1)

where D = OTD̂O is the positive definite and symmetric dif-fusion tensor, D̂ = diag(D1,D2, . . . ,DN ) denotes its diago-nalized form with the diffusion coefficients Di belonging tothe principal axes, and O is an orthogonal matrix which de-scribes the orientation of the principal axes relative to theframe of reference.

For the simulation of such processes, an alternative de-scription exists, where the trajectories are evolved by theLangevin equation

dxdt

=√

2Dξ (t) (2)

with√

D = OT√

D̂O and√

D̂ = diag(√

D1,√

D2, . . . ,√DN ). The vector ξ (t) = [ξ1(t), . . . , ξN (t)]T denotes

Gaussian white noise in N dimensions with 〈ξ (t)〉 = 0 and〈ξ i(t)ξ j(t′)〉 = δijδ(t − t′) ∀ i, j ∈ {1, 2, . . . , N}.

Assuming time-translation invariance Eq. (1) is simpli-fied to the probability density p(x′ + r, τ |x′) of displace-ments r = x − x′ by substituting τ = t − t′. This conditionalprobability density is averaged by the equilibrium distribu-tion p0(x′) given by the Boltzmann distribution to obtain theensemble-averaged probability density

p(r, τ ) =∫

dNx′ p(x′ + r, τ |x′)p0(x′)

= (2π )−N2√

det �exp

(−1

2rT�−1r

)(3)

of a displacement r = (r1, . . . , rN )T in the time interval τ .Thus, p(r, τ ) is a N-dimensional Gaussian distribution withzero mean and covariance tensor � = 2τD.

Expressions with dimensionality N > 3 may be interest-ing for simultaneous observation of d particles correspond-ing to an extended many-particle state space x(x1, . . . , xd ).The simplest case is the two-dimensional motion of two non-interacting particles with different diffusion coefficients D1

and D2. From the sequence of simultaneous positions, we ob-tain a four-dimensional displacement vector r and the diffu-sion tensor D̂ = diag(D1,D1,D2,D2). This is a limiting caseof a more interesting situation where particles interact, e.g.,head and tail of a polymer, where the distance between bothpositions depends on the conformation of the polymer.

B. Distribution of diffusivities

By observing a trajectory x(t) of an arbitrary stochasticprocess in N dimensions, individual displacements during agiven time lag τ can simply be measured for a certain particle.Moreover, it is natural to relate each displacement to a single-particle diffusivity

Dt (τ ) = [x(t + τ ) − x(t)]2

2Nτ. (4)

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184105-3 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

This simple transformation of displacements to diffusivitiesoffers the advantage to compare these quantities for differ-ent experimental setups and different τ . Since for a fixedtime lag τ the single-particle diffusivity is fluctuating alonga trajectory, an important quantity is given by the probabilitydensity p(D, τ ). Therefore, the distribution of single-particlediffusivities19 is defined as

p(D, τ ) = 〈δ[D − Dt (τ )]〉, (5)

where 〈· · · 〉 either denotes a time average 〈· · · 〉= limT →∞ 1/T

∫ T

0 · · · dt , which is typically accessibleby SPT, or an ensemble average as measured by otherexperimental methods, such as nuclear magnetic resonance.37

For ergodic systems, as considered here, time average andensemble average coincide. Since the displacements arerescaled by τ , one can easily compare the distributions ofprocesses observed on different time scales. It should benoted that other definitions of diffusivity distributions exist inthe literature.38

For time-homogeneous systems, i.e., when the distribu-tion of displacements p(r, τ ) is independent of t, Eq. (5) canbe rewritten as

p(D, τ ) =∫

dNr δ

(D − r2

2Nτ

)p(r, τ ) (6)

transforming p(r, τ ) into the distribution of diffusivities.The literature discusses the influence of correlations be-

tween segments of length τ defining the displacements for thedetermination of the mean squared displacement.39, 40 Specif-ically, the displacements with time lag τ are correlated if thedifference δ between the initial times of the displacements issmaller than τ and vanishes only if the shift δ of the segmentsis larger than τ . These correlations can be calculated explic-itly for the case of simple Brownian motion for which our dis-tribution of diffusivities is independent of the time lag. Thismeans that our distribution of diffusivities is not affected bythe relation of time lag τ and shift δ of the segments. Thisimplies that calculating the distribution p(D, τ ) from over-lapping or non-overlapping segments yields the same result.Whereas for homogeneous systems the distribution is station-ary and independent of τ , which means that the first momentor mean D is stationary as well, for inhomogeneous systemsthe distribution and its moments become stationary only forlarge τ . Thus, we encourage experimentalist to vary τ in theirmeasurements to detect inhomogeneities.

For data from SPT experiments, displacements froma trajectory are transformed to diffusivities according toEq. (4) and the distribution of diffusivities is obtained by bin-ning these diffusivities into a normalized histogram accordingto Eq. (5). Due to the nature of experiments, the number ofdisplacements may be small. As a consequence, this leads tofluctuations in the distribution which are related to the insuf-ficient statistics of rare displacements. However, adapting thebin width of the histogram may reduce these fluctuations.

For homogeneous isotropic processes in N dimensions,the msd grows linearly with τ , since it obeys the well-knownEinstein relation 〈r2(τ )〉 = 2NDcτ , where Dc is the diffusioncoefficient of the process. Due to the transformation of dis-placements to diffusivities by Eq. (4) the linear dependence

on τ is removed. Hence, the corresponding distribution ofdiffusivities becomes stationary and comprises single-particlediffusivities fluctuating around Dc. For N-dimensional homo-geneous isotropic processes, the distribution of diffusivities

pNdDc

(D) =(

N

2Dc

) N2 D

N2 −1

�(N2 )

exp

(− N

2Dc

D

)(7)

is obtained, where �(x) denotes the gamma function.Equation (7) is identified as a χ2-distribution of N degreesof freedom and results directly from the sum of the squares ofN independent and identically distributed Gaussian randomvariables with variance Dc/N and vanishing mean.41 Sincethese variables are the squared and rescaled components ofthe displacement vector r2

i (τ )/(2Nτ ), their sum correspondsto the diffusivity.

For inhomogeneous isotropic diffusion processes whichare ergodic, Eq. (7) provides a further useful application.Since for normal diffusion in N dimensions the Einstein re-lation holds for large τ , p(D, τ ) converges to the stationarydistribution given by Eq. (7). In this case, Dc is the mean dif-fusion coefficient of the process.

C. Moments

The distribution of diffusivities is fully characterized byits corresponding moments

Mm(τ ) = 〈D(τ )m〉 =∞∫

0

dD Dm p(D, τ ). (8)

It should be noted that the first moment for large τ is knownas the mean diffusion coefficient, which is obtained by awell-defined integration. This is in contrast to msd measure-ments, where the mean diffusion coefficient is determinedby a numerical fit to the slope of the msd. If the numberof displacements obtained in the experiment is small, thedistribution may have strong fluctuations due to insufficientstatistics. However, the most interesting low-order momentsof the distribution can also be calculated directly from thediffusivities independent of the bin width of the distribution,which yields more reliable values of the moments, even forsmall samples.

