Propagator representation of anomalous diffusion: the orientational structure factor formalism in NMR.
ABSTRACT The radial Fourier transform for the isotropic space with a fractal dimension is discussed. The moments of diffusive displacements with non-Gaussian propagators arising as solutions of fractional diffusion equations are calculated. The Fourier propagator is applied to NMR correlation and spectral density functions in context with the orientational structure factor formalism. It is shown that the low-frequency molecular fluctuations of liquids in porous media with strong or forced adsorption at surfaces are due to reorientations mediated by translational displacements caused by surface diffusion of the adsorbate molecules. In terms of this formalism, field-cycling NMR experiments provide information on the static and dynamic fractal dimensions related to surface diffusion. The experimental results for liquids in porous silica glass can be explained by a surface fractal dimension df=2.5, where the mean squared displacement scales as <r(2)(t)> proportional, variantt(2/dw) with dw=1 (ballistic transport), if the surface population can exchange with the bulklike phase in the pores, and with dw=2, if the bulklike phase is frozen. The former dynamics is interpreted in terms of bulk-mediated surface diffusion.
Article: Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal.[show abstract] [hide abstract]
ABSTRACT: This report introduces a novel method to characterize the diffusion-time dependence of the diffusion-weighted magnetic resonance (MR) signal in biological tissues. The approach utilizes the theory of diffusion in disordered media where two parameters, the random walk dimension and the spectral dimension, describe the evolution of the average propagators obtained from q-space MR experiments. These parameters were estimated, using several schemes, on diffusion MR spectroscopy data obtained from human red blood cell ghosts and nervous tissue autopsy samples. The experiments demonstrated that water diffusion in human tissue is anomalous, where the mean-square displacements vary slower than linearly with diffusion time. These observations are consistent with a fractal microstructure for human tissues. Differences observed between healthy human nervous tissue and glioblastoma samples suggest that the proposed methodology may provide a novel, clinically useful form of diffusion MR contrast.Journal of Magnetic Resonance 01/2007; 183(2):315-23. · 2.14 Impact Factor
Article: Q-ball imaging.[show abstract] [hide abstract]
ABSTRACT: Magnetic resonance diffusion tensor imaging (DTI) provides a powerful tool for mapping neural histoarchitecture in vivo. However, DTI can only resolve a single fiber orientation within each imaging voxel due to the constraints of the tensor model. For example, DTI cannot resolve fibers crossing, bending, or twisting within an individual voxel. Intravoxel fiber crossing can be resolved using q-space diffusion imaging, but q-space imaging requires large pulsed field gradients and time-intensive sampling. It is also possible to resolve intravoxel fiber crossing using mixture model decomposition of the high angular resolution diffusion imaging (HARDI) signal, but mixture modeling requires a model of the underlying diffusion process.Recently, it has been shown that the HARDI signal can be reconstructed model-independently using a spherical tomographic inversion called the Funk-Radon transform, also known as the spherical Radon transform. The resulting imaging method, termed q-ball imaging, can resolve multiple intravoxel fiber orientations and does not require any assumptions on the diffusion process such as Gaussianity or multi-Gaussianity. The present paper reviews the theory of q-ball imaging and describes a simple linear matrix formulation for the q-ball reconstruction based on spherical radial basis function interpolation. Open aspects of the q-ball reconstruction algorithm are discussed.Magnetic Resonance in Medicine 01/2005; 52(6):1358-72. · 2.96 Impact Factor
Propagator representation of anomalous diffusion: The orientational structure factor formalism
Tatiana Zavada,1Norbert Su ¨dland,2Rainer Kimmich,1and T. F. Nonnenmacher2
1Sektion Kernresonanzspektroskopie, Universita ¨t Ulm, 89069 Ulm, Germany
2Abteilung fu ¨r Mathematische Physik, Universita ¨t Ulm, 89069 Ulm, Germany
?Received 3 March 1999?
The radial Fourier transform for the isotropic space with a fractal dimension is discussed. The moments of
diffusive displacements with non-Gaussian propagators arising as solutions of fractional diffusion equations
are calculated. The Fourier propagator is applied to NMR correlation and spectral density functions in context
with the orientational structure factor formalism. It is shown that the low-frequency molecular fluctuations of
liquids in porous media with strong or forced adsorption at surfaces are due to reorientations mediated by
translational displacements caused by surface diffusion of the adsorbate molecules. In terms of this formalism,
field-cycling NMR experiments provide information on the static and dynamic fractal dimensions related to
surface diffusion. The experimental results for liquids in porous silica glass can be explained by a surface
fractal dimension df?2.5, where the mean squared displacement scales as ?r2(t)??t2/dwwith dw?1 ?ballistic
transport?, if the surface population can exchange with the bulklike phase in the pores, and with dw?2, if the
bulklike phase is frozen. The former dynamics is interpreted in terms of bulk-mediated surface diffusion.
