How to compare diffusion processes assessed by single-particle tracking and pulsed field gradient nuclear magnetic resonance
aa r X i v : . [ phy s i c s . d a t a - a n ] O c t How to compare diffusion processes assessed by single-particle tracking and pulsedfield gradient nuclear magnetic resonance
Michael Bauer and G¨unter Radons ∗ Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
Rustem Valiullin † and J¨org K¨arger Institute of Experimental Physics I, University of Leipzig, 04103 Leipzig, Germany (Dated: June 20, 2018)Heterogeneous diffusion processes occur in many different fields such as transport in living cellsor diffusion in porous media. A characterization of the transport parameters of such processes canbe achieved by ensemble-based methods, such as pulsed field gradient nuclear magnetic resonance(PFG NMR), or by trajectory-based methods obtained from single-particle tracking (SPT) experi-ments. In this paper, we study the general relationship between both methods and its applicationto heterogeneous systems. We derive analytical expressions for the distribution of diffusivities fromSPT and further relate it to NMR spin-echo diffusion attenuation functions. To exemplify the ap-plicability of this approach, we employ a well-established two-region exchange model, which haswidely been used in the context of PFG NMR studies of multiphase systems subjected to inter-phase molecular exchange processes. This type of systems, which can also describe a layered liquidwith layer-dependent self-diffusion coefficients, has also recently gained attention in SPT experi-ments. We reformulate the results of the two-region exchange model in terms of SPT-observablesand compare its predictions to that obtained using the exact transformation which we derived.
PACS numbers: 05.40.-a, 82.56.Lz, 87.80.Lg, 87.80.NjKeywords: heterogeneous diffusion, distribution of diffusivities, single-particle tracking, PFG NMR
I. INTRODUCTION
Diffusion is one of the omnipresent phenomena in na-ture involved in most physico-chemical and biologicalprocesses [1]. Often media, where the molecules per-form their chaotic Brownian motion, do include differenttypes of compartments, regions of different densities ordomains surrounded by semi-permeable membranes. Dif-fusion properties in these spatially separated regions may,in general, be different. Altogether, this typically givesrise to very complex processes of diffusive mass trans-port including regimes of anomalous diffusion. To modelsuch inhomogeneous systems, they may be represented toconsist of a number of domains with different local dif-fusivities subjected to exchange processes between them.The most simple two-phase exchange model with an ex-ponential exchange kernel has often been used to describeexperimental results obtained using pulsed field gradientnuclear magnetic resonance (PFG NMR) technique [2].Such examples include, e.g. diffusive exchange betweentwo pools of guest molecules in zeolite crystals and sur-rounding gas atmosphere [3] and between extra- and in-tracellular water [4] in biosystems.Recently, a new type of experimental approach, namelysingle-particle tracking (SPT) has emerged [5]. It pro-vides an alternative method for studying diffusion pro-cesses and for measuring their properties as well as someproperties of the surrounding medium [6]. In contrast ∗ [email protected] † [email protected] to PFG NMR, where an ensemble of diffusing parti-cles is investigated, SPT only observes individual tracerparticles. In particular, fluorescent dye molecules, likerhodamine B, in a solvent, e.g. tetrakis(2-ethylhexoxy)-silane (TEHOS), which arranges in ultra-thin liquid lay-ers [7], are excited by laser radiation. The emitted lightof the dyes is captured with a wide-field microscope andrecorded by a CCD camera. Hence, the obtained moviesshow diffusing spots representing a two-dimensional pro-jection of the three-dimensional motion of the dyes. Froma statistical point of view, such processes are knownas observed diffusion [8–10] or hidden Markov models[11, 12] leading in general to the loss of the Markov prop-erty. A tracking algorithm detects the positions of thespots and connects them to trajectories [13]. A basicquantity for the characterization of diffusion processes isobtained by taking two positions x ( t ) and x ( t + τ ) from atrajectory separated by a time lag τ and by consideringthe rescaled squared displacement [ x ( t + τ ) − x ( t )] /τ .This quantity is a local or microscopic diffusivity whichfluctuates along a given trajectory or in an ensembleof diffusing particles. It is natural to extract the cor-responding distribution of diffusivities from experimentsby forming histograms of the observed rescaled squareddisplacements [14]. Note that other definitions of diffu-sivity distributions may be found in the literature [15].For homogeneous diffusion processes the distribution ofdiffusivities is independent of the time lag τ , whereas forheterogeneous systems a non-trivial τ -dependence is ob-served. Therefore in analyzing heterogeneous systems thedistribution of diffusivities provides advantages over ananalysis via mean squared displacements (msd) becausein addition to its mean value it contains all informationabout the fluctuations [16]. Furthermore, quantities suchas the mean diffusion coefficient, obtained as the first mo-ment of the distribution of diffusivities, are well defined,and thus time-dependent diffusion coefficients and theirfluctuations can be calculated.The objective of this work is to investigate the con-nection between the two different techniques of measur-ing diffusion. SPT and PFG NMR are clearly relatedto each other, since both measure displacements of dif-fusing particles. For instance, the time lag between theobservation of two positions in SPT corresponds to thetime interval between two gradient pulses in PFG NMR.In both SPT and PFG NMR this time lag τ is a param-eter, which can be tuned by varying the time betweensnapshots and by altering the temporal distance betweengradient pulses, respectively. Furthermore, the signal at-tenuation in PFG NMR is related to the propagator inFourier space, from which the distribution of diffusivi-ties can be calculated. At first, it seems to be sufficientto compare the propagators obtained from both typesof experiments directly. However, if diffusion processeswith heterogeneities or anomalous behavior are consid-ered, access to the propagator will be complicated oreven hindered. In such cases, the distribution of diffusiv-ities offers a well-defined analysis of the processes and acomparison of data from the two approaches is feasible.Moreover, it becomes possible to contrast results fromtime-averaged and ensemble-averaged quantities and de-tect anomalous diffusion leading to ergodicity breaking asreported recently [17]. More generally, an improvementin the analysis of heterogeneous diffusion could benefitfrom the link between single-particle analysis and ensem-ble methods. Hence, analytical expressions for one- upto three-dimensional processes are derived which trans-form PFG NMR signal attenuation into the distributionof single-particle diffusivities from SPT.For simple systems with heterogeneous diffusion thetwo-region exchange model of PFG NMR offers an