By inserting Eq. (6) into Eq. (8), the integration over Dyields as a result the moments

Mm(τ ) = 1

(2Nτ )m

∫dNr r2mp(r, τ )

= (2Nτ )−m〈r2m〉. (9)

They are directly related to the moments of the distributionof displacements and, thus, to the moments of the propagatorp(x, t |x′, t ′).

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III. PROPERTIES OF THE DISTRIBUTION OFDIFFUSIVITIES FOR HOMOGENEOUS ANISOTROPICBROWNIAN MOTION

A. Distribution of diffusivities

For homogeneous anisotropic diffusion in N dimensions,where p(r, τ ) is a Gaussian distribution with zero mean givenby Eq. (3), the computation of the distribution of diffusivi-ties, its moments, or its characteristic function is simplifiedby reformulating the integral of Eq. (6). Applying the coordi-

nate transformation r = Qq with Q = √2τOT

√D̂ gives for

the distribution of diffusivities

pNdD̂

(D) =∫

dq1 · · ·∫

dqN

× δ

(D − 1

N

N∑i=1

Diq2i

)N∏

j=1

p(0,1)(qj ), (10)

where p(0,1)(qj ) = 1√2π

exp(− 12q2

j ). Thus, the distribution ofdiffusivities is calculated by integration over independentstandard normally distributed variables with zero mean andunit variance. Since the msd for homogeneous anisotropic dif-fusion again grows linearly as in the homogeneous isotropiccase, the τ dependency in the distribution of diffusivitiesvanishes.

By obtaining the distribution of diffusivities, for instance,from displacements along a single trajectory, informationabout the orientation of the diffusion tensor is lost. However,all directions contribute to the distribution and, thus, it stillcontains information about the diffusion coefficients corre-sponding to the principal axes, i.e., the eigenvalues of D.

B. Characteristic function, cumulants, and moments

With the transformation Eq. (10), the moments and thecharacteristic function of the distribution of diffusivities ofanisotropic Brownian motion can be calculated. For the mo-ments, given by Eq. (8), this yields

MNdm = 1

Nm

∫dq1 · · ·

∫dqN

(N∑

i=1

Diq2i

)m N∏j=1

p(0,1)(qj ).

(11)So, for instance, the first moment of the distribution of diffu-sivities is given by

MNd1 = 1

N

N∑i=1

Di = 〈D(τ )〉 , (12)

which is simply the arithmetic mean of all the diffusion co-efficients Di and coincides with the slope of the msd. Forhigher moments of the distribution of diffusivities, it is easierto calculate its characteristic function GNd

D̂(k) = 〈exp(ikD)〉

= ∫∞0 dD exp(ikD)pNd

D̂(D) by substituting pNd

D̂(D) from

Eq. (10) and performing the Fourier transform to obtain

GNdD̂

(k) =N∏

j=1

∫dqj exp

(ik

Djq2j

N

)p(0,1)(qj )

=N∏

j=1

(1 − ik

2Dj

N

)− 12

. (13)

From the characteristic function Eq. (13), the cumulantsof the distribution pNd

D̂(D) are obtained as

κm = 1

im∂m ln GNd

D̂(k)

∂km

∣∣∣∣∣k=0

= 2m−1(m − 1)!

Nm

N∑i=1

Dmi (14)

for m > 0. The moments are recursively related to thecumulants by

Mm =m−1∑k=0

(m − 1

k

)κm−kMk (15)

with initial value M0 = 1.42

It should be noted that the characteristic functionin Eq. (13) is a product of different characteristic func-tions in Fourier space. Hence, the distribution of diffusiv-ities of a N-dimensional anisotropic system is determinedby inverse Fourier transform of the characteristic func-tion pNd

D̂(D) = F−1[GNd

D̂(k)] = F−1[

∏Ni=1 G1d

Di/N(k)], where

G1dDi/N

(k) = F[p1dDi/N

(D)] is the Fourier transform ofthe one-dimensional distribution of diffusivities p1d

Di/N(D)

= 1/√

2πDDi/N exp(−ND/(2Di)) with diffusion coeffi-cient Di/N. Correspondingly, the distribution of diffusivitiesof a N-dimensional anisotropic system is obtained by convo-lution of N one-dimensional distributions of diffusivities

pNdD̂

(D) = {p1d

D1/N∗ p1d

D2/N∗ · · · ∗ p1d

DN/N

}(D)

=∞∫

0

d1 · · ·∞∫

0

dN

× δ

(D −

N∑i=1

i

)N∏

j=1

p1dDj /N

(j ), (16)

which follows directly from Eq. (10). Thus, with Eqs. (10),(13), and (16), we provide three equivalent expressions to de-termine the distribution of diffusivities in terms of the eigen-values Di of D. Depending on the considered experimentalsystem, each representation offers its own advantages.

C. Asymptotic decay

In the following, we present the asymptotic behavior ofthe distribution of diffusivities for homogeneous anisotropicBrownian motion. We show how the anisotropy of the processcan be identified.

Considering a M-fold degeneracy of the largest diffusioncoefficient with D1 = · · · = DM > DM + 1 ≥ · · · ≥ DN thedistribution of diffusivities of the homogeneous anisotropic

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184105-5 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

system is obtained from the convolution

pNdD̂

(D) = {pMd

D1/N∗ p1d

D(M+1)/N∗ · · · ∗ p1d

DN /N

}(D), (17)

where pMdD1/N

(D) is the distribution of diffusivities of the M-dimensional isotropic system Eq. (7) with diffusion coeffi-cient Dc = D1/N, which results from the convolution of Midentical one-dimensional distributions p1d

D1/N(D).

For D � D1DM + 1/(D1 − DM + 1), an asymptotic expan-sion for large D is performed and yields the asymptotic be-havior of Eq. (17)

pNdD̂

(D)D→∞∼

(N

2D1

)M2 D

M2 −1

�(

M2

)× exp

(− N

2D1D

) N∏j=M+1

√D1

D1 − Dj

. (18)

Thus, the leading behavior in the logarithmic representationis given by

log pNdD̂

(D)D→∞∼ − N

2D∞D, (19)

with D∞ = max (D1, D2, . . . , DN), i.e., an exponential decayinvolving the largest diffusion coefficient of the anisotropicsystem.

In homogeneous isotropic systems D∞, which describesthe asymptotic decay, is equal to the isotropic diffusioncoefficient Dc, which further coincides with the first mo-ment 〈D〉. The corresponding distribution of diffusivities isa χ2-distribution given by Eq. (7). This is in contrast tothe anisotropic case, where 〈D〉 < D∞. Thus, a discrep-ancy between 〈D〉 and D∞ leads to deviations from theχ2-distribution and rules out a homogeneous isotropic pro-cess. In general, this can be exploited to detect that the ob-served system comprises more than one diffusion coefficient.By further assuming homogeneity such a system is identifiedas an anisotropic one.