PACS number?s?: 05.40.?a, 61.43.Hv, 68.35.Ct, 68.35.Fx
The anomalous transport in disordered media can be dis-
cussed either in terms of a random walk ?or statistical? ap-
proach or on the basis of a fractional diffusion equation
?FDE?. Fractional or enhanced diffusion is characterized by
the mean squared displacement law
where the dynamic parameter dw?usually called the anoma-
lous diffusion exponent? of the corresponding ?anomalous?
random process deviates from Einstein’s classical result dw
?2. Non-Gaussian propagators arise as the solutions of frac-
tional diffusion equations, which are mostly discussed for
dw?2 ?see Refs. ?1–3??. However, it can be shown ?4–6?
that this type of equation can be solved for the superdiffusive
regime 1?dw?2 as well.
With respect to the exponent dw, anomalous diffusion
can be classified ?5,6? as follows.
?i? dw?2 corresponds to the classical Brownian motion
described by a Gaussian propagator. From the statistical
point of view, this random process is characterized by fixed
step length and waiting time ?1?.
?ii? dw?2 corresponds to the dispersive diffusion regime
described by a non-Gaussian displacement probability den-
sity. In the frame of the continuous time random walk
?CTRW? model, processes of this kind are due to temporal
disorder, characterized by power-law distributions of the
waiting times ?1?.
?iii? 1?dw?2 defines the intermediate region of super-
diffusion. Enhanced transport of this sort results from long-
tailed step-length and waiting-time distributions ?i.e., spatial
or/and temporal disorder of the random process?.
?iv? dw?1 indicates ballistic transport. In this case, the
fractional diffusion equation adopts the form of a wave equa-
In the present study, the reciprocal-space solutions for
displacement distributions are employed in the frame of the
orientational structure factor formalism ?7,8? in order to cal-
culate the NMR correlation and intensity functions and the
frequency dependence of the spin-lattice relaxation rate for
molecular dynamics dominated by reorientation mediated by
translational displacements ?RMTD? along surfaces ?9?. The
static and dynamic fractal dimensions, dfand dw, respec-
tively, which are related to diffusion of strongly adsorbed
molecules along fractal surfaces of porous glass, are evalu-
ated. The experimental data provide evidence for superdiffu-
sive surface displacements with dw?1 if the surfacelike and
bulklike phase are in fast exchange. This result is consistent
with the model of bulk-mediated surface diffusion developed
by Bychuk and O’Shaughnessy ?10?. That is, on the time
scale of the so-called retention time, surface diffusion takes
place in the form of Le ´vy walks.
II. ANOMALOUS DIFFUSION: A FRACTIONAL
Fick’s second diffusion law reads
where ? is equal to the diffusion coefficient, and “ is the
nabla operator. P(r,t) is the probability density of diffusive
displacements r in a time t ?the propagator?. The well-known
solution of Eq. ?2? is the Gaussian probability density.
Anomalous diffusion processes on fractal structures can
be treated using the fractional diffusion equation ?4,6?. The
integral representation of the FDE reads
PHYSICAL REVIEW EAUGUST 1999VOLUME 60, NUMBER 2
1063-651X/99/60?2?/1292?7?/$15.001292© 1999 The American Physical Society
The memory kernel ?(t?t?) reflects temporal disorder of an
anomalous transport process. Assuming ?(t?t?)?? (t
?t?)??1/?(?), one gets ?4,6?
The units of the transport constant ? are related to that of the
ordinary diffusion coefficient D0: ?????D0/???1?. The
differential form of this equation can be expressed by the
following time-fractional diffusion equation
??is the Riemann-Liouville integral operator ?11?.
?P(r,0)t??/?(1??), accounting for the initial-value con-
dition P(r,0)??(r), is often not mentioned explicitly in the
literature ?3,5,12,13?. Actually, this term can be omitted only
in the case ??N0but is essential if the diffusion equation
really has a fractional character. Obviously, for ??1, we
have (?/?t)P(r,0)?0, and hence the conventional diffusion
(?2/?t2)P(r,0)?0, Eq. ?5? takes the form of a wave equa-
tion and describes the ballistic transport regime.
The geometry of the fractal structure and the dynamics of
the corresponding random process are described by the static
fractal dimension dfand by the dynamic fractal dimension
dw?i.e., the anomalous diffusion exponent?, respectively.
