Anomalous diﬀusion
December 8, 2000
In this lecture we discuss stochastic models of correlated random walks.
By “correlated” we mean that that if a particle is headed in one direction
then there is nonzero probability that it continues in that same direction forsome time and this probability fades to zero as the time interval increases.
This is, of course, the situation envisaged by Taylor (1921).
The distinction between normal and anomalous diﬀusion made in lecture
1 can be understood by examining the rate at which velocity correlation decrease to zero. Normal diﬀusion occurs if the velocity correlation decrease
rapidly while anomalous diﬀusion results from processes in which particles
move coherently for long times with infrequent changes of direction. Roughly
speaking, this distinction is quantiﬁed by the tail behaviour of the velocity
autocorrelation function. For example, if the correlation function decays exponentially then there is normal diﬀusion, whereas if the correlation function
decays algebraically then there is the possibility of anomalous diﬀusion.
The deﬁnition of anomalous diﬀusion is based only on the behaviour ofthe second moment, x2 . But we usually want to know more about the
distribution of a tracer than simply the second moment. In the case of
normal diﬀusion, detailed information concerning the tracer distribution is
obtained by solving the diﬀusion equationct = Dcxx .

(1)

Can we obtain continuum models, analogous to (1), which provide the samedetailed information for anomalously diﬀusing tracer? The main goal of thislecture is to develop partial diﬀerential equation models which can be used
for this purpose.

11Superdiﬀusion and subdiﬀusion

1.1

Taylor’s formula and long tails

Yet again we recall Taylor’s formula which relates the growth of position
variance to an integral of the Lagrangian velocity autocorrelation function,corr(t), tdx2 
(2)= 2 corr(t ) dt .
dt
0In order to obtain x2  we must integrate (2). Standard manipulations turnthe resulting double integral of corr(t) into a single integral
 t
2
(3)
x  = 2 (t − t )corr(t ) dt .
0

The result (3), which is not in Taylor’s original paper, will prove to be very
useful.
We usually have in mind situations in which corr(t) decreases to zeroas t → ∞ so that the integrals in (2) and (3) converge to nonzero values.
An example is the renovating wave model, with its “triangular” correlation
function, from lecture 2. Later in this lecture I will introduce the telegraphmodel which has an exponentially decaying correlation function, corr(t) =U 2 exp(−2αt). These are both examples in which correlations decrease very
rapidly so that normal diﬀusion occurs. But now consider the possibility
that corr(t) decreases so slowly that the integrals in (3) diverge.
Suppose, for instance, that as t → ∞, corr(t) ∼ t−η with 0 < η < 1.
Even though the diﬀusivity no longer exists, it still follows from (3) thatx2  ∼ t2−η .

(4)In this case there is superdiﬀusion: the variance of the particle displacement
grows faster than linearly with time because 2 − η > 1.
Taylor’s formula also contains the possibility of subdiﬀusion. This case issubtle because, like the example of the sea-surface mentioned in lecture 1, it
requires that the integral deﬁning D is zero. But suppose additionally thatthe remaining integral in (3) diverges. This can happen∼ ct−η with
 ∞ if corr(t)
1 < η < 2. The condition that 1 < η ensures that 0 corr(t ) dt converges
t(to zero). The second inequality, η < 2, ensures that 0 t corr(t ) dt diverges.
2

Using (3), we again ﬁnd the scaling law in (4). However this time, because
2 − η < 1, there is subdiﬀusion.
At ﬁrst glance two possibilities above appear as unlikely exceptions to themore natural cases in which both integrals in (3) converge. However there
are examples in ﬂuid mechanics in which either subdiﬀusion or superdiﬀusion
is observed experimentally or computationally. Thus (4) cannot be dismissed
as an unlikely pathology.