A quantitative measure for the discrepancy between 〈D〉and D∞ is given by

η = D∞〈D〉 − 1 (20)

which characterizes the deviation from the homogeneousisotropic case. Thus, for homogeneous systems it quantifiesthe anisotropy of the process. In cases where both values coin-cide, i.e., the system is isotropic, η becomes zero. In contrast,if one diffusion coefficient is much larger than all others, 〈D〉→ D∞/N resulting in η = N − 1, which denotes the largestpossible anisotropy in N dimensions. Thus, η is a measureof the anisotropy, but it is not suitable to compare systemsof different dimensionality N. It should be noted that similarmeasures exist.16, 43

From experimental data, both quantities for theanisotropy measure Eq. (20) can be determined easily. Themean diffusion coefficient 〈D〉 corresponds to the first mo-ment of the distribution of diffusivities and is obtainedby averaging the diffusivities. The decay for large D isobtained from a fit to f (D) = c exp(−λfitD) to calculateD∞ = N/(2λfit). The actual dimensionality Neff of processes

observed in N ≥ Neff dimensions can be estimated withNeff = 2〈D〉λfit leading to η = N/Neff − 1. For example, if anobserved N-dimensional motion yields the largest anisotropyvalue of η = N − 1, the process is effectively a one-dimensional motion.

D. Reconstruction of the diffusion tensor

For experiments, it is of great interest to reconstruct thediffusion tensor D from measurements. If complete informa-tion about the trajectories is available, the diffusion tensor ofthe homogeneous anisotropic process can be estimated via thedisplacements. By defining tensorial diffusivities analogouslyto Eq. (4)

Dijt (τ ) = [xi(t + τ ) − xi(t)][xj (t + τ ) − xj (t)]

2τ, (21)

where xi(t) denotes the ith component of the N-dimensionaltrajectory x(t), the linear τ dependence of the mixed displace-ments is removed. These tensorial diffusivities are simplyaveraged

Dij = ⟨D

ijt (τ )

⟩(22)

providing an estimator for the corresponding elements of D.Here, 〈· · · 〉 either denotes a time average or an ensemble av-erage depending on the available data.

IV. PROJECTION TO A M-DIMENSIONAL SUBSPACE

Due to experimental restrictions, the complete trajectoryis often not accessible but its projection on a M-dimensionalsubspace can be measured. Such processes are commonlyknown as observed diffusion.44, 45

The projection of the distribution of displacementsEq. (3) on the considered subspace is the marginal probabilitydensity

p(rMα , τ ) =

∫drα1 · · ·

∫drαN−M

p(r, τ ), (23)

where rαi, i = 1, . . . , (N − M) denotes (N − M) arbitrar-

ily chosen directions which are integrated out. The vectorα = (α1 · · · αN−M ) contains the indices αi describing whichelements of r are omitted. Alternatively, the projected distri-bution of displacements is computed by the M-dimensionalinverse Fourier transform of the characteristic functionof p(r, τ ) where the components, kαi

= 0, ∀i ∈ {1, . . . ,

N − M}, which correspond to the chosen directions, are dis-carded. Hence, the distribution of displacements of the sub-space is

p(rMα , τ

) =∫

dMkMα

1

(2π )Mexp

[−i(kM

α

)TrMα

]G(kM

α

)(24)

with the characteristic function of the projected propa-gator G(kM

α ) = exp[−(kMα )T�M

α kMα ]. The vector kM

α is aM-dimensional sub-vector of the complete k-space and �M

α

denotes a principal M × M submatrix of � obtained by dele-tion of rows and columns with corresponding indices αi.

The distribution of diffusivities of such a projecteddiffusion process is calculated analogously to Eq. (6) by

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184105-6 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

integrating over rMα . Since � = 2τD is a symmetric, posi-

tive definite matrix for τ > 0, all principal submatrices �Mα

are symmetric, positive definite matrices as well and can bediagonalized. Hence, the projected distribution of diffusivi-ties has the M-dimensional form of the generic expressionEqs. (10), (13), or (16). However, it depends on the eigen-values DM

k,α, k = 1, . . . ,M of the projected diffusion tensorDM

α = �Mα /(2τ ). If the eigenvalues of D are identified as

D1 ≥ D2 ≥ · · · ≥ DN (25)

and the eigenvalues of DN−1

( α )are

DN−1

1,( α )≥ DN−1

2,( α )≥ · · · ≥ DN−1

N−1,( α ), ∀α ∈ {1, . . . , N},

(26)the well-known interlacing inequalities46 require

Dk ≥ DN−1

k,( α )≥ Dk+1, ∀k ∈ {1, . . . , N − 1} (27)

for all α ∈ {1, . . . , N}. This expression is applied recursively(N−M) times to obtain a relation for the eigenvalues of theprincipal M × M submatrix47

Dk ≥ DMk,α ≥ Dk+N−M, ∀k ∈ {1, . . . ,M} (28)

for arbitrary α. By implication, if at least two eigenvaluesof the submatrix DM

α differ, i.e., the projected process isanisotropic, Eq. (27) states recursively that the complete pro-cess is anisotropic as well. Thus, the distribution of diffu-sivities of the projected N-dimensional anisotropic Brownianmotion may already indicate the anisotropy of the completeprocess as well as the magnitude of one of the involved diffu-sion coefficients. However, a single projection is not sufficientto obtain the underlying diffusion coefficients.

Nevertheless, it is possible to estimate the bounds of thediffusion coefficients. The lower bound of the eigenvalues isgiven by zero, due to the positive semidefiniteness of D. Anupper bound for the largest eigenvalue can be found if enoughprojections or submatrices are available to comprise all diago-nal elements of D. By use of the relation between the trace ofa N × N matrix A and its eigenvalues λi, tr A = ∑

i λi , subto-tals of the trace of D are given by the sum of the eigenvaluesof the respective submatrices. If the non-overlapping orthog-onal projections of D defined by α compose a partition of theset {1, . . . , N}, the trace of the tensor is given by

tr D =N∑

i=1

Di =∑

α

tr DMα =

∑α

∑k

DMk,α (29)

with⋃̇

α = {1, . . . , N}, where the partition elements α donot necessarily have identical dimensionality.

For example, if one measures the eigenvalues of two non-overlapping projections of a 3 × 3 diffusion tensor D, thetrace of D is given by

tr D = tr D1(1 3) + tr D2

(2)

= D11,(1 3) + D2

1,(2) + D22,(2). (30)

Thus, the eigenvalue inequalities for that example using therelations above are given by

tr D ≥D1 ≥ max(D1

1,(1 3),D21,(2)

) ≥ D2

D2 ≥ min(D1

1,(1 3),D22,(2)

) ≥ D3 ≥ 0, (31)

which allows a rough estimation of the diffusion coefficientsfrom the given projections.

V. SPECIFIC SYSTEMS

A. Two-dimensional systems

The distribution of diffusivities of a two-dimensional ho-mogeneous anisotropic system can be calculated explicitly,for instance, via Eq. (16), resulting in

p2dD̂

(D) =∞∫

0

d1

∞∫0

d2 δ [D − (1 + 2)]

×p1dD1/2(1)p1d

D2/2(2)

=exp

[− 1

2

(1

D1+ 1

D2

)D]

√D1D2

I0

[1

2

(1

D1− 1

D2

)D

],

(32)

where I0(x) denotes the modified Bessel function of the firstkind. The first two moments of this distribution, as given byEq. (8), yield

〈D〉 = M1 = 1

2(D1 + D2) (33)

and

〈D2〉 = M2 = 1

4

(3D2

1 + 2D1D2 + 3D22

). (34)

Hence, the mean diffusion coefficient coincides with the arith-metic mean of the diffusion coefficients belonging to thetwo directions of the anisotropic system as expected fromEq. (12). Solving the simultaneous Eqs. (33) and (34) yieldsthe expression

D1,2 = M1 ±√

M2 − 2M21 (35)

to obtain the diffusion coefficients D1 and D2 from themoments.