ana-lytical expression for the spin-echo diffusion attenuation[18]. In conjunction with our transformation, this modelprovides an example, where the distribution of single-particle diffusivities can be calculated exactly and alsothe non-trivial time-lag dependence can be investigated.In this context we consider a two-layer liquid film on ahomogeneous surface characterized by two distinct diffu-sion coefficients [16]. This two-layer system correspondsexactly to the two-region exchange model of PFG NMR.In particular, its condition of exponential waiting timesis fulfilled since a change in the diffusion coefficient ispossible at any time and independent of the particles’current positions. For a system comprising an arbitrarynumber of layers, exact asymptotic results for the disper-sion of particles in the long-time limit have already beenprovided [19]. We substantiate our findings by analyzingdata from simulated single-particle trajectories of hetero-geneous diffusion. To evaluate experimental limitations,we study the influence of a signal attenuation bounded to a certain range of k -values. The impact on the distri-bution of single-particle diffusivities will also be pointedout.The remainder of the paper is organized as follows. InSec. II we recall the basic principles of PFG NMR andunderline the differences to SPT experiments. In partic-ular, we discuss properties of both approaches and theconnection between them. In this context, we introducethe distribution of single-particle diffusivities and provideexpressions for the well-known case of homogeneous diffu-sion. To apply our new concepts to some more elaboratedsystems, we consider in Sec. III heterogeneous diffusionin two-region systems, where analytical expressions of thePFG NMR signal attenuation exist. We outline the prin-ciples of the simulation of such systems in Sec. IV. Inorder to provide a simple relation between signal attenu-ation and distribution of diffusivities, we suggest an ap-proximation in Sec. V to avoid the inconvenient Fouriertransformation. This approximation is then compared tothe exact expressions of the relation in Sec. VI for simu-lated data of the two-region system. Finally, in Sec. VIIwe address the issue of finite intensity of the magneticfield gradient pulses in the PFG NMR experiment andillustrate its influence on our exact transformation intothe distribution of diffusivities. II. SIGNAL ATTENUATION ANDDISTRIBUTION OF DIFFUSIVITIES
Diffusion measurement by PFG NMR is based on ob-serving the transverse magnetization of nuclear spins ina constant magnetic field. Offering the highest sensitiv-ity and occurring in numerous chemical compounds, inmost cases the nuclei under study are protons. By su-perimposing, over two short time intervals, an additionalmagnetic field with a large gradient, the displacement ofthe nuclei (and hence of the molecules in which they arecontained) in the time span between these two “gradientpulses” is recorded in a phase shift of their orientationin the plane perpendicular to the magnetic field with re-spect to the mean orientation. Hence, the distribution ofthe diffusion path lengths appears in the distribution ofthese phase shifts and, consequently, in the vector sumof the magnetic moments of the individual spins, i.e., inthe magnetization [2, 20–22]. Since it is this magneti-zation which is recorded as the NMR signal, moleculardiffusion leads to an attenuation of the signal intensityduring the PFG NMR experiments which is the largerthe larger the displacements in the time interval betweenthese two gradient pulses are.The signal attenuation from PFG NMR may be shownto obey the relation [2, 18, 20, 23]Ψ( τ, k ) = Z d r p ( r , τ ) exp(i kr ) (1)with the ensemble-averaged conditional probability den-sity p ( r , τ ) = Z d x p ( x + r , τ | x ) p ( x ), (2)where p ( x + r , τ | x ) is the stationary probability densityof a displacement r = ( r , . . . , r d ) T in d dimensions in thetime interval τ and p ( x ) refers to the equilibrium distri-bution given by the Boltzmann distribution. Further, τ is the time interval between the two gradient pulses andrepresents the diffusion time in the PFG NMR experi-ment. According to the PFG NMR experiment signalattenuation is measured in the direction of the appliedfield gradients. Thus, k in Eq. (1) is given by k = k ˆ e ,where ˆ e denotes the unit vector in that direction. Thequantity k is a measure of the intensity of the field gradi-ent pulses. Assuming an isotropic system, without loss ofgenerality, an arbitrary direction ˆ k = ( k, , . . . , T maybe considered. Obviously, the scalar product in the ex-ponential of Eq. (1) picks out only the component r ofthe displacement r . Then, the signal attenuationΨ ( τ, k ) = Ψ( τ, ˆ k = ( k, , . . . , T )= + ∞ Z −∞ d r p ( r , τ ) exp(i kr ) (3)depends only on scalar values corresponding to the cho-sen direction and p ( r , τ ) is the projection of the prob-ability density Eq. (2) on the considered direction, givenby p ( r , τ ) = Z · · · Z d r · · · d r d p ( r , τ ). (4)In NMR p ( r , τ ) in Eq. (3) is known as the mean prop-agator, i.e., the probability density that, during τ , anarbitrarily selected molecule has been shifted over a dis-tance r in the direction of the applied field gradients.However, it should be noted that for heterogeneous sys-tems, such as systems with regions of different mobil-ity, p ( r , τ ) may not be called propagator since it can-not evolve the system due to the loss of Markovianity.The reason is that, in general, p ( r , τ ) of such systemsdoes not satisfy the Chapman–Kolmogorov equation [24].Non-Markovian behavior, besides others, may also arisein systems which can be modeled by fractional Brownianmotion [25] or by certain fractional diffusion equations[26]. Further, the mean propagator in Fourier space asgiven by Eq. (1) corresponds to the incoherent interme-diate scattering function. The details of this connectionare given in Appendix A for clarification.In contrast to the PFG NMR technique, which isensemble-based as described above, SPT experiments al-low to follow the trace of individual diffusing molecules.Therefore by considering the displacement of a particularmolecule in d dimensions it is natural to define a micro-scopic single-particle diffusivity D t ( τ ) by the relation D t ( τ ) = [ x ( t + τ ) − x ( t )] / (2 d τ ), (5) where x ( t ) denotes the trajectory of an arbitrary stochas-tic process. Note that the term “microscopic” has beenused before by Kusumi and co-workers [27] to charac-terize the short-time behavior of averaged squared dis-placements equivalent to the small τ limit of our meandiffusivity defined in Eq. (10) below. In the context ofMarkovian diffusion processes this limit also correspondsvia jump moments to the diffusion terms appearing inFokker-Planck equations [24]. Here we use the term “mi-croscopic” in analogy to the statistical physics conceptof microstates to distinguish it from ensemble based av-erages. For a given time lag τ , the microscopic single-particle diffusivity is a fluctuating quantity along a tra-jectory x ( t ) and we now ask for the probability p ( D )d D that, under the so far considered conditions of normal dif-fusion, D t ( τ ) attains a value in the interval D . . . D +d D .Therefore, the