1.2

The Texas experiments

An experiment illustrating anomalous diﬀusion has been conducted in Swinney’s laboratory at University of Texas; see Solomon, Weeks & Swinney(1994) and Weeks, Urbach & Swinney (1996). These investigators study thedispersion of particles in an almost two-dimensional ﬂow in annular tank(see ﬁgure 1). The tank is rotating at about 1 or 2 Hertz and the bottom
is sloped to simulate the β-eﬀect. Because of the rapid rotation the ﬂow isquasi two-dimensional.
The ﬂow is forced by pumping ﬂuid through the tank. If the pumping rateis suﬃciently large then this azimuthal ﬂow is unstable to a vortex-forming
instability. A typical ﬂow pattern in the rotating frame is shown in ﬁgure 2.
Evident also in this ﬁgure is the azimuthal jet which runs all the way aroundthe tank. The vortex pattern can be perturbed experimentally by making
the strength of the pumping depend on azimuth. In this fashion, one candrive an unsteady ﬂow and observe chaotic particle trajectories.
Automated image processing techniques are used to follow nearly neutrally buoyant tracer particles suspended in such ﬂow. Typical particle trajectories are shown in ﬁgure 3. Particles within a vortex remain trapped forvery long time (stick ). Particles in the azimuthal jet experience prolonged
ﬂights around the circumference of the tank. Because the vortex pattern is
not perfectly stationary particles alternate, apparently randomly, between
ﬂying in jets and sticking in vortices.
One can change the pattern of jets and vortices by altering the diametersof the circular barriers which conﬁne the ﬂow. Thus it is possible to create
a ﬂow with two oppositely directed jets separated by a vortex chain. In thiscase the dispersion process is more symmetric than in ﬁgure 4 because theﬂights go in both directions around the tank.
During a ﬂight the angular displacement is proportional to the time3

video camera

r1

r2
h

O

I

I

d

O

d

2d

Figure 1: A sketch of the rotating annulus; the rotation rate is about 1Hz. Flow isforced by pumping water in through the ring of holes marked by I and withdrawing
the same volume through the other ring marked O. As a consequence of the strong
Coriolis forced acting on the radial ﬂow between these concentric rings there is anazimuthal ﬂow around the annulus. The experiment is viewed from above using avideo camera. Figure courtesy of Eric Weeks.

4Figure 2: Streaks formed by 100s trajectories of 12 particles reveal four vortices.

Weeks et al. show that the motion of these coherent vortices is chaotic. That is, avelocity spectrum, obtained by measuring velocity with a hot ﬁlm probe, is broadband. Figure courtesy of Eric Weeks.