Note that “ in the FDE Eq. ?5? represents the nabla operator
in D dimensions ?3,5?. The order of the fractal time deriva-
tive, ?, is related to dwby ??2/dwand is independent of the
static parameter df.
Occasionally, the operator ??is discussed in the literature
?6,12? instead of ?2. The FDE is then certainly treatable in
one dimension, and leads to solutions in the form of Le ´vy
distributions ?6?. However, the interpretation of the fractional
spatial derivative in more dimensions remains unclear,
whereas the use of the operator ?2in Eqs. ?4? or ?5? is
motivated by the radial symmetry of the ?fractal? space under
consideration: In the reciprocal space, Eqs. ?4? or ?5? are
The square of the wave number, k2??k?2, is equal to the sum
of the d components of k if the space in which the random
motion takes place is defined by d Euclidean dimensions. On
the other hand, if the random-walk space is of a fractal di-
mensionality, the number of the k components is defined by
the global dimension D ?e.g., D?2 for a fractal surface, D
?3 for a fractal volume?.
Schneider and Wyss ?4? solved Eq. ?5? after rescaling it so
that ??1. However, the parameter ? is important for experi-
ments since it is related to the diffusion coefficient D0, and,
as a consequence, to the temperature dependence of the dif-
The solution ?4? of Eq. ?6? is given by the Mittag-Leffler
where ??2/dwcan be found with the approach suggested by
Metzler and Nonnenmacher ?5?. The inverse D-dimensional
Fourier transformation ?see Appendix? of the characteristic
function given in Eq. ?7? leads to the real-space propagator in
terms of Fox’s H function ?14?,
Note that the space dimension D does not enter in the ex-
pression for p(k,t) ?Eq. ?7??, in contrast to the real-space
propagator ?Eq. ?8??. Thus, in this model, p(k,t) exclusively
provides the dynamical information indicated by dw, regard-
less of the structure of the r space. For D?2, Eq. ?8? repre-
sents our surface propagator.
The mth moment of the propagator at Eq. ?8? reads ?see
so that the second moment takes the form
In the limit ?t2/dwk2?1, the propagator given in Eq. ?7?
With this expression, one finds
PRE 601293PROPAGATOR REPRESENTATION OF ANOMALOUS . . .
in general, and
for the second moment. Comparing Eqs. ?10? and ?13?, one
realizes that the asymptotic form of the Mittag-Leffler propa-
gator given in Eq. ?11? can be used to calculate the mean
squared displacement instead of the exact solution at Eq. ?7?.
This finding is of relevance for practical applications since
the asymptotic form of p(k,t) is more obvious than the exact
Mittag-Leffler representation of the propagator.
In Ref. ?6?, an alternative approach to superdiffusion was
considered for the fractional diffusion equation in one di-
mension. In that case, the k-space solution turns out to be
equal to the characteristic Kohlrausch-Williams-Watts func-
where (a?0,t?0). By means of p(k,t) given in Eq. ?14?,
Le ´vy distributions p˜(r,t) in the real space can be generated
with the aid of the inverse Fourier transform ?6?. For large r,
one obtains p˜(r,t)?t/r??1as the limiting form of the Le ´vy
?-stable process in a one-dimensional space.
III. THE RMTD RELAXATION MECHANISM
AND THE ORIENTATIONAL
Proton spin-lattice relaxation in liquids is predominantly
due to fluctuations of the intramolecular dipole-dipole inter-
action among the spin-bearing nuclei. That is, molecular dy-
namics reorients the molecules so that dipolar coupling is
modulated. In context with adsorbate diffusion along rough
surfaces, fluctuations slow compared with bulk correlation
times are governed by reorientations mediated by transla-
tional displacements along rough and curved surfaces ?8,15–
Molecular fluctuations are described by the autocorrela-
tion function G(t), and, in the frequency domain, by the
intensity function I(?). The latter is defined as the cosine
Fourier tranform of G(t),
The spin-lattice relaxation rate is given by
A technique suitable to record the frequency dependence
of this function over several orders of magnitude is field-
cycling NMR relaxometry ?9?. The information probed in
this way refers to the autocorrelation function via the inten-
The correlation function G(t) virtually reflects the corre-
lation of molecular orientations at the moments t? and t?
?t. Molecular reorientations in bulk liquids are a conse-
quence of rotational diffusion. If the molecule is adsorbed on
a surface, this rotational diffusion is hindered and incomplete
with respect to the solid angle range covered. That is, re-
sidual orientational correlations persist on a time scale much
longer than that of ordinary rotational diffusion. The mecha-
nism coming at longer times into play is ‘‘reorientation me-
diated by translational displacements’’ along rough surfaces.