The asymptotic behavior of Eq. (32) for large D is givenby Eq. (18) and yields

p2dD̂

(D)D→∞∼

exp(− D

D∞

)√|D1 − D2|πD

(36)

with D∞ = max (D1, D2). Thus, the asymptotic behavior inthe logarithmic representation is given by

log p2dD̂

(D)D→∞∼ − D

D∞, (37)

which corresponds to the decay of the distribution of diffu-sivities in two-dimensional homogeneous isotropic systemswith diffusion coefficient D∞, i.e., an exponential decay withthe largest diffusion coefficient of the anisotropic system.Accordingly, the smallest diffusion coefficient is given by

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2〈D〉 − D∞. From the asymptotic decay and the mean diffu-sion coefficient, the anisotropy of the system is characterizedby Eq. (20) and corresponds to the ratio

η = |D1 − D2|D1 + D2

=√

M2 − 2M21

M1, (38)

which is also related to the moments.The diffusion coefficients D1, D2 can also be obtained

from the asymptotic behavior for vanishing D. Since

limD→0

p2dD̂

(D) = (D1D2)−12 , (39)

the corresponding value in experimental data is determined byextrapolating the distribution of diffusivities in a log-log plottowards D = 0. In conjunction with an estimate of the largestdiffusion coefficient from a fit to Eq. (37), both diffusion co-efficients can be identified. This provides a consistency checkfor the calculation via the moments of the distribution of dif-fusivities given in Eq. (38).

To substantiate our analytical expressions by results fromsimulations, a random walk was performed by numerical inte-gration of the Langevin equation, Eq. (2), in two dimensionsusing the diffusion tensor

D =(

4√

3√

3 2

)(40)

with eigenvalues D1 = 5 and D2 = 1. The obtained trajec-tory of the two-dimensional homogeneous anisotropic diffu-sion process consisted of 105 displacements and its distribu-tion of diffusivities is depicted in Fig. 1. The agreement ofthe normalized histogram from simulated data with the an-alytic distribution Eq. (32) is obvious. Deviations betweensimulation and the analytic curve for large D are due to in-sufficient statistics from the finite number of displacements.

p(D

)

D

10-4

10-3

10-2

10-1

100

0 5 10 15 20 25 30

FIG. 1. The distribution of diffusivities (histogram) from a simulated trajec-tory of a homogeneous anisotropic diffusion process in two dimensions withdiffusion tensor D given by Eq. (40) agrees well with the analytic distributionof diffusivities (solid line) from Eq. (32) with D1 = 5 and D2 = 1 denotingthe eigenvalues of D. Additionally, the asymptotic function Eq. (36) (dot-ted line, D∞ = 5) agrees reasonably for large D. Furthermore, a distributionof diffusivities (dashed line) of two-dimensional isotropic diffusion with thesame mean diffusion coefficient Dc = 〈D〉 = (D1 + D2)/2 = 3 is shown forcomparison. The different asymptotic decays are clearly visible and allow thedistinction from homogeneous isotropic processes.

Moreover, Fig. 1 shows the mono-exponential behavior cor-responding to isotropic diffusion in two dimensions for com-parison. Although the mean diffusion coefficients of bothprocesses coincide, the asymptotic decays of the distribu-tions differ. The reason is the asymptotic behavior given byEq. (36) in the anisotropic case which decays exponentiallywith the largest eigenvalue for large D as depicted in the fig-ure. In contrast, for the isotropic system the asymptotic de-cay corresponds to the mean diffusion coefficient resultingin the observed quantitative difference. Furthermore, the dis-tributions are qualitatively different for small D. A charac-teristic difference between isotropic and anisotropic systemsis the convex shape in the logarithmic representation of theanisotropic distribution of diffusivities. This intuitively resultsfrom the two different exponential decays related to the dis-tinct diffusion coefficients D1 and D2. In a more rigorous way,since d2

dD2 log p2dD̂

(D) ≥ 0, with the equal sign being valid onlyfor isotropic diffusion, the anisotropic distribution of diffusiv-ities is a superconvex function.48

For experimental data, it is easy to calculate the first twomoments M1 and M2 by averaging the short-time diffusivi-ties of Eq. (4) and their squares, respectively. The averagingis accomplished either along a single trajectory or from an en-semble of trajectories avoiding any numerical fit. The first twomoments are sufficient to calculate D1 and D2 by Eq. (35).

For the sample trajectory used in Fig. 1, the first twomoments are determined to be M̃1 = 2.987 and M̃2 = 21.72.According to Eq. (35), the underlying diffusion coefficientsyield D̃1 = 4.956 and D̃2 = 1.018. These values agree wellwith the eigenvalues of the tensor Eq. (40), which was usedas input parameter of the simulation. The resulting value ofη = 2/3 indicates a considerable anisotropy of the process.

1. Limiting cases

In the case of identical diffusion coefficients for both di-rections, the anisotropy vanishes as discussed for Eq. (38).The resulting isotropic diffusion process is characterized by asingle diffusion coefficient Dc = D1 = D2. Hence, Eq. (32)simplifies to the well-known distribution of single-particlediffusivities of two-dimensional isotropic diffusion19

p2dDc

(D) =exp

(− D

Dc

)Dc

(41)

given by an exponential function.If, on the contrary, the anisotropy is large, diffusion in

one direction will be suppressed. Without loss of generality,this is accomplished by sending one of the diffusion coeffi-cients to zero. Thus, by taking the limit of vanishing D2, thedistribution of diffusivities Eq. (32) is simplified to

p1dD1

(D) = limD2→0

p2dD̂

(D) =exp

(− D

D1

)√

πD1D, (42)

which has the structure of the distribution of diffusivitiesof one-dimensional diffusion.19 Since diffusion into the per-pendicular direction is prohibited, as expected, it qualita-tively leads to the observation of a one-dimensional process.

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This can be identified by the characteristic factor D−1/2 dueto which the distribution of diffusivities diverges for smallD. Applying Eq. (8) the first moment of Eq. (42), i.e., themean diffusion coefficient, yields 〈D〉 = D1/2. The factor of1/2 results from the single-particle diffusivities Eq. (4) withN = 2 assuming that a two-dimensional process is observed.However, due to the suppression of one direction this assump-tion is no longer valid and N = 1 should have been used in-stead. This conclusion is also obtained from the anisotropyvalue η = 1, which is equal to its maximum value fortwo-dimensional anisotropic processes since effectively one-dimensional motion is observed.

2. Reconstruction of D

In addition to the eigenvalues, it is sometimes of interestto determine the orientation of the principal axes of the systemrelative to the given frame of reference. This is achieved bythe reconstruction of the diffusion tensor

D =(

D11 D12

D12 D22

), (43)

where the off-diagonal elements are labeled identically dueto symmetry reasons. The reconstruction is accomplished intwo ways either by considering the complete two-dimensionaltrajectory or by using one-dimensional projections of thetrajectory.