5. .. ..... ....... ..... .........
....... . .. .. . ......
.
. ............ ........ ............
.
.
.
.
.
..
.... .... . .. ......... .. .. .. . ...
....... . .............. ..... ....... .
.
.
.
.
.
.. .......... . . .........
..
...
. .
.......
.....
.
.
.
.
..
.
..
..
.
......
...
.
.
.
.
......
.
.
..
.......
.
.
................................. ............... . ... ...........
............
...........
..........................
.... ................... . ..............
...
...
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. ....
........ ...... ...........
......
...... ........ ... ........ ....... . ..... .....
...... ... ... ... ................ ....
...
.. ......... ................... ............................................ ... ........... ....
.
.
.
.
.
..
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
......... . ........
...
.........
... .... ...........
............. ........... .... .......... ......
.
.
.
.
.
.
.
.
.
.
.
.
........... ....
. . .. ..
.... .................. .......... ..
................
.......... ....
.................. .
............
........
....... .......... ....
......
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..... ...
.......... .......... . ............ ........ ...
.. ......................................
........ ........ ..
.
.
.
.
.
... ..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
................... ...... .....
..
..
..
....................
................ ... ..
... ... ........... .......
.............. ..... ...
..
...
... .. .... . ...
. .. .. . . .
... ...... ........... .......
......
...... ............ ........ ....
... . ... .. .. ..
....
.. .... ... ..... .....
...
.. .... ..... .. ... ...
.. .... .. ..... ....
..
.. . .. . .. ..
... ....... ... ... ... .......
..
.. ... ... .. .. ..
.....
... .... ... ... .... ... ... ...
.. .. ... .. .. ..
..
.. .. .. .. .. ..... ...
.... ... .. ... ..
.
...
. .. ..... ... ....
..
..
......
.. ....
...
............. ... ... ....
... .... ...... .... .... ..
.
.
.
.
.
.
.
.
. .
... .................................... ... .. ...
.. .. ............... ... ................
....
. . .
......
... .. ... ......... .... .............................. ......
...... .......................... .... ... ...
.. .. .... .. ... ... ................................ ...
...... .....
.
.
.. ... ..
. .... ..
.. ..
. . . . . ... . .
... ... ....
........................
.. .... ........ ......... ........................................
.
.... ..
... .......... ...... .. ......... ... .. ............
... .... ...
..... .... .......
.. .. ....
.
.
.. .. ..... ..... ... .. ..... ......
.
. ... ...
... .... .....
.. ... ........... .... ... .. ........ ..... ........
.
. .
.
.
.
.
.
.
.
.
.
.. ...... ...... . .........
. ... .
...... ..... ...
...... ......
.
.
... ..... ......... ..... ......
.
.
.. ... ... .. .. .. ..
... ...... ...... ............................................ .......
..... ..... .... ....... ... ..... ... .... ....
.
... ...... ....... ..............
.
.
.
..
.
.
. ..
... ........ .......... ............
.. .
............................... ...... ... ..... .... .... ...
..........
....................
....
.... ......
... . .. . ........ .......................
......... .. ....
...........
....................
.......
...................
.
..
.
.
.
.
.
.
.
.
.
..
..
.
.
... ....... ...........
......... ...... ........................
..
.
.
.
.
.
.
.
.
.
.
................ . . .... . .. .. . . ..
....
.....
.. ....... ...............
..... .... . .. . . . .. .
... ............... .. .................... ......... ....................................... .......
.... ..... ....... .............
........... ............... ... ... ... .... ... ... ....
... .. .... ............. ............. ................. ....... .......... ...... .. . ...........................
.
.
.
.
.
.. . . .
.
.
.
.
.
.. ........ .................. ............
.
......................... ... .................. ...................... .........
.. .... ...
...... .. ....... .. ..... ......... ..........
.. .. ................ ... ......... ........ .
.... ... .. .. ........ .......... ...... ......... ...... .... ..... . . ........
.
.
...... .... ............. ..... .... ... ...... ........ ....
....... ........... ... .......... ...
... ..... ........... ... ............ ...... .......... ... .............................................................................. ........................
....... .... ................ .. ........... .............................. ........................... . ......... ... . .
. .. . ..... .................. ...................................................................................... .. ... ............... .
.. ........ ................ ................................... ..... ....... . .. .. .. ......... .
. .... .... ................ . ...... .. . .. ... . .. . . ... .. ...... .
.......... . .................... ... . .. .. .... . ........ ...
......... . ..
. ......... .. . ... . . .......... ... . ..... .

Figure 3: Trajectories of three tracer particles in the ﬂow shown in ﬁgure 2. The
beginning of each trajectory is indicated by a triangle and the end by with a circle.
In (b) the particle spends most of its life trapped in a single vortex. However, this
vortex wobbles erratically because the ﬂow is chaotic. In parts (a) and (c) the
particles experience several episodes of trapping within a vortex and ﬂight aroundthe tank in the jet. Figure courtesy of Eric Weeks.

50

(a)
40

(b)

20

(c)

(rad)

30

10

0
0

300

600

900

t (s)

Figure 4: Angular displacement, θ(t) for the trajectories in ﬁgure 3. There is an
obvious distinction between the ﬂights and the sticking events. The small oscillations during the sticking events correspond to particle motion within a vortex.
Figure courtesy of Eric Weeks.

6elapsed since the ﬂight began:∆θ ≈ U t.

(5)

The displacement, ∆θ, is essentially zero during a sticking event (see ﬁgure
4).
The experiments show that the dispersion of an ensemble of particles issuperdiﬀusive. That is(θ − θ)2  ∼ tγ

(6)where γ > 1; typical values are γ ≈ 1.4 to 1.7 depending on the experimental conﬁguration. (It is also possible to observe normal diﬀusion, γ = 1,
by strongly forcing the ﬂow and breaking the azimuthal symmetry of theforcing.)
To characterize the motion Solomon et al. used sticking and ﬂying PDFs:
PF (a)da = Probability that a ﬂight has a duration ∈ (a, a + da) .