The formalism is described in more detail in Refs.
?7–9,15,16,18?. Thus, on a correspondingly long time scale,
nuclear spin-lattice relaxation is dominated by surface diffu-
Displacements of the adsorbed molecule along the surface
are characterized by the propagator P(s,t) on the one hand,
and the orientational correlation at sites separated by the dis-
tance s on the other. The latter is described by the surface
orientation correlation function g(s) which can be expressed
in terms of second-order spherical harmonics Y2,?1(?),
A? d2s0? d?0? d?sY2,?1??0?
The quantity A is the surface area accessible by surface
diffusion on a time scale of the order T1. The vectors ?0and
?sdenote the surface orientations at the initial and final
positions on the surface, s0and s0?s, respectively. The func-
tion ?(?0,?s,s)d?0d?s is the conditional probability
that the surface orientation at the position s is within ?sand
?s?d?sif the surface orientation at the position s0is
within ?0and ?0?d?0. Expressing ?(?0,?s,s) by the
product of ? functions averaged over all possible initial po-
Random surfaces may be discussed by considering a one-
dimensional surface profile ?18?. Furthermore, it can be
shown that the surface correlation function given above in
terms of second-order spherical harmonics essentially decays
the same way as the correlation function of the normal vec-
tors, g(x)??n(x0)•n(x0?x)?x0. On the basis of fractal scal-
ing relations, the proportionality g(x)?xH?1can be ob-
tained, where H is the roughness exponent of the surface
profile related to the surface fractal dimension by H?3
On these grounds and assuming radial symmetry, we sug-
gest that for fractal surfaces characterized by the fractal di-
mension df, the surface correlation function scales as
1294PRE 60ZAVADA, SU¨DLAND, KIMMICH, AND NONNENMACHER
where s is the curvilinear displacement within the two-
dimensional ?2D? space of the second-order base plane rela-
tive to which the surface roughness is considered. Equation
?19? is valid in the scale-invariance range ?0?s??1, where
?0is of the order of the molecular diameter, and ?1is of the
order of the mean pore size.
For the analysis in the following, we recall that g(s) re-
flects the geometry of the surface, whereas P(s,t) accounts
for the dynamics on the surface. That is, we have to set D
?2 in Eq. ?8?. The same applies to the correlation function
G(t), which is calculated in the base-plane space of the di-
mension D?2. In the isotropic case, G(t) can be expressed
This is the real-space variant. In the reciprocal space, the
correlation function reads
where the orientational structure factor S(k) is introduced as
a counterpart to the surface correlation function g(s). The
two functions are related by the spatial Hankel transform
with J0(ks) the Bessel function of zeroth order. The Hankel
transform is a special case of the radial Fourier transform for
an isotropic space with two dimensions ?see, also, Appen-
Actually, S(k) in the RMTD model is analogous to the
static structure factor used in scattering theories. The only
difference is that the orientational structure factor in context
of NMR reflects orientational rather than material density
correlations ?18?. For fractal surfaces, S(k) is a power law
S(k)?kdf?3, leading to power-law decays of G(t) and I(?).
Equation ?21? stipulates the availability of p(k,t) in the
whole wave number range. However, the orientational struc-
ture factor is a power law only in the scale-invariance range
of the surface. Therefore, the decays of the correlation and
intensity functions calculated below apply in correspond-
ingly limited time and frequency ranges, respectively. This
should be kept in mind when contemplating the examples in
The correlation function G(t) can be calculated for the
RMTD mechanism with the help of Eq. ?20? using the exact
representation of a real-space propagator Eq. ?8?. Alternately,
the calculation can be performed in the k space ?Eq. ?21??
using the asymptotic k-space distribution given by Eq. ?11?.
Thus, in terms of the fractional-time diffusion equation ap-
proach, one finds
provided that the displacements along the surface correspond
to the scale-invariance length scale of the surface.
On the other hand, in context with Le ´vy walks and a
fractal-space diffusion equation, one gets
This suggests that ? has the character of a fractal dimension
as already pointed out by Klafter et al. ?1? for the same
parameter in Eq. ?14?. Comparing G(t) calculated in terms
of the time-fractal and space-fractal FDE, Eqs. ?23? and ?24?,
respectively, indicates indeed that the exponent ? in Eq. ?14?
corresponds to dwin the context of the fractional diffusion
theory. This conclusion thus elucidates the nature of a very
The counterpart of the correlation function G(t) is the
intensity function. In the present context, it reads
according to Eq. ?15?.