distribution of single-particle diffusivitiesis defined as p ( D, τ ) = h δ ( D − D t ( τ )) i , (6)where h . . . i denotes an average, which can be evaluatedeither as a time-average h . . . i = lim T →∞ /T R T . . . d t ,which is accessible by SPT, or, as an ensemble aver-age, appropriate for NMR measurements. Note, how-ever, that in SPT, T is usually limited by the finitenessof the trajectory and complications arise due to the blink-ing and bleaching of the fluorescent dyes [28]. However,advanced tracking algorithms in SPT reduce these ef-fects [13, 29] and arbitrary time lags τ between snap-shots, which are only limited below by the inverse framerate of the video microscope, can be accomplished. Forexperimental SPT data, the distribution of diffusivitiesis obtained by binning diffusivities into a normalized his-togram according to Eq. (6).For ergodic systems, as considered here, time averageand ensemble average coincide. By additionally assumingtime invariance, Eq. (6) can be rewritten as p ( D, τ ) = Z d r δ (cid:18) D − r d τ (cid:19) p ( r , τ ) (7)with the probability density Eq. (2) given by p ( r , τ ) = h δ ( r − r ( τ )) i . By performing the angular integration,Eq. (6) or (7) can also be expressed as p ( D, τ ) = ∞ Z d r δ (cid:18) D − r d τ (cid:19) p r ( r, τ ) (8)in terms of the radial propagator p r ( r, τ ), which is theprobability density p ( r , τ ) integrated over the surface ofa d -dimensional sphere with radius r .The delta functions in Eqs. (7) and (8) simply describea transformation of the coordinates from displacementsto diffusivities. Hence, the distribution of diffusivities is arescaled version of the propagator. This becomes obviousby expanding for r > δ [ D − r / (2 d τ )] = p d τ / (2 D ) δ [ r −√ d τ D ] which yieldsthe relation p ( D, τ ) = r d τ D p r ( √ d τ D, τ ). (9)Furthermore, it should be noted that the distributionof single-particle diffusivities is closely related to the selfpart of the van Hove function given in Appendix A, whichcoincides with p ( r , τ ) given by Eq. (2) for identical par-ticles. Hence, the distribution of diffusivities is also arescaled version of the van Hove self-correlation functionand offers some beneficial properties for our investiga-tions.The diffusivity h D i results as the mean of the micro-scopic single-particle diffusivities Eq. (5). Therefore, forclarity, we denote it in the following as mean diffusivity.According to the definition of the distribution of diffu-sivities the mean diffusivity has to obey the relation h D ( τ ) i = ∞ Z d D D p ( D, τ ). (10)It is thus obtained as the first moment of the probabil-ity density of diffusivities by a well-defined integration,avoiding any numerical fit. Obviously it may also dependon the time lag τ .In the special case of free diffusion of a particle x ( t + τ ) − x ( t ) = R t + τt d t ′ ξ ( t ′ ) is a fluctuating quantity takenfrom one realization of the Gaussian white noise ξ ( t ) withvariance proportional to the diffusion coefficient. WithEq. (3), the mean propagator and the signal attenuationare seen to be interrelated by Fourier transformation [18,23]. In the case of normal diffusion in one dimension onehas p ( r , τ ) = (4 π h D i τ ) − / exp (cid:18) − r h D i τ (cid:19) (11)where h D i stands for the diffusivity. To avoid confusionwe deviated from the usual way of denoting the diffusivitysimply by D . This is because we use this notation torefer to microscopic single-particle diffusivities D t ( τ ). Byinserting Eq. (11) into Eq. (3) the signal attenuation inPFG NMR experiments is seen to obey the well-knownexponential relationΨ ( τ, k ) = exp( − k h D i τ ). (12)Let us now consider a molecular random walk in atwo-dimensional plane. Eq. (11) describes the probabil-ity of a molecular displacement in any arbitrarily chosendirection. For the probability that radial molecular dis-placements are within the interval r . . . r +d r one obtains,therefore, p r ( r, τ ) d r = 14 π h D i τ exp (cid:18) − r h D i τ (cid:19) πr d r . (13) The mean squared displacement14 τ h r ( τ ) i = 14 τ ∞ Z d r p r ( r, τ ) r = h D i (14)obeys the well-known Einstein relation for normal dif-fusion in two dimensions. Inserting the correspondingpropagator of homogeneous diffusion in two dimensionsEq. (13) into Eq. (7) yields the distribution of single-particle diffusivities p ( D ) = h D i − exp( − D/ h D i ). (15)In general, for homogeneous diffusion in d dimensions,the distribution of diffusivities is found to be p d ( D ) = N d D (cid:18) D h D i (cid:19) d/ exp (cid:18) − d D h D i (cid:19) , (16)where N d can be obtained from the normalization condi-tion and is explicitly given by N d = / √ π for d = 11 for d = 23 √ / √ π for d = 3 . (17)Since the system is governed by only one diffusion con-stant, the dependence on τ vanishes in Eq. (16). Howeverfor heterogeneous diffusion, the distribution of single-particle diffusivities additionally depends on the time lag τ . Then, p ( D, τ ) cannot generally be expressed by a sim-ple exponential function as in Eq. (16).By inserting Eq. (15), the first moment of the distri-bution of diffusivities Eq. (10) ∞ Z d D D/ h D i exp( − D/ h D i ) = h D i (18)is easily seen to be fulfilled for homogeneous systemsand equals the mean squared displacement obtained inEq. (14). Hence, with p ( D, τ ), which is a rescaled vanHove self-correlation function, it becomes possible to de-termine the mean diffusion coefficient of the system byordinary integration.With Eq. (15) for diffusion in two dimensions, the dis-tribution of the single-particle diffusivities in homoge-neous systems is seen to result in an exponential. Thesemi-logarithmic plot of the number of trajectory seg-ments governed by a particular single-particle diffusivityversus these diffusivities is correspondingly expected toyield a straight line. Its negative slope is defined as thereciprocal value of the mean diffusivity. Fig. 1 depictsthe distribution of diffusivities of a homogeneous diffu-sion process in two dimensions. The data are obtainedfrom simulations of a system with diffusion coefficient h D i = 0 . p ( D ) D10 -5 -4 -3 -2 -1
0 2 4 6 8 10 p ( D ) D10 -5 -4 -3
6 6.5 7 7.5 8 8.5 9
Figure 1. Distribution of diffusivities from a simulated tra-jectory of a homogeneous diffusion process in two dimensions.The distribution agrees well with the exponential behaviorexpected from Eq. (15) and is independent of τ . The insetdepicts deviations between simulation and Eq. (15) for large D due to insufficient statistics from finite simulation. histogram. The inset of Fig. 1 shows deviations betweensimulated data and Eq. (15) for large D due to insufficientstatistics originating from the finite sample in simulation.It is interesting to note that the shape of the distri-bution of diffusivities of homogeneous diffusion is similarto that of the attenuation function of PFG NMR diffu-sion measurements (Eq. (12)). One has to note, however,that now, in contrast to Eq. (12), the mean diffusivity h D i appears in the denominator of the exponent. Froma semi-logarithmic plot of the PFG NMR signal attenu-ation versus k the mean diffusivity thus directly resultsas the slope, rather than its reciprocal value.In the simple cases of isotropic and homogeneous dif-fusion both the signal attenuation from PFG NMR andthe distribution of diffusivities from SPT resulted in well-known and easily obtainable expressions. In the follow-ing we investigate a more elaborated two-region systemexhibiting inhomogeneous diffusion. III. HETEROGENEOUS DIFFUSION INTWO-REGION SYSTEMS