In the first approach, the tensorial diffusivities of Eq. (21)are used to obtain the tensor entries of D. In accordance withEq. (22), the tensor elements are estimated by averaging thetensorial diffusivities along a trajectory or over an ensemble.Moreover, the eigenvalues of the tensor D are expressed by itsentries

D1,2 = 1

2

(D11 + D22 ±

√(D11 − D22)2 + 4D2

12

)(44)

and correspond to the diffusion coefficients of the system.For the sample trajectory used in Fig. 1, the measured

values D̃ij yield the diffusion tensor

D̃ =(

3.983 1.7191.719 1.990

), (45)

which agrees reasonably with the input parameters of thesimulation. The eigenvalues from this measured tensorD̃1 = 4.973 and D̃2 = 1.000 show a good agreement with theexact eigenvalues of the input tensor D1 = 5 and D2 = 1.

The second approach determines the tensor D exclusivelyfrom one-dimensional projections of the trajectory. In orderto obtain results, at least three different projections are nec-essary. For simplicity, it is preferable to use projections alongtwo perpendicular axes, which define the frame of referencefor D. Furthermore, a projection onto an axis is requiredwhich is rotated about an angle θ relatively to the frame of ref-erence. In such a setup, the first moments of the distributionof diffusivities related to the first two projections are identicalto the averaged tensorial diffusivities 〈D11

t (τ )〉 and 〈D22t (τ )〉.

Thus, they yield the two diagonal elements of D. The firstmoment of the third projection measures the leading diago-nal element Dθ

11 of the rotated tensor Dθ = R(θ )TDR(θ ) with

rotation tensor R(θ ) = ( cos θ − sin θsin θ cos θ

). This additional value is

sufficient to obtain the off-diagonal element of D from

D12 = Dθ11 − D11 cos2 θ − D22 sin2 θ

sin(2θ ). (46)

For the calculation, any projection of the trajectory onto anarbitrary one-dimensional axis, i.e., any θ , can be used ex-cept directions perpendicular or parallel to axes of the frameof reference, i.e., angles θ which are multiples of π /2. Itshould be emphasized that the reconstruction from the distri-bution of diffusivities of projected trajectories is possible al-though the definition of the diffusivities omit any directionalinformation.

In the example with D̃11 = 3.983, D̃22 = 1.990, and ameasured D̃

5π/1211 = 2.983, the off-diagonal element yields

D12 = 1.719, which is in good agreement with the value√3 ≈ 1.732 appearing as input parameter of the simulation.

In conclusion, it depends on the constraints of the exper-iment which of both approaches is more practicable. In eitherway, the complete diffusion tensor D is reconstructed reason-ably well.

B. Three-dimensional systems

Analogous to the two-dimensional case, it is possible tocalculate the distribution of diffusivities for three-dimensionalsystems either by inverse Fourier transform of the generalcharacteristic function Eq. (13) or by the convolution Eq. (16).In both cases, the analytical integration cannot be performedcompletely. However, the integration can be accomplished nu-merically. By integrating two variables, Eq. (16) is reducedto

p3dD̂

(D) =∞∫

0

d1

∞∫0

d2

∞∫0

d3 δ [D − (1 + 2 + 3)]

×p1dD1/3(1)p1d

D2/3(2)p1dD3/3(3)

=D∫

0

d1

(3

2

)3/2 1√πD1D2D31

× exp

{−3

4

[(1

D2+ 1

D3

)(D − 1) + 21

D1

]}

× I0

[3

4

(1

D3− 1

D2

)(D − 1)

]. (47)

For further simplification, a series expansion of the modifiedBessel function I0(x) can be applied, which allows performingthe last integration. However, this only results in a convergingsum, which cannot be simplified any further.

By using the general expression of the cumulants Eq. (14)and the relation between cumulants and moments Eq. (15), thefirst three moments of the distribution of diffusivities of three-dimensional homogeneous anisotropic diffusion processes

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are

M1 = 1

3(D1 + D2 + D3), (48)

M2 = 1

9

[(D1 + D2 + D3)2 + 2

(D2

1 + D22 + D2

3

)], (49)

and

M3 = 1

9

[5D3

1 + 3D21(D2 + D3)

+D1(3D2

2 + 2D2D3 + 3D23

)+ (D2 + D3)

(5D2

2 − 2D2D3 + 5D23

)]. (50)

These expressions are similar to Eqs. (33) and (34) and relatethe moments of the distribution of diffusivities to diffusioncoefficients D1 to D3 of the anisotropic process. By solvingsimultaneously Eqs. (48)–(50), the underlying diffusion coef-ficients are determined by the measured moments of the distri-bution. The solution comprises six triplets (D1 to D3), whichare permutations of the three diffusion coefficients. Due tothe cubic contributions in Eq. (50), the expressions are toolengthy to be shown here but can be easily obtained, for in-stance, using symbolic computer algebra systems.

The asymptotic behavior of Eq. (47) for large D is givenby Eq. (18), which assumes D1 > D2 > D3, and results in

p3dD̂

(D)D→∞∼

√3D1 exp

(− 3D

2D1

)√

2π (D1 − D2)(D1 − D3)D. (51)

Thus, the behavior in the logarithmic representation is deter-mined by

log p3dD̂

(D)D→∞∼ − 3D

2D∞, (52)

which corresponds to the asymptotic decay of a three-dimensional isotropic distribution of diffusivities withD∞ = max (D1, D2, D3). The anisotropy measure Eq. (20)in the three-dimensional case corresponds to

η = (D∞ − D1) + (D∞ − D2) + (D∞ − D3)

D1 + D2 + D3, (53)

which considers the differences of the individual diffusion co-efficients to characterize the anisotropy. It is obvious that thelargest anisotropy yields η = 2.

In order to substantiate our results by simulated data, thesimulation of a three-dimensional homogeneous anisotropicrandom walk was performed using the diffusion tensor

D =

⎛⎜⎜⎜⎝

4 −√

32 − 1

2

−√

32

134

3√

34

− 12

3√

34

74

⎞⎟⎟⎟⎠. (54)

The obtained trajectory consists of 105 displacements and itsdistribution of diffusivities is depicted in Fig. 2. The distri-bution of diffusivities from the simulated trajectory shows agood agreement with the curve obtained from numerical inte-gration of Eq. (47). The deviations for larger values of D resultfrom the finite simulation time, i.e., its insufficient statistics.

p(D

)

D

10-4

10-3

10-2

10-1

100

0 5 10 15 20 25

p(D

)

D

10-610-510-410-310-2

15 20 25 30 35 40

FIG. 2. The distribution of diffusivities (histogram) from one simulated tra-jectory of a homogeneous anisotropic diffusion process in three dimensionswith diffusion tensor D given by Eq. (54) agrees well with the distribution ofdiffusivities (solid line) obtained from numerical integration of Eq. (47), us-ing the eigenvalues D1 = 5, D2 = 3, and D3 = 1 of tensor D. For comparison,the distribution of diffusivities (dashed line) of an isotropic diffusion processin three dimensions, given by Eq. (55), is shown, where the same mean dif-fusion coefficient Dc = 〈D〉 = (D1 + D2 + D3)/3 = 3 as in the anisotropicprocess was used. The different asymptotic decays are clearly visible and al-low the distinction from homogeneous isotropic processes. In the inset, theasymptotic function Eq. (51) (dotted line) agrees reasonably for large D.