In the case of normal two-dimensional diffusion, i.e., dw
???2, the RMTD correlation and intensity functions scale
respectively. Normal 2D diffusion is expected, for example,
in the thin interfacial liquid layer arising between the matrix
and the frozen bulklike adsorbate at temperatures below the
freezing point. In that case, bulk-mediated surface diffusion
is prevented, and we are dealing with ordinary diffusion in a
On the other hand, at temperatures above the freezing
point and under strong-adsorption conditions, the adsorbate
molecules perform random walks along the surface as a con-
sequence of intermittent excursions into the bulklike phase.
Surface diffusion on this basis was shown to be anomalous
within the so-called retention time ?see Refs. ?10,19??. The
character of surface diffusion then turns out to be of the
ballistic type, i.e., dw?1. This sort of random displacement
is also known as Le ´vy walk. The corresponding RMTD
IV. COMPARISON WITH EXPERIMENTAL DATA
The RMTD low-frequency spin-lattice relaxation mecha-
nism links dynamic properties of adsorbate molecules with
the structural details of the adsorbent surface, characterized
by the dynamic fractal parameter dwof the random process
and, for fractal surfaces, by the static fractal dimension dfof
the surface, respectively. According to Eq. ?25?, these param-
eters can be evaluated from the power-law low-frequency T1
dispersion curves, which were observed for polar liquids in
PRE 601295PROPAGATOR REPRESENTATION OF ANOMALOUS . . .
porous glasses ?16?, for instance.
A clear example for a power-law frequency dependence
of the spin-lattice relaxation time T1is shown in Fig. 1. It
refers to the system dimethylsulfoxide ?DMSO? filled into
porous silica glass with a mean pore dimension of 10 nm.
Experimental details can be found in Ref. ?8?. Figure 1 also
shows the proton T1dispersion measured in the adsorbate
diluted by its deuterated form so that any intermolecular di-
polar interactions are reduced. The coincidence of the two
data sets proves that the spin interactions dominating the
low-frequency spin-lattice relaxation in DMSO are of an in-
Two different temperatures have been examined. At 270
K the bulklike adsorbate in the pores is frozen and does not
perceptibly contribute to the spin-lattice relaxation rate. The
observed T1dispersion is rather caused by the nonfreezing
interfacial liquid existing in the form of a one to two mo-
lecular diameter thick nonfreezing surface layer. In such a
situation, one expects that diffusion along the surface is nor-
mal, that is, dw?2.
On the other hand, at 291 K when all adsorbate molecules
are in the liquid state, the bulklike adsorbate phase contrib-
utes, and the ‘‘bulk-mediated surface diffusion’’ mechanism
can occur ?10?. As already outlined above, the consequence
is that in the strong-adsorption limit ?which is pertinent here?
and for surface displacements short relative to diffusion in
the bulk the dynamic parameter dw?1 applies for the propa-
gation of adsorbate molecules along the surface.
At both temperatures a power-law behavior is observed
over three to four decades of the frequency ???0/2? (?0
is the Larmor frequency?. The results are
T?270 K, dw?2
T?291 K, dw?1.
Note that similar T1dispersion slopes have also been ob-
served ?16? with several other polar organic liquids in a po-
rous glass with 30 nm pores. This indicates that the surface
structure acts on all adsorbate liquids the same way.
As another example, Fig. 2 shows malononitrile in the
same nanoporous glass at 275 K ?8 K below the melting
region of the bulklike liquid in the pores? and at 291 K ?18 K
above the freezing point?, respectively. The power-law fre-
quency dispersions of the spin-lattice relaxation time are
T?275 K, dw?2
T?291 K, dw?1.
Interpreting these power laws according to Eq. ?25? and the
propagator Eq. ?8? suggests a common orientational structure
factor of this particular porous glass independent of the ad-
sorbate species. The result is
The surface fractal dimension to be inferred from this is
FIG. 1. Frequency dependence of the proton spin-lattice relax-
ation time of dimethylsulfoxide ?DMSO? in porous glass B10 above
and below the freezing temperature of the bulklike liquid. Data for
an isotopically diluted sample ?80% DMSO-d6) are also shown.
The relaxation times of the partially frozen sample at 270 K refer to
the slowly decaying component of the NMR signal corresponding
to the nonfreezing surface layers.
FIG. 2. Frequency dependence of the proton spin-lattice relax-
ation time of malononitrile in porous glass B10 above and below
the freezing temperature of the bulklike liquid. The relaxation times
of the partially frozen sample at 275 K refer to the slowly decaying
component of the NMR signal corresponding to the nonfreezing
1296 PRE 60
ZAVADA, SU¨DLAND, KIMMICH, AND NONNENMACHER