Let us now consider molecular diffusion in an isotropictwo-region system. With the respective probabilities π i ,the molecules are assumed to propagate with either thediffusivity D or D and to remain with the mean dwelltimes τ m ( m = 1 ,
2) in each of these states of mobility.Thus, the observed diffusion process exhibits dynamicheterogeneities emerging as a time-dependent diffusioncoefficient due to the exchange of particles between tworegions with different diffusion coefficients. For such het-erogeneous systems, the behavior of the distribution ofsingle-particle diffusivities, in general, deviates from themono-exponential decay. This is attributed to a super- position of many different exponentials of type Eq. (15)originating from trajectory segments which include layertransitions during the time lag τ . Thus, we denote thedistribution of single-particle diffusivities by p ( D, τ ) em-phasizing its dependence on τ . Further, the superpo-sition and accordingly the characteristics of the distri-bution of diffusivities strongly depend on the relation ofdwell times and the time lag τ between observed positions[14]. For short time lags compared to the dwell times theexchange rates are very low. Then, the two diffusion pro-cesses can be separated into the two underlying processes.As a result, the probability density is the weighted su-perposition of the mono-exponential decays belonging tohomogeneous diffusion inside each region. In the oppo-site case, for time lags much larger than both dwell times,the observation only reveals a long-term diffusion processwith the mean diffusion coefficient of the system. Hence,the probability density is given by a mono-exponentialdecay parameterized by this mean diffusivity.In the case of a two-region system, the PFG NMRspin-echo diffusion attenuation (and hence the Fouriertransform of the mean propagator) has been shown toresult as a superposition of two terms of the shape ofEq. (12) [2, 18]:Ψ ( τ, k ) = p ′ ( k ) exp( − k D ′ ( k ) τ )+ p ′ ( k ) exp( − k D ′ ( k ) τ ) (19)with D ′ , ( k ) = 12 D + D + 1 k (cid:18) τ + 1 τ (cid:19) ∓ ((cid:20) D − D + 1 k (cid:18) τ − τ (cid:19)(cid:21) + 4 k τ τ ) (20) p ′ ( k ) = 1 − p ′ ( k ) p ′ ( k ) = 1 D ′ ( k ) − D ′ ( k ) ( π D + π D − D ′ ( k )). (21)It should be noted that the primed quantities inEqs. (20) and (21) depend on the intensity of the mag-netic field gradient being related to k and, thus, on theFourier coordinate. Therefore, Eq. (19) cannot be consid-ered as a superposition of separated populations of thetwo regions. It is rather the total interference of spin-echo attenuations observed from both regions. Further,the initial condition of a process described by Eqs. (19)to (21) has to be chosen in such a manner that for theinitial time t = 0 the particles are located at a given po-sition x and are already distributed stationarily betweenthe regions. This is obvious since neither p ′ ( k ) nor p ′ ( k )depends on t which would be necessary to converge to thestationary distribution. For any other initial distributionEq. (19) will only be valid in the limit of t → ∞ .The signal attenuation can also be considered for thelimiting cases. For τ →
0, i.e., τ ≪ τ , τ , the signalattenuationΨ ( τ, k ) = π exp( − k D τ ) + π exp( − k D τ ) (22)decomposes into the superposition of two signal attenua-tions corresponding to each region. As discussed, twocompletely separated diffusion processes are observed.Hence, the inverse Fourier transformation leads to a su-perposition of the distribution of diffusivities of each re-gion. In contrast, for τ → ∞ , i.e., τ ≫ τ , τ , the mixingof the two regions leads to the observation of an effectivemean diffusion process with a signal attenuationΨ ( τ, k ) = exp( − k ( π D + π D ) τ ) (23)containing the mean diffusion coefficient. Analogously,its inverse Fourier transform, i.e., the distribution of dif-fusivities, is only characterized by the mean diffusion co-efficient h D i = π D + π D . A detailed deviation of thelimiting cases is given in Appendix C. IV. SIMULATION OF TWO-REGION SYSTEMS
In order to simulate heterogeneous diffusion we con-sider a system with two regions where particles propagatewith different diffusivities and can change their state ofmobility. Following the experiment with rhodamine inTEHOS [16, 30], this two-region system is modeled bya bi-layer system with layer-dependent diffusion coeffi-cients D and D , respectively. Such processes can for-mally be described as composite Markov processes [31] orequivalently as multistate random walks [32, 33], whichare known to be widely applicable. A recent biophysicalapplication consists of changes in the diffusive behaviorof molecules in membranes due to random changes of themolecules’ conformation [34]. In the case of two statesor regions the probability density of finding the particleat position x at time t is determined by the evolutionequations ∂∂t ˆ p ( x , t ) = w ˆ p ( x , t ) − w ˆ p ( x , t ) + D ∇ ˆ p ( x , t ) ∂∂t ˆ p ( x , t ) = w ˆ p ( x , t ) − w ˆ p ( x , t ) + D ∇ ˆ p ( x , t )(24)for each region with corresponding diffusion coefficients D and D . Within each region the motion of themolecules is accomplished by ordinary two-dimensionaldiffusion, i.e., random walkers experiencing shifts of thepositions distributed according to a Gaussian with a vari-ance defined by the diffusion coefficient in the region.The exchange between these two diffusive regions is sim-ulated by a jump process governed by a master equationwith jump rates w nm , which describe a transition fromregion m to n ( m, n = 1 , w nm yields the mean dwell time τ m τ = 1 w and τ = 1 w (25)for which particles remain in region m . Further, the sta-tionary distribution between the regions π = w w + w and π = w w + w (26) is also dictated by the jump rates. With the station-ary distribution the mean diffusion coefficient of the two-region system is given by h D i = π D + π D , (27)which is the weighted average of the diffusion coefficientsbelonging to each region [19]. -1 0 1 2 3 4 5 6 7 -2 -1 0 1 2 3 4 5 6 7 y x 1 2 (a)(b) x y l a y e r Figure 2. Single-particle trajectory from simulation of diffu-sion in a bi-layer system. (a) The particle performs diffusionwith corresponding diffusion coefficients and jumps betweenthe layers. (b) Projection of the trajectory shown in (a) ontothe x - y -plane as usually observed by single-particle tracking.Information of the layer and the corresponding diffusion coef-ficient is lost in the projection and can only be identified dueto the color code. To investigate the effects of heterogeneous diffusion,simulation of the two-region system is performed with thefollowing system parameters. The diffusion coefficientswithin each of the two regions are given by D = 0 . D = 1 .
0. The jump rates w = 8 and w = 4 yieldthe dwell times τ = 0 .
125 and τ = 0 .
25, respectively.Hence, the stationary distribution between the regionsresults in π = and π = and a mean diffusion co-efficient h D i = 0 . t = 0 .