(7)

Later in this lecture we will refer to a as the “lifetime” of a particle in the
ﬂying or sticking state. We ﬁguratively speak of a particle being born into theﬂying state and moving coherently for a lifetime a so that the total angulardisplacement during the ﬂight is ∆θ ∞= U a.
The PDF PF is normalized by 0 PF (a)da = 1 and ∞τF =aPF (a) da = average duration of a ﬂight .
(8)0

The PDF of sticking times, PS (a), and the average sticking time, τS , aredeﬁned analogously.
Experiments show that as a → ∞, PF and PS have algebraically decaying
tails:PF (a) ∼ a−µF ,PS (a) ∼ a−µS ,

(9)

with
2 < (µF , µS ) < 3.

(10)

Because of this slow algebraic decay the variance of the lifetimes, deﬁned by ∞2
a2 PF,S (a) da ,(11)a F,S ≡07

diverges.
The divergence of a2 F is signiﬁcant because invoking Einstein’s formulafor the diﬀusivityD=(∆θ)2 ,τ(12)

and using ∆θ = U a, we conclude that D ∝ (∆θ)2  = U 2 a2 F = ∞. The
divergence of D is symptomatic of superdiﬀusion.
Notice that the denominator τ in (12) is related to the average ﬂying andsticking times, τF and τS , which are both ﬁnite. Thus, in the Texas experiments, we can say that anomalous diﬀusion occurs because the numerator of(12) is divergent. In other cases it is the denominator which causes trouble.
The Texas experiments show that anomalous diﬀusion occurs in realistic
and geophysically relevant systems. Several theoretical questions suggest
themselves. How do the algebraic tails of PS and PF arise, and can we makea microscopic models which exhibits this phenomenon? Can we relate theexponents γ, µF and µS ? (From section 4, the answer to the last question isγ = 4 − µF .)2

The telegraph model

The key issue raised by anomalous diﬀusion is decay of velocity correlations.
Thus our goal is to formulate models for which we can explicitly calculate
velocity statistics and understand the decay of correlations. Our ﬁrst attempt is not very ambitious: we begin with the telegraph model, which is thesimplest example of a continuous-time correlated random walk.

2.1

The Lagrangian formulation of the telegraph model

In a telegraph process the velocity of particle n, denoted by un (t), can have
only one of two possible values, +U and −U . The velocity of each particle,
un (t), ﬂips randomly back and forth between ±U with a transition probability
α per time. This means that in a time dt a fraction αdt of the ensemble
switches velocity. Because the transition rate, α, is constant we can say thata particle has no “memory” of when it ﬁrst arrived in its present state. Thus
this telegraph model is Markovian.

8

We refer to the prescription for constructing a telegraph process as model
A. There is a variant, model B, discussed below.
With the prescription above, the velocity of a particle is a discontinuousfunction of time as shown in ﬁgure 5. The correlation function and thediﬀusivity are ∞U2
2 −2α|t|corr(t) = U e,D=corr(t) dt =,(model A) . (13)2α0Notice that the corrtt is inﬁnite at t = 0; this is because the acceleration is
inﬁnite at the discontinuities in ﬁgure 5.
To obtain (13), return to the deﬁnition of the correlation function
1 corr(t) =un (0)un (t),
N n=1
N

(14)

where N is the total number of particles in the ensemble. Suppose that at t
the sum on the right hand side has P (t) positive terms, all equal to U 2 , andN − P (t) negative terms, all equal to −U 2 . Thuscorr(t) =U2[2P (t) − N ] .
N(15)In a time dt, P αdt of the positive terms become negative and (N − P )αdt
of the negative terms become positive. Thus, at t + dt,
P (t + dt) = P (t)(1 − 2αdt) + N αdt ,(16)and the analog of (15) is:corr(t + dt) =U2[2P (t)(1 − 2αdt) + 2N αdt − N ] .
N(17)Taking the limit dt → 0 in (17) gives corrt = −2αcorr; the solution of thisdiﬀerential equation is (13).
An alternative telegraph process (model B) is constructed by imaginingthat at random instants each particle ﬂips at coin. The ﬂipping rate is α sothat in a time dt, there are N αdt coin ﬂips. After each ﬂip, the velocity is+U if there is a head and −U if a tail. With this prescription, a particle willchange direction on average once out of every two tosses. On the other tosses9C(t)=U2exp[-2 α|t|]