Furthermore, Fig. 2 shows the distribution of an isotropic sys-tem where a qualitative distinction at the crossover from themaximum peak to the exponential decay becomes apparent.This behavior of the curvature in the logarithmic represen-tation depends on the observed system and is discussed inSec. V B 2. The deviating asymptotic decay of the anisotropicprocess is clearly visible in Fig. 2 and allows the distinctionfrom homogeneous isotropic processes. Thus, in conjunctionwith the mean diffusivity the asymptotic decay provides ameasure of the anisotropy. In addition, the asymptotic behav-ior given by Eq. (51) is depicted and provides a reasonableapproximation for large D. The eigenvalues of D for exper-imental data are easily determined by measuring the lead-ing moments of the diffusivities. For the sample trajectoryused in Fig. 2, the first three moments result in M̃1 = 2.995,M̃2 = 16.68, and M̃3 = 140.2. By solving the simultaneousEqs. (48)–(50), the underlying diffusion coefficients are ob-tained as D̃1 = 4.884, D̃2 = 3.153, and D̃3 = 0.948. Thesevalues agree reasonably well with the eigenvalues of the ten-sor Eq. (54), which was used as input parameter of the simula-tion. The value of η = 2/3 indicates a considerable anisotropyof the process.

1. Limiting cases

If the diffusion coefficients of all three directions coin-cide with Dc = D1 = D2 = D3, the distribution of diffusivitiesfor the three-dimensional isotropic system19

p3dDc

(D) = 3

√3

2π

D

D3c

exp

(− 3D

2Dc

)(55)

will be obtained from Eq. (47) in agreement with Eq. (7).If exactly two diffusion coefficients coincide, one usu-

ally refers to diffusion processes of uniaxial molecules.35 In

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184105-10 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

this case, the general distribution of diffusivities of three-dimensional homogeneous anisotropic diffusion Eq. (47) sim-plifies to

p3duni(D) = 3

2

exp(− 3D

2D(2)

)erf

(√32

(1

D(1) − 1D(2)

)D)

√D(2)(D(2) − D(1))

, (56)

where D(1) and D(2) are the eigenvalues of D with multiplic-ity one and two, respectively. In general, a distinction be-tween the oblate case (D(2) > D(1), disc) and the prolate case(D(2) < D(1), rod) is made for uniaxial molecules. In the pro-late case, both square roots in Eq. (56) yield complex num-bers. However, with erf(

√−x)/√−y = erfi(

√x)/

√y for x, y

> 0 and x, y ∈ R, Eq. (56) remains a real-valued function.Hence, a distinction between the two cases for the diffusioncoefficients is not required for the distribution of diffusivities.

In the uniaxial case, the first three moments simplify to

M1 = 1

3(D(1) + 2D(2)), (57)

M2 = 1

9

(3D(1)2 + 4D(1)D(2) + 8D(2)2)

, (58)

and

M3 = 1

9

(5D(1)3 + 6D(1)2

D(2) + 8D(1)D(2)2 + 16D(2)3).

(59)Thus, the eigenvalues of D are calculated by

D(1) = M1 ∓√

3M2 − 5M21 (60)

and

D(2) = M1 ± 1

2

√3M2 − 5M2

1 , (61)

where the sign in the equations depends on the constraint ofpositive diffusion coefficients. None of the eigenvalues willbecome complex since with Eqs. (57) and (58) the expressionunder the square root 3M2 − 5M2

1 = 49 (D(1) − D(2))2 > 0 is

always positive and, hence, M2 > 53M2

1 . It should be notedthat for 5

3M21 < M2 < 2M2

1 both signs in Eqs. (60) and (61)yield positive diffusion coefficients. In this case, the third mo-ment has to be exploited in order to decide the correct pairof diffusion coefficients by comparing Eq. (59) with the mea-sured value. Hence, there exist distributions of diffusivitieswith identical moments M1 and M2, which result from differ-ent diffusion coefficients. In this case, the distinct M3 deter-mines the corresponding diffusion coefficients of the system.In the limit M2 → 5

3M21 , D(1) and D(2) approach each other.

In this particular case, the decision for the correct pair cannotbe made accurately since both pairs yield approximately thesame M3 from Eq. (59). However, this limit corresponds to theisotropic system and, hence, the single diffusion coefficient isdirectly given by the first moment of the distribution.

Fig. 3 depicts examples of such distributions for the gen-eral anisotropic, the prolate, and the oblate case. The differ-ences can be identified qualitatively. In the general and in theprolate case, the decay after the maximum peak has a con-vex curvature in the logarithmic representation, whereas in the

p(D

)

D

10-3

10-2

10-1

100

0 5 10 15 20 25

p(D

)

D

10-9

10-8

10-7

10-6

45 50 55 60 65

FIG. 3. Distribution of diffusivities (lines with open symbols) of differenthomogeneous anisotropic diffusion processes in three dimensions. A qualita-tive distinction between the oblate case (♦; D(1) = 1, D(2) = 5), the prolatecase (�; D(1) = 5, D(2) = 1), and a general anisotropic case (©; D1 = 5, D2= 3, D3 = 1) is possible since the decay after the maximum peak shows aconcave curvature in the first case and a convex curvature in the latter cases.The inset shows that each anisotropic case obeys the same asymptotic de-cay given by the largest diffusion coefficient (dotted line as a guide to theeye). For comparison, the isotropic case with the same asymptotic decay (�;Dc = 5) is given, which always has a concave shape. Thus, it is qualitativelyindistinguishable from the oblate case. However, a comparison of the firstmoment with the asymptotic decay offers a simple distinction between bothcases.

oblate case it decays in a purely concave manner. This quali-tative change is obtained from d2

dD2 log p3duni(D) and discussed

in Sec. V B 2. In all cases, the exponential decay for large Dis determined by the largest diffusion coefficient as given byEq. (51). However, since the first decay after the peak is domi-nated by the smallest diffusion coefficient, the curve is shiftedto the left for the prolate case in contrast to the oblate casewhen D2 is changed from D3 to D1. As expected from thefirst moment, the general case lies in between. A better dis-tinction between the different cases is achieved quantitativelyby determining the moments and calculating the diffusioncoefficients.