01, whichis much smaller than the dwell times to ensure diffusivemotion of the particles within the regions.Simulation of Eq. (24) is depicted in Fig. 2 (a). Itshows the trajectory of a particle in a bi-layer system,where the particle jumps between the layers. In eachlayer, diffusion is governed by a different diffusion coef-ficient denoted by the color of the trajectory segments.Since in experiments with video microscopy only a two-dimensional projection of the process is observed, thetrajectory is projected onto the x - y -plane in Fig. 2 (b).As a consequence, information about the layer is ob-scured and can only be identified due to the color codingin the figure. Hence, in the projection it is unknownwhich diffusion coefficient currently governs the process.A description of such observed diffusion processes by theFokker-Planck equation with time-dependent diffusioncoefficient would become possible if all trajectories jumpsynchronously. Since in our bi-layer system the particlesmove independently, the process is more complicated. Asa result of the projection, the observed process does notpossess the Markov property anymore since, in general,the Chapman–Kolmogorov equation cannot be satisfied.The simulation provides an approach to study proper-ties of an N -layer system, which is closely related to asystem where the diffusion coefficient varies continuouslywith the z -coordinate.To avoid transient behavior in our simulation, the par-ticle positions are initialized with their correspondingstationary distributions between the layers given by π i .It should be noted, however, that experimental resultswill be influenced by such transient effects if the tracermolecules require a sufficiently long time to distribute be-tween the layers of the solvent. On the other hand, suchslow relaxation is related to low exchange rates leadingto almost complete separation of the two diffusive regions[14]. This would allow for an appropriate bi-exponentialfit of our distribution of diffusivities although the weightsdo not correspond to the stationary distributions yet.To investigate the connection between spin-echo signaldiffusion attenuation, as measured by PFG NMR, anddistribution of single-particle diffusivities, as assessedby SPT, we simulated one particle. Next, we recordedsquared displacements along the simulated trajectory of10 time steps. The squared displacements are calculatedfrom the changes of the particle positions and are dividedby the time lag τ elapsed between the observations of thetwo positions. Hence, we obtain scaled squared displace-ment with the dimension of a diffusion coefficient. Thethus obtained diffusivity is a fluctuating quantity along atrajectory. Finally, we gather them in a histogram count-ing their occurrences. The histogram contains data froma moving-time average since the diffusivities originatefrom single trajectories. Note that for ergodic systemsensemble averaging will yield identical results. After nor-malizing the histogram we obtain a probability densityreferred to as the distribution of diffusivities. The distri-bution of diffusivities contains all information about thediffusivities of the process and their fluctuations. Follow- ing the experiment, only a fraction of the time steps isavailable for the distribution depending on the selectedtime lag. Thus, our resulting distributions of diffusivitiesdepicted in log-linear plots have their lower boundary at10 − since data below suffer from insufficient statistics. V. APPROXIMATION OF DIFFUSIVITYDISTRIBUTIONS
Since an exact relation of the PFG NMR signal atten-uations to distributions of diffusivities requires inverseFourier transformation we are now going to use the setof Eqs. (19) to (21) for an approximation of the proba-bility distribution of the single-particle diffusivities in atwo-region system. We proceed in analogy with our treat-ment of the simple system with only one (mean) diffusiv-ity. In either case the information about the probabilitydistribution p ( D, τ ) of the single-particle diffusivities D isclearly contained in the propagator. For the system withone diffusivity this propagator is given by Eq. (11). ItsFourier transform (Eq. (12), which is nothing else thanthe PFG NMR spin-echo diffusion attenuation curve) wasfound to coincide with the shape of the probability distri-bution of the single-particle diffusivities (Eq. (15)) withthe only difference that the mean diffusivity, which repre-sents the slope in the semi-logarithmic attenuation plots,appears in the denominator of the exponent in the dis-tribution function p ( D ).In the two-region system, the PFG NMR spin-echo dif-fusion attenuation (and hence the Fourier transform ofthe propagator) is now found to be given by two expo-nentials (Eq. (19)) of the form of Eq. (12). Formally wemay refer, therefore, to two populations with the relativeweights p ′ i and the effective (mean) diffusivities D ′ i asquantified by Eqs. (20) and (21). Following the analogyof our simple initial system, as a first attempt, the result-ing probability function of the single-particle diffusivitiesmay be approximated by a corresponding superpositionof two exponentials of the type of Eq. (15) p ( D, τ ) ≃ ˜ p ( D, ˜ k ( τ )) = p ′ (˜ k ) 1 D ′ (˜ k ) exp( − D/D ′ (˜ k ))+ p ′ (˜ k ) 1 D ′ (˜ k ) exp( − D/D ′ (˜ k ))(28)with the parameters p ′ i (˜ k ) and D ′ i (˜ k ) as given by Eqs. (20)and (21). Since this approximation avoids Fourier trans-formation, a proper τ -dependence of ˜ k has to be chosenfor the primed quantities. It should be noted that thetransformation of Eq. (19) from Fourier space will onlyresult in a superposition of two exponentials in real spaceif the primed quantities in Fourier space are independentof ˜ k . Hence, Eq. (28) could only serve as a rough ap-proximation of the observed process. However, insertingEq. (28) into Eq. (10), the mean diffusivity of the two-region system results in h D i = p ′ (˜ k ) D ′ (˜ k ) + p ′ (˜ k ) D ′ (˜ k ) = π D + π D (29)with the second equality resulting from the applicationof Eqs. (20) and (21). This is exactly the result which iswell-known [19] and it should be noted that it does notdepend on τ .Further on, we may consider the limiting cases ˜ k → k → ∞ which can be translated to r → ∞ and r → r → ∞ arerelated to long observation times τ → ∞ and vice versa.This relation is substantiated by keeping ˜ k τ constant(see also Eq. (32)) where ˜ k → τ →∞ and vice versa. Due to this, the respective limits of˜ p ( D, ˜ k ( τ )) and p ( D, τ ) should coincide. As a result weobtain the expected expressionslim ˜ k →∞ ˜ p ( D, ˜ k ( τ )) = lim τ → p ( D, τ )= π D − exp( − D/D ) + π D − exp( − D/D ) (30)and lim ˜ k → ˜ p ( D, ˜ k ( τ )) = lim τ →∞ p ( D, τ )= h D i − exp( − D/ h D i ). (31)Since the diffusivities and probabilities D ′ i and p ′ i oc-curring in Eqs. (19) to (21) depend on the Fourier coor-dinate ˜ k , we have referred to the probability density inthis context as an approximated one, ˜ p ( D, ˜ k ( τ )). Hence,Eq. (28) in the given notation is unable to provide anapproximation of the probability distribution function ofthe single-particle diffusivities over the whole diffusivityscale. This is in perfect agreement with previous results[14] where it has been shown that the distribution of dif-fusivities, in general, cannot be represented by a weightedsuperposition of the underlying homogeneous diffusionprocesses. However, such an approximation of the prob-ability density might become possible by inserting an ap-propriately selected value for the Fourier coordinate. Asa first trial, one may put˜ k − = h D i τ , (32)which ensures highest sensitivity with respect to thespace scale covered during the experiments. Note thatin PFG NMR experiments the exponent in the signal at-tenuation Eq. (12) is of the order of 1, which yields aneasily observable PFG NMR spin-echo diffusion attenu-ation.Fig. 3 depicts the distribution of diffusivities from asimulated two-dimensional trajectory in a two-region sys-tem with mean dwell times τ = 0 .
125 and τ = 0 . τ = 0 .
01, 0 . .
0. Further, theapproximation of the distribution of diffusivities fromEq. (28) is investigated for corresponding ˜ k . Thus, thelimiting case of completely separated diffusion processesfound for τ → τ = 0 . ≪ τ , τ andcompared with Eq. (28) for ˜ k → ∞ , i.e., Eq. (30). On τ =0.01k ~ →∞ τ =0.2k ~ ≈ p ( D , τ ) D τ =1.0k ~ → -3 -2 -1
0 1 2 3 4 5 6
Figure 3. Comparison of distribution of diffusivities (coloredhistograms) from a simulated two-dimensional trajectory withnumerical approximation via Eq. (28) (solid lines) of a two-region system for time lags τ = 0 .
01, 0 . . τ = 0 .
125 and τ = 0 .