A random telegraph process
1.2

1

1

C(t)/U2

u(t)/U

0.5
0

0.8
0.6
0.4

-0.5

0.2

-1
0

0.2

0.4

0.6

0.8

0
-1

1

t

-0.5

0

αt

0.5

1

Figure 5: An example of a telegraph time series, and the telegraph correlation
function.

the particle continues in the same direction and the result is as if nothing
happened. Thus with model B we simply replace α by α/2 in our earlier
calculations and consequently the correlation function and diﬀusivity are ∞U2
2 −α|t|corr(t) = U e,D=corr(t) dt =,(model B).
(18)α0
The diﬀerence between model A and model B is trivial. However the distinction between the two cases will plague us later.
If we are searching for a model of anomalous diﬀusion then the telegraphmodel is a disappointment: the exponentially decaying correlation function
ensures that D is ﬁnite and that the displacement variance ultimately grows
diﬀusively. We continue our investigation of the telegraph model in order tobetter understand “ultimately” and because in section 4 the telegraph modelis used as the foundation of more elaborate models which do show anomalous
diﬀusion.

2.2

The Eulerian formulation of the telegraph modelNow we ignore the Lagrangian information contained in the correlation function (14) and instead we give an Eulerian formulation of the telegraph process. Let R(x, t) denote the density (particles/length) of particles moving
to the right with velocity +U and L(x, t) denote the density of left-moving
10

particles with velocity −U . The coupled conservations laws are
Rt + U Rx = α(L − R),

Lt − U Lx = α(R − L).

(19)

These equations should be self-evident...
We can put (19) into a revealing alternative form by deﬁning the total
concentration, C(x, t), and the ﬂux, F (x, t), asC ≡ R + L,

F ≡ U (R − L) .

(20)

In terms of these new variables the model is
Ct + Fx = 0 ,Ft + 2αF = −U 2 Cx .

(21)

The ﬁrst equation is conservation of particles and the second equation is theﬂux-gradient relation.
Notice that in (21) Fick’s law does not apply — the ﬂux F is not instantaneously related to the gradient Cx . Equation (21b), which might becalled Cattaneo’s law (see the 1989 review by Joseph and Preziosi), can besolved as a ﬁrst-order diﬀerential equation for F (x, t). Thus, the ﬂux at x is
expressed as weighted integral over the past history of the gradient at x: t2e−2α(t−t ) Cx (x, t ) dt .
(22)
F (x, t) = −U−∞

The ﬂux has a “fading memory” of the gradient and the exponential in (22)is the fading factor which strongly weights the most recent values of thegradient.
Next, if we eliminate F from (21), we obtain
Ctt + 2αCt − U 2 Cxx = 0 .

(23)This is the telegraph equation; the diﬀusion equation is obtained only as an
approximation which applies to the low frequency and wavenumber components of C(x, t). On these large and slowly evolving scales one can neglect
the term Ctt in (23) and so obtain the approximation
Ct ≈ DCxx ,

D=

U2
.
2α

The diﬀusivity D in (24) was anticipated in (13).
11

(24)