In Fig. 4, the distribution of diffusivities for differentratios

r = D(1)/D(2) (62)

is shown, ranging from oblate cases (r < 1) to prolate cases.It can be seen that in the limit D(1) → 0 and, thus, r → 0,the distribution converges to the two-dimensional isotropiccase with Dc = 2/3D(2). For r → 1, the distribution con-verges to the three-dimensional isotropic case. In the prolatecases, the distribution separates significantly from the three-dimensional isotropic case for increasing r. For further in-creasing ratios (r → ∞), the distribution converges to theone-dimensional isotropic case with Dc = 1/3D(1). In con-trast, the oblate cases converge rapidly to the two-dimensionalisotropic case for decreasing r. A qualitative distinction mayonly be possible for small D, where the distribution still de-viates from the mono-exponential behavior of the isotropicsystem. However, quantitatively the anisotropy is character-ized by Eq. (20), which results in η = 1−r

2+rand η = 2(r−1)

2+r

for oblate and prolate cases, respectively. Thus, in the oblate

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p(D

)

D

r = 0.1r = 0.5r = 2.0

r = 10.0

10-4

10-3

10-2

10-1

100

101

0 2 4 6 8

FIG. 4. Distribution of diffusivities (lines with open symbols) for differentratios r given by Eq. (62) and fixed D(2) = 1. The crossover from oblate cases(r < 1, solid lines) to prolate cases (r > 1, dashed lines) shows a broadeningof the peak for increasing ratios. Again, the behavior after the peak changesfrom concave to convex, respectively. For comparison, the distribution of dif-fusivities of the limiting isotropic cases are depicted for two-dimensional (�;Dc = 2/3) and three-dimensional processes (�; Dc = 1). The distinction ofprolate cases from the isotropic limits is simpler than for the oblate cases.

case the largest possible anisotropy emerges at small r, whichyields η = 1/2 and clearly indicates the anisotropy. In the pro-late case, the largest anisotropy will be obtained, if only onedirection is preferred. Then, the anisotropy measure η = 2 ismaximal, which corresponds to one-dimensional motion in athree-dimensional system.

2. Curvature of the distribution of diffusivities

As noticed in Fig. 3, the convex or concave curvature ofthe probability density in the logarithmic representation de-pends on the observed system and, thus, on the structure ofthe diffusion tensor. In the literature, such a concave curva-ture is known as log-concavity of functions which is a com-mon property of probability distributions and has been studiedextensively.49–51 However, in the case of log-convex functionsthere are much less properties known. In the following, wediscuss the curvature of the distribution of diffusivities in thelogarithmic representation, which can be exploited to deter-mine characteristic properties of the observed processes.

For anisotropic processes, the asymptotic curvature of thedistribution of diffusivities in the logarithmic representation isobtained from the uniaxial case Eq. (56) since Eq. (47) doesnot provide a closed-form expression. For isotropic diffusion,the curvature of the distribution of diffusivities is determinedfrom Eq. (55).

The asymptotic expansion of the second derivative forsmall D yields

d2

dD2log p3d

D̂(D)

D→0∼ −1/(2D2), (63)

which coincides with the curvature of three-dimensionalisotropic systems. Analogously, we perform the asymptotic

expansion of the second derivative for large D

d2

dD2log p3d

D̂(D)

D→∞∼

⎧⎪⎪⎨⎪⎪⎩

1/(2D2) D1 > D2 = D3,

− a3/2√πD

exp(−aD) D1 = D2 > D3,

−1/(2D2) D1 = D2 = D3,

(64)

with positive a = 3/2(1/D(1) − 1/D(2)). The different resultsdepend on the multiplicity of the largest eigenvalue for pro-late, oblate, and isotropic cases, respectively. In the generalanisotropic case with D1 �= D2 �= D3, the system is dominatedby the largest diffusion coefficient for large D. Hence, in thiscase the asymptotic curvature is identical to the prolate case(D1 > D2 = D3) and can also be obtained from Eq. (18). Asexpected, a degeneracy of the smaller eigenvalues does notcontribute to the asymptotic curvature. Hence, in all systemswhere the largest eigenvalue is not degenerated, for instance,in anisotropic two-dimensional and also one-dimensional sys-tems, we obtain the same behavior for large D, which is gov-erned by the largest eigenvalue of the system.

The curvature of the distribution of diffusivities in thelogarithmic representation for small D is always concave asgiven by Eq. (63). However, for large D it depends on theobserved system showing either a convex or a concave behav-ior as given in Eq. (64). Hence, the sign of the curvature canchange with D. In the prolate case, the corresponding point ofinflection is found to be approximately at 1.504D(1)D(2)/(D(1)

− D(2)) by numerical evaluation of the root of d2

dD2 log p3dD̂

(D).For anisotropic systems, only in the oblate case the curva-ture does not change its sign and the distribution is a log-concave function. If the anisotropy measure becomes zero andthe distribution is a log-concave function, a three-dimensionalisotropic diffusion process is observed. This qualitative dif-ference in the curvature of distributions with the same asymp-totic decay can clearly be identified in Fig. 3.

Furthermore, it is interesting to note in Eq. (64) that in theoblate case, where the largest eigenvalue exhibits a twofolddegeneracy, the asymptotic behavior of the curvature still de-pends on the diffusion coefficients of the system. In all othercases, the dependence on the diffusion coefficients vanishes.

As noted above, for two-dimensional anisotropic pro-cesses the asymptotic behavior for large D in the loga-rithmic representation is identical to the prolate case inEq. (64). However, the asymptotic behavior for small D isgiven by 1/8(1/D1 − 1/D2)2 and clearly differs from that ofthe three-dimensional process. Since the sign of the curva-ture does not change with D the curvature is always convexin two-dimensional anisotropic systems. However, for two-dimensional isotropic systems the distribution of diffusivi-ties in the logarithmic representation is just a straight line forall D.

3. Reconstruction of D

As discussed for two-dimensional processes, the diffu-sion tensor will be easily obtained by measuring the averagedtensorial diffusivities according to Eq. (22) if the complete

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184105-12 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

three-dimensional trajectory of the homogeneous anisotropicprocess is available. For the sample trajectory used in Fig. 2,the measured values D̃ij yield the diffusion tensor

D̃ =

⎛⎜⎜⎝

3.994 −0.862 −0.492

−0.862 3.233 1.296

−0.492 1.296 1.758

⎞⎟⎟⎠, (65)

which agrees reasonably with the input parameters of thesimulation Eq. (54). Further, the eigenvalues from this mea-sured tensor D̃1 = 4.983, D̃2 = 2.998, and D̃3 = 1.004 showa good agreement with the exact eigenvalues of the input ten-sor D1 = 5, D2 = 3, and D3 = 1.

However, if only a projection of the complete trajectoryis available, e.g., from SPT, only the properties of the re-spective submatrix of D can be measured. For instance, if thetwo-dimensional projection onto the x-y-plane of the sampletrajectory is available, the first two moments of the distribu-tion of diffusivities are determined to be M̃1,z = 3.613 andM̃2,z = 26.95. Using Eq. (35), the eigenvalues of the prin-cipal submatrix D2

z are computed to be D̃21,z = 4.531 and

D̃22,z = 2.695. Hence, the eigenvalue inequalities of Eq. (27)

provide the estimate

D1 ≥ D̃21,z = 4.531 ≥ D2 ≥ D̃2

2,z = 2.695 ≥ D3 ≥ 0 (66)

of the diffusion coefficients. As explained in Sec. IV, anyfurther observed projection improves the estimates of theeigenvalues of D. An additional projection onto the x-z-plane, for instance, yields the moments M̃1,y = 2.876 andM̃2,y = 18.00 resulting in the eigenvalues D̃2

1,y = 4.083 and

D̃22,y = 1.668. Since with two orthogonal two-dimensional

projections of the three-dimensional process all diagonal el-ements of D are available, an upper bound for the largesteigenvalue is found to be D1 ≤ tr D ≤ D̃2

1,z + D̃22,z + D̃2

1,y

+ D̃22,y = 12.977. Hence, the eigenvalue inequalities yield

12.977 ≥ D1 ≥ max(D̃2

1,z, D̃21,y

)= 4.531,

min(D̃2

1,z, D̃21,y

) = 4.083 ≥ D2 ≥ max(D̃2

2,z, D̃22,y

)= 2.695,

min(D̃2

2,z, D̃22,y

) = 1.668 ≥ D3 ≥ 0. (67)

If additionally the projection onto the y-z-plane is avail-able, the eigenvalues of D are estimated more precisely sim-ilar to the previous steps. To improve the upper bound ofD1, the trace of D is calculated from all these eigenval-ues by tr D = 1

2 (D̃21,x + D̃2

2,x + D̃21,y + D̃2

2,y + D̃21,z + D̃2

2,z),where the prefactor arises from the overlapping diagonal ele-ments of the submatrices.