25. The limiting casesof ˜ k → k → ∞ approximate the simulated data rea-sonably. However, for τ = 0 . k ≈ .
67, as suggested in Eq. (32), does notapproximate the density sufficiently. the other hand, the second limiting case of mean diffu-sion emerging for τ → ∞ is obtained from simulationwith τ = 1 . ≫ τ , τ and comparison with Eq. (28) for˜ k →
0, i.e., Eq. (31). Fig. 3 clearly shows that simulateddata from both limiting cases are recovered reasonably byEq. (28) for corresponding ˜ k . In contrast, the distribu-tion of diffusivities reveals a more complicated behaviorin the intermediate exchange regime between the limitingcases. Since the time lag τ = 0 . k ( τ ). This isobvious, since with such an estimate of ˜ k the dependenceon k of the primed quantities Eqs. (20) to (21) in Fourierspace is neglected. Then, the inverse Fourier transfor-mation of Eq. (19) as well as the transformation to thedistribution of diffusivities would yield a simple super-position of two exponentials again. In general, this doesnot provide appropriate results for arbitrary dwell timesand time lags [14]. As a consequence, a general expres-sion requires inverse Fourier transformation of the PFGNMR attenuation curve. VI. EXACT RELATION BETWEEN SIGNALATTENUATION AND DISTRIBUTION OFDIFFUSIVITIES
In Sec. V, the approximation of the distribution of dif-fusivities by Eq. (28) was shown to reproduce the limit-ing cases of time lag τ as well as the correct mean value.Cases in between the limits did not deliver appropriateresults. In order to produce proper results for arbitrary τ we derive general formulae for the transformation ofPFG NMR signal attenuations to distributions of single-particle diffusivities.Quite formally two steps have to be accomplished toderive a general expression of p ( D, τ ) from Ψ( τ, k ). As afirst step, inverse Fourier transformation of Eq. (1) yieldsthe propagator in real space. Further, the shift r betweenpositions, as given by the propagator, can be translatedinto diffusivities via scaled squared displacements leadingto the distribution of diffusivities as defined in Eq. (7).The two steps can be combined to directly obtain theprobability density from signal attenuation. Dependingon dimensionality d , the distribution of diffusivities isgiven by p ( D, τ ) = Z d r δ (cid:18) D − r d τ (cid:19) × π ) d Z d k Ψ( τ, k ) exp( − i kr ). (33)With the rescaled coordinates r ′ = r √ d τ and k ′ = k √ d τ (34)it is further simplified to p ( D, τ ) = Z d k ′ Ψ (cid:18) τ, k ′ √ d τ (cid:19) π ) d S d ( k ′ , D ), (35)with S d ( k , D ) being the Fourier transform of a uniformdensity on the surface of a d -dimensional sphere of radius √ D S d ( k , D ) = Z d r δ ( D − r ) exp( − i kr ). (36)Since Eq. (36) can be expressed analytically [35] by S d ( k , D ) = π a +1 D a a J a ( | k |√ D )( | k |√ D ) − a (37)with a = d/ − J a ( x ) denoting the Bessel functionof the first kind, the exact transformation of signal atten-uations Ψ( τ, k ) to distributions of diffusivities p ( D, τ ) isaccomplished without applying an inverse Fourier trans-formation.For isotropic systems, the signal attenuation Ψ( τ, k )depends only on the absolute value of k , i.e., the radialintensity of the field gradient k . Without loss of gen-erality, an arbitrary direction k = ( k, , . . . , T may beconsidered and the corresponding signal attenuation isdenoted by Ψ ( τ, k ) = Ψ( τ, k = ( k, , . . . , T ). Then thefollowing expressions are obtained for the distribution ofdiffusivities depending on the dimensionality of the sys-tem. For one-dimensional systems Eq. (33) reduces to p ( D, τ ) = 1 π √ D ∞ Z d k Ψ (cid:18) τ, k √ τ (cid:19) cos( k √ D ). (38) The transformation for d = 2 can be written as p ( D, τ ) = 12 ∞ Z d k Ψ (cid:18) τ, k √ τ (cid:19) kJ ( k √ D ), (39)and for d = 3 one obtains p ( D, τ ) = 1 π ∞ Z d k Ψ (cid:18) τ, k √ τ (cid:19) k sin( k √ D ) (40)using polar and spherical coordinates, respectively. Thegiven transformations move the whole dependence ontime lag τ to the signal attenuation. This is achievedby rescaling the k coordinate by √ d τ .Hence, a signal attenuation of an ensemble diffusingin a two-dimensional plane measured by PFG NMR istransformed into a distribution of single-particles diffu-sivities via Eq. (39). For homogeneous diffusion Eq. (39)yields the expected probability of single-particle diffusiv-ities Eq. (15) by inserting the simple exponential relationEq. (12) as signal attenuation.Furthermore, the limiting cases of time lag τ are repro-duced exactly by the presented transformations Eqs. (38)to (40): For τ → τ → ∞ , the resulting distributionof single-particle diffusivities for the given dimensionalityis also of type Eq. (16), respectively, and depends only onthe mean diffusion coefficient of the system. A detailedderivation of the limiting cases is given in Appendix C.To examine the transformations, the same parameters,for which the approximation via Eq. (28) failed, are usedagain, now applying Eq. (39) for an exact transformationof the PFG NMR signal attenuation relation, Eq. (19),into the distribution of single-particle diffusivities. Theresults are depicted in Fig. 4 and again the distribution ofdiffusivities from a simulated two-dimensional trajectoryis given for comparison. For each of the chosen τ = 0 . .
2, 0 . . τ and reveals a transition from anon-exponential behavior to a mono-exponential decay.For small τ corresponding to diffusion in separated re-gions it deviates considerably from a mono-exponentialbehavior. However, for long-term observations ( τ → ∞ )only a mean diffusion process is observed due to aver-aging of the motion in both regions. Consequently, thisyields a mono-exponential decay of the distribution ofdiffusivities. This transition reveals the heterogeneity ofthe diffusion process [14]. Hence, in order to characterizediffusive motion the distribution of diffusivities has to beinvestigated for its dependence on the time lag τ .0 τ =1.0 τ =0.5 τ =0.2 p ( D , τ ) D τ =0.0510 -2 -1
0 1 2 3 4
Figure 4. Comparison of distributions of single-particle diffu-sivities from a simulated two-dimensional trajectory (coloredhistograms) with distributions obtained by applying Eq. (39)for an exact transformation of the PFG NMR spin-echo sig-nal diffusion attenuation Eq. (19) of a two-region system fortime lags τ = 0 .
05, 0 .
2, 0 . . τ = 0 .
125 and τ = 0 .
25 (solid lines). The data agree wellwith each other for each τ . Further, the dependence on τ is apparent, which is typical for diffusion in heterogeneousmedia. VII. INFLUENCE OF EXPERIMENTALLYBOUNDED k PFG NMR spin-echo diffusion attenuation functionscan only be measured up to a finite intensity k of themagnetic field gradient pulses. However, to generate thedistribution of diffusivities exactly, the signal attenuationhas to be given over the whole intensity scale. Hence, theeffect of an experimentally bounded Fourier coordinate k has to be considered.Fig. 5 illustrates the influence of finite k on the distri-bution of single-particle diffusivities obtained for τ = 0 . k max the respective spin-echosignal is not sufficiently attenuated, the transformationof the signal attenuation from a finite interval will yieldsignificant deviations from the expected probability dis-tribution. As a consequence, the first moment, i.e., themean diffusion coefficient of the system, is altered ac-cordingly. Furthermore, due to the bounded signal at-tenuation, the inverse Fourier transformation introducesoscillations since only a limited range of the spectrumcontributes to the values in real space. The reason is theintegrand in Eqs. (38) to (40) which will only vanish forlarge k if Ψ decays faster than the remainder.This effect may clearly be identified in Fig. 5. In or-der to obtain reasonable results, the signal must be at-tenuated to a sufficient extent. Simulated data of two-dimensional diffusion processes have shown that the at-tenuation should fall below 10 − of its maximum at k max to suppress oscillations. This has to be considered whendealing with experimental data. p ( D , . ) D k max =2k max =5k max =8k max =10k max =12k max =1510 -3 -2 -1
0 1 2 3 4 5 6 7 8
Figure 5. Transformation Eq. (39) of PFG NMR spin-echosignal diffusion attenuation by integration up to k max (solidlines) due to experimentally bounded intensity k of the fieldgradient pulses. The distribution of single-particle diffusiv-ities (colored histogram) from a simulated two-dimensionaltrajectory will only be obtained reasonably if k is given overthe whole intensity scale. For smaller intervals of k deviationsbecome clearly visible as well as oscillations introduced by theinverse Fourier transformation. The necessity of fast decaying Ψ becomes especiallyimportant for large time lags τ . In the case of small timelags τ → k coordinate in Eqs. (38)to (40) leads to k/ √ τ → ∞ in the second argument ofΨ ( τ, k/ √ d τ ). Thus, for small τ , signal attenuationbecomes more pronounced and reduces the influence ofthe bounded k . Moreover, signal attenuation is closelyrelated to the incoherent structure factor [36], as demon-strated in Appendix A, dealing with similar limitations.A possible solution is to split the integral into two parts,integrating numerically up to the experimental limit k max and assuming an analytical expression for the remainingpart.Since the oscillations in the approximate densities ofFig. 5 seem to be induced by the hard cut-off at thewavelength k = k max , a possible strategy in reducingthese oscillations may lie in applying an appropriate win-dow function as in spectrum estimation procedures. Wetested this option by applying a half Hann window tosmoothen the cut-off. The best results were obtained fora window decaying from the value one at k = 0 to zeroat k = k max . The obtained results are very convincing ifthe cut-off value k max is not too small as can be seen inFig. 6. VIII. CONCLUSIONS
We investigated the connection between the signal at-tenuation measured by pulsed field gradient nuclear mag-netic resonance and the distribution of single-particle dif-fusivities obtained from single-particle tracking. Due to1 p ( D , . ) D k max =2k max =5k max =8k max =10k max =12k max =1510 -3 -2 -1
0 1 2 3 4 5 6 7 8
Figure 6. Same situation as in Fig. 5, but the densities arenow obtained by replacing the sharp cut-off at k = k max by asmooth cut-off resulting from applying a half Hann window. Aconsiderable improvement is achieved, especially if the value k max is not too small. their interrelations with the diffusion propagator of thesystem, the distribution of diffusivities is expressed bya general transformation of the signal attenuation. Inthe special case of a system involving two different statesof diffusive mobility, the two-region exchange model ofPFG NMR offers analytical expressions and allows fora comparison of analytical and simulated data. An ap-proximation of the distribution of single-particle diffu-sivities via two populations with relative weights avoidsthe inverse Fourier transformation. Even in this sim-ple system, such an approximation will only yield ap-propriate results if the time lag is much larger or muchsmaller than the dwell times. These cases correspond toan observation of the mean diffusion of the system anda process of completely separated diffusive motion with-out transition between the regions, respectively. Thus, ingeneral, to obtain a proper distribution of single-particlediffusivities for diffusion in two-region systems, the ex-act transformation of the respective NMR signal atten-uations is necessary. Only in this way we found perfectagreement of the experimental and analytical data. How-ever, since PFG NMR data in some systems cannot bemeasured over a sufficiently large dynamic range, the in-verse Fourier transformation may introduce deviationsand oscillations. In these cases, the data analysis hasto be performed with care and may require the use ofadditional information.In summary, the investigated connection between twopopular methods to experimentally observe and analyzediffusive motion offers new approaches for the evalua-tion of data. Hence, the methods of analysis may benefitfrom each other. This becomes especially relevant forsystems with heterogeneities, where the distribution ofdiffusivities exhibits a dependence on the time lag. Formore elaborated processes it may even not become sta-tionary and enables to assess non-trivial properties of such systems. Since the distribution of diffusivities canbe measured easily and contains more information fromthe propagator than well-established methods it shouldbe used for future analysis of experimental data. ACKNOWLEDGMENTS
We gratefully acknowledge financial support from theDeutsche Forschungsgemeinschaft (DFG) for funding ofthe research unit FOR 877 “From Local Constraints toMacroscopic Transport”. We also thank the anonymousreferees for their valuable suggestions, which helped toimprove the paper considerably.
Appendix A: Correspondence between incoherentintermediate scattering function and signalattenuation
The signal attenuation of PFG NMR and the incoher-ent intermediate scattering function as well as the dy-namic structure factor are closely related. In this ap-pendix, their correspondence is illustrated briefly andfurther details can be found in Refs. 18, 37, and 38.The observed motion of tracer particles can be ana-lyzed by the self part of the van Hove time-dependentpair correlation function G s ( r , τ ) = * N N X i =1 δ (cid:16) r − (cid:0) x i ( τ ) − x i (0) (cid:1)(cid:17)+ (A1)describing the correlation of N individual particles [39].Its spatial Fourier transformation S ( k , τ ) = Z d r G s ( r , τ ) exp(i kr ) (A2)leads to the incoherent intermediate scattering function S ( k , τ ) = 1 N N X i =1 (cid:10) exp(i k ( x i ( τ ) − x i (0))) (cid:11) , (A3)which is linked to the velocity autocorrelation functionof the particles. Furthermore, the incoherent intermedi-ate scattering function S ( k , τ ) is related to the dynamicstructure factor S ( k , ω ) known from neutron scatteringvia Fourier transformation in τ , i.e., the power spectrumof S ( k , τ ), where ω denotes a frequency.For ergodic systems, S ( k , τ ) can be obtained from anarbitrary particle S ( k , τ ) = (cid:10) exp(i k ( x ( τ ) − x (0))) (cid:11) = 1 V Z Z d x d x ′ exp(i k ( x − x ′ ) p ( x , τ, x ′ , V is the normalization and p ( x , τ, x ′ ,