In the case of availability of all orthogonal two-dimensional projections of the process, the tensorial diffusiv-ities offer an advanced approach to determine the diffusiontensor. Since their first moments yield the entries of the prin-cipal submatrices D2

x , D2y , and D2

z , the underlying diffusiontensor D is completely defined.

To summarize, the experimental setup influences theavailable data and affects how many parameters of the un-derlying process can be restored. A single two-dimensionalprojection may already hint at the anisotropy of the process.However, it is not sufficient to give an upper bound for the

largest eigenvalue. An additional orthogonal two-dimensionalprojection or even a one-dimensional projection in the miss-ing direction determines this upper bound and narrows theranges of the eigenvalues. For a reconstruction of the com-plete tensor, either the complete trajectory or three orthogonaltwo-dimensional projections of the process are necessary.

VI. CONCLUSIONS

To investigate N-dimensional homogeneous anisotropicBrownian motion, we applied the distribution of diffusivitiesas, e.g., obtained from single-particle tracking data. We in-troduced an anisotropy measure depending on the asymptoticdecay of the distribution and the mean of the diffusivities,which both are easily determined from experimental data. Ingeneral, if this anisotropy measure is larger than zero, the dis-tribution deviates from the χ2-distribution, which we obtainfor homogeneous isotropic diffusion. Thus, the observed pro-cess involves more than one diffusion coefficient attributed toan inhomogeneity or an anisotropy of the system. For homo-geneous processes, we concluded that those systems have tobe anisotropic. Furthermore, from the general expression ofthe distribution of diffusivities we derived relations betweenits moments or cumulants and the eigenvalues of the diffu-sion tensor D. Since, due to experimental restrictions, oftenonly projections of the trajectories are observed we furtherdiscussed the consequences and provided an estimate for thebounds of the involved diffusion coefficients.

After our general considerations, we applied the results tospecific systems with high relevance to experiments. In partic-ular, we investigated two-dimensional and three-dimensionalsystems as well as uniaxial molecules in three dimensions. Ina two-dimensional homogeneous anisotropic system, the dis-tribution of diffusivities comprises a modified Bessel func-tion and allows a qualitative distinction from the mono-exponential decay observed in isotropic systems. Moreover,the first two moments of the distribution are sufficient to cal-culate the diffusion coefficients corresponding to the principalaxes. Even the orientation of the principal axes and, thus, thecomplete diffusion tensor D can be determined by using ten-sorial diffusivities or three one-dimensional projections of thetrajectory. For three-dimensional processes, the general ex-pression of the distribution of diffusivities is more elaboratedand one integration has to be evaluated numerically. However,we expressed the first three moments in terms of the diffu-sion coefficients belonging to the principal axes. Conversely,these expressions offer a method to calculate the diffusion co-efficients from the moments measured in experiments, whereother analysis fails. It is further shown that the isotropic andanisotropic systems differ in the logarithmic representationof the distribution of diffusivities, i.e., the asymptotic decayrate is proportional to the inverse slope of the msd and tothe inverse of the largest diffusion coefficient, respectively.Thus, the distribution of diffusivities for anisotropic diffusionasymptotically decays slower than for isotropic diffusion withthe same mean diffusion coefficient. The deviation betweenthe asymptotic decay and the first moment provides a suit-able measure for the anisotropy of the process. For uniaxialmolecules diffusing in three dimensions, the third integration

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184105-13 Heidernätsch, Bauer, and Radons J. Chem. Phys. 139, 184105 (2013)

was accomplished and the resulting distribution of diffusivi-ties involves an error function. In this case, the diffusion co-efficients along the direction of the principal axes depend onthe first two moments of the distribution. For different ratiosof the diffusion coefficients, we distinguish between oblateand prolate cases, which show a concave and a convex cur-vature in the logarithmic representation, respectively. Finally,we offer a guide to quantify the eigenvalues of D from pro-jected observations and to reconstruct the diffusion tensor inthree dimensions from the moments of the tensorial diffusiv-ities. The reconstruction from projected observations is pos-sible although any directional information is discarded whendetermining the distribution of diffusivities.

In summary, the distribution of diffusivities provides anadvanced analysis of anisotropic diffusion processes. The dis-tribution is easily obtained from measured trajectories or fromensemble measurements such as NMR and allows for a char-acterization of the processes. For homogeneous diffusion pro-cesses, this distribution is stationary, which allows us to com-pare experiments conducted on different time scales. The firstmoment of the distribution corresponds to the mean of thediffusivities and coincides with the slope of the mean squareddisplacement. From the discrepancy between the asymptoticdecay of the distribution and the mean of the diffusivities, itis easy to identify systems which are not sufficiently charac-terized by a single diffusion coefficient. Hence, we encourageexperimentalists to determine these simple quantities in or-der to detect a discrepancy and to verify their assumptionsabout homogeneous isotropic processes. Furthermore, if thesystem is homogeneous and anisotropic, the diffusion coeffi-cients can be reconstructed from the moments of the distribu-tion. Beyond that, the concept of diffusivities as scaled dis-placements is extended to tensorial diffusivities, which allowthe reconstruction of the diffusion tensor from their first mo-ments. Hence, the distribution of diffusivities complementswell-established methods, such as investigating mean squareddisplacements, for the analysis of diffusion data.

In future publications, we will address the distinctionbetween anisotropic and heterogeneous diffusion processes,which also involve more than one diffusion coefficient. Themore general case of heterogeneous anisotropic diffusion mayalso be of interest for applications. This case requires thecombination of our previous results for time-heterogenousprocesses19 with the results presented here. Furthermore,since the eigenvalues of the tensor D are invariant to orthog-onal transformations, we will apply our distribution of dif-fusivities to systems where the diffusion tensor changes itsorientation in space and time, such as diffusion of ellipsoidalparticles in isotropic media and diffusion in liquid crystallinesystems with an inhomogeneous director field.

ACKNOWLEDGMENTS

We thank Sven Schubert for stimulating discussions andvaluable suggestions. We gratefully acknowledge financialsupport from the Deutsche Forschungsgemeinschaft (DFG)for funding of the research unit FOR 877 “From Local Con-straints to Macroscopic Transport.” We also appreciated valu-

able suggestions of the anonymous referees, which helped toimprove the paper considerably.

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