0) denotesthe joint probability of a particle to be located initially2at x ′ and at time τ at position x . The joint probabilitycan be expressed by the conditional probability p ( x , τ, x ′ ,
0) = p ( x , τ | x ′ , p ( x ′ ). (A5)Since during time τ the particle accomplished a displace-ment r , its positions are interrelated by x = x ′ + r . Due totranslation invariance, without loss of generality, x ′ = leads to the propagator in Fourier space1 V Z d r exp(i kr ) p ( r , τ ) = Ψ( τ, k ) (A6)corresponding to the signal attenuation in PFG NMRas introduced in Eq. (1). Hence, signal attenuationand incoherent intermediate scattering function coincide.Furthermore, for identical particles without restrictionsby the boundaries the averaging over the particles inEq. (A1) can be omitted and G s ( r , τ ) is equal to p ( r , τ )given by Eq. (2).For isotropic systems the self part of the radial vanHove time-dependent pair correlation function G s ( r, τ ) = * N N X i =1 δ ( r − | x i ( τ ) − x i (0) | ) + (A7)considers only absolute values of the displacements.Again, for identical particles without restrictions by theboundaries an arbitrary particle can be considered and G s ( r, τ ) is equal to p r ( r, τ ). Appendix B: Relation between evolution equationsand PFG NMR signal attenuation
For Eq. (24), i.e., the evolution equations of the prob-ability density to find a particle at position x at time t ,the moments of the random variable x can be obtainedvia the characteristic functions. By introducing the vec-tor p ( k , t ) comprising the characteristic functions of eachregion and the matrix W ( k ) consisting of the elements W ( k ) nm = w nm + − D n k − X l w ln ! δ nm , (B1)the Fourier transform of Eq. (24) can be written elegantlyas dd t p ( k , t ) = W ( k ) p ( k , t ) (B2)where p ( k , t ) = exp( t W ( k )) p ( k ,
0) (B3)is easily seen to be the solution. For the two-region sys-tem the initial distribution p ( k ,
0) = ( π , π ) T is givenby the equilibrium distribution between the regions. Applying the spectral decomposition the matrix ex-ponential in Eq. (B3) for the two-region system can bewritten asexp( t W ( k )) = X α =1 exp( tµ α ( k )) A α ( k ), (B4)where µ , ( k ) = 12 ( − D k − D k − λ ± D ( k )) (B5)denote the eigenvalues and A , ( k ) = 12 D ( k ) (cid:18) D ( k ) ± η ( k ) ± w ± w D ( k ) ∓ η ( k ) (cid:19) (B6)represent the corresponding matrices from the dyadicproduct of the right- and left-eigenvectors with λ = w + w , (B7) η ( k ) = − D k + D k − w + w , and (B8) D ( k ) = { ( D k + D k + λ ) − D D k − D k w − D k w } . (B9)Finally, the signal attenuation obtained from PFGNMR corresponds to the projection of the characteris-tic functionΨ( τ, k ) = (cid:0) (cid:1) exp( τ W ( k )) (cid:18) π π (cid:19) , (B10)where k = k ˆ e is measured in the direction of the ap-plied field gradient denoted by the unit vector ˆ e . Sincefor isotropic systems an arbitrary direction can be con-sidered, Eq. (B10) results in the expressions given inEqs. (19) to (21) for the two-region system. Appendix C: Exact transformation of limiting cases
By choosing k = k ˆ e , the isotropic signal attenuationsfor dimensionality d in Eqs. (38) to (40)Ψ (cid:18) τ, ku √ τ (cid:19) with u = √ d = 12 for d = 2 √ d = 3 ,are considered in an arbitrary direction of the ap-plied field gradient with intensity k . The exponent ofEq. (B10) is given by τ W (cid:18) k ˆ e u √ τ (cid:19) = τ (cid:18) − w w w − w (cid:19) − k u (cid:18) D D (cid:19) .(C1)Based on these expressions the limiting cases are dis-cussed separately.3 Limiting case τ → In the limiting case of τ →
0, only the diagonal ma-trix on the right hand side of Eq. (C1) remains. Hence,the matrix exponential can be expressed by the exponen-tiation of the diagonal elements and Eq. (B10) reducestoΨ (cid:18) τ, ku √ τ (cid:19) = (cid:0) (cid:1) exp (cid:16) − k u D (cid:17)
00 exp (cid:16) − k u D (cid:17) (cid:18) π π (cid:19) (C2)yielding a superposition of two exponentials correspond-ing to separated regions. This is in agreement with previ-ous findings since for short times τ no exchange betweenthe regions occurs. Obviously, this result is not restrictedto the two-region system but holds for an arbitrary num-ber of diffusion states.Applying the presented transformations Eqs. (38) to(40) for dimensionality d to the obtained signal attenua-tion results in a distribution of diffusivities which is thesuperposition of two distributions of diffusivities for ho-mogeneous diffusion in each region as given by Eq. (16),respectively. Limiting case τ → ∞ In the limiting case of τ → ∞ , the situation is morecomplicated. Arguing analogously to the case of τ → A , (cid:18) k ˆ e u √ τ (cid:19) τ →∞ −−−−→ A , ( ) A ( ) = 1 λ (cid:18) w w w w (cid:19) (C3) A ( ) = 1 λ (cid:18) w − w − w w (cid:19) . (C4)Due to the projection in the signal attenuation Eq. (B10) (cid:0) (cid:1) A ( ) = (cid:0) (cid:1) (C5)the contribution from A ( ) vanishes. Thus, for τ → ∞ only eigenvalue µ contributes to the spectral decompo-sition. Moreover, µ = 0, which explains that the contri-bution from the diagonal matrix in Eq. (C1) cannot beneglected.Then, according to Eq. (B4), the exponential of τ µ ( k ˆ e / ( u √ τ )) is required, which is given by τ µ (cid:18) k ˆ e u √ τ (cid:19) = 12 (cid:16) − a − λτ + p ( a + λτ ) − b − cτ (cid:17) (C6) with a = D k u + D k u (C7a) b = 4 D D k u (C7b) c = (4 D w + 4 D w ) k u . (C7c)The square root in Eq. (C6) can be rewritten as p ( a + λτ ) − b − cτ = λτ s (cid:18) aλ − cλ (cid:19) τ + a − bλ τ = λτ (cid:18) (cid:18) aλ − cλ (cid:19) τ + O (cid:18) τ (cid:19)(cid:19) . (C8)After further simplification, Eq. (C6) reduces to τ µ (cid:18) k ˆ e u √ τ (cid:19) ≃ (cid:16) − a − λτ + λτ + a − c λ (cid:17) = − c λ , (C9)which results in τ µ (cid:18) k ˆ e u √ τ (cid:19) ≃ − ( π D + π D ) k u = −h D i k u (C10)by applying Eq. (C7c) and Eqs. (26) and (29). Hence inthe limiting case of τ → ∞ , the signal attenuationΨ (cid:18) τ, ku √ τ (cid:19) = exp (cid:18) −h D i k u (cid:19) (C11)depends only on the mean diffusion coefficient of the two-region system.By integrating the signal attenuation Eq. (C11) for thelimiting case τ → ∞ with the presented transformationsEqs. (38) to (40) for dimensionality d , as expected, therespective distributions of diffusivities Eq. (16) are ob-tained, which correspond to homogeneous diffusion withthe mean diffusion coefficient h D i .To conclude, the derivation of the two limiting casesreveals the properties of the distribution of single-particlediffusivities and its dependence on τ . Starting from thelimiting case τ → ∞ , where only eigenvalue µ con-tributes, the weight of µ increases for decreasing τ . Thisis reflected in the distribution of diffusivities by the de-pendence on τ as presented in Fig. 4. It describes thetransition from a mean diffusion process to two com-pletely separated diffusion processes for τ → ∞ and τ →
0, respectively. It should be noted that for theself part of the van Hove function the limiting cases can-not be determined. However, for the distribution of dif-fusivities, which is a rescaled van Hove self-correlationfunction, both limits are well-defined.4 [1] P. Heitjans and J. K¨arger, eds.,
Diffusion in Condensed Matter , 2nd ed. (Springer,Berlin, 2005).[2] W. S. Price,
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