2

0.07

1
0
-40 -20

0

20

40

0.35
1

0.8
0.6
0.4
0.2
0
-40 -20

0

20

40

20

1

0.5

0.5

0.5

0
-40 -20

0

20

40

0.42

1

0
-40 -20

0.8
0.6
0.4
0.2
0
40 -40 -20

0.91

0.6

1.5

1

0
-40 -20

0

1

20

40

0.49

0

20

40

20

40

0.98

0.6

0

0.8

0.7

0

0
-40 -200-40 -20

20

40

0.77

0
-40 -20
0.6

0.4

0.4

0.2

0.2
0

20

40

1.05

0.6

0
-40 -20

0.4

0.4

0.2

0.2

0.2

0.2

0.6

20

40

0
-40 -20
0.6

0

1.19

20

40

0
-40 -20

0

1.26

20

40

0
-40 -20

0.4

0.4

0.2

0.2

0.2

0.2

20

40

1.47

0
-40 -20

0.4

0.4

0.2

0.2

0
-40 -20

0

20

40

1.75

0.4

20

40

1.54

0

20

40

0

20

40

1.82

0.4

0
-40 -200-40 -20

0

20

40

1.61

0.4

0
-40 -20

0

20

40

1.89

0.4

20

40

0

0
-40 -20

20

40

1.12

0

20

40

0
-40 -20

0

20

40

1.68

0.4

0
-40 -20

0

20

40

1.96

0.4
0.2

0.2

0

40

0.2

0.2

0.2

0.2
0
-40 -200-40 -20

0

20

1.4

0.4

0

40

0.84

1.33

0.4

0
-40 -20

0

0.6

0.4

0

20

0.56

0.8

0.4

0
-40 -20

0

1

0.6

0
-40 -20

0.28

0.5

0.5

0.63

0

0.21

1.5

1

0.5

0.5
0
-40 -20

0.14

1.5

0

20

40

0
-40 -200

20

40

Figure 6: Solution of the telegraph equation. αt is in the top corner of the panel.
At t = 0, R = L = exp(−x2 /50). The solid curve is C = R + L, and R and L are
shown as dotted and dashed curves.

12Figure 6 shows a numerical solution of (23) starting with an initial condition of the form
R(x, 0) = L(x, 0) = e−µ x .
2 2

(25)

At small times the density C develops a double peaked structure as the left
and right going populations separate. This behaviour is transient, and at
longer times the central part of the concentration relaxes to the well-known
Gaussian solution of the diﬀusion equation.
According to (23) the disturbance travels at a ﬁnite speed: these are the“heat waves” discussed by Joseph and Preziosi (1989), and also evident inﬁgure 6. The approximate diﬀusion equation (24) makes the unrealistic prediction that disturbances are propagated at inﬁnite speed. This unphysicalconsequence of the diﬀusion equation motivated Cattaneo to propose (21b)
as an alternative to Fick’s law.
These considerations shows that one cannot blithely assert the validity ofthe diﬀusion equation (24) as an exact description of dispersion. The diﬀusion
equation applies only as an approximate description of low frequencies andlong wavelengths.

2.3

Discretization of the telegraph modelThis section is a digression. Read on if you want to learn how to solve thetelegraph equation using a simple numerical scheme. (This is how I drewﬁgure 6.)
We reformulate the telegraph model in terms of discrete variables: divide
the x-axis is divided into segments of length δx separated by “scatteringsites” at xn = nδx. Time is also discretized in units of δt so that t = T δtwhere T is an integer T = 0, 1, 2 · · · The walkers move along the x–axis witha velocity that is either +δx/δt or −δx/δt. In terms of the continuous modelin (19)
U=

δx
.
δt

(26)

When a walker reaches the scattering site at xn = nδx he is “backwards scattered” or reﬂected with probability b and “forward scattered” or transmittedwith probability 1 − b. Because the probability of a change in direction, b, isthe same for left as for right moving walkers there is no mean velocity along
the line.
13

R

L
n

bR + (1-b)L(1-b)R + bL

Figure 7: A steady state with constant ﬂux, f = U (1 − b)(R − L), passing throughsite n.

Let Rn (T )δx be the number of right walkers in the segment n, nδx < x <(n + 1)δx. The number of left walkers, Ln (T )δx, is deﬁned analogously. Rn
and Ln are the discrete analogs of the continuous densities used in (19).
With these rules and deﬁnitions, the discrete evolution equations for theensemble areRn (T ) = (1 − b)Rn−1 (T − 1) + bLn (T − 1) ,Ln (T ) = (1 − b)Ln+1 (T − 1) + bRn (T − 1) .

(27)
(28)

For instance, in the ﬁrst equation above, the number of right movers insegment n is equal to the number in segment n − 1 at the previous time thatsuccessfully passed through scattering site n, plus the number of left movers
previously in segment n that were reﬂected at this same site. Figure 6 shows
the result of iterating the discrete system above.
One exact solution of the diﬀerence equations above is
Rn = Ln = Ln+1 = Rn+1 = · · ·(29)This solution is steady: Rn (T ) = Rn (T − 1). In fact, (29) is the discrete
analog of the equilibrium solution of the diﬀusion equation. The distributionof walkers is spatially uniform with equal numbers going left and right ineach interval and there are no concentration gradients. An individual walker
is moving to and fro, but the ensemble is in steady state.
Next, we consider the constant-ﬂux solution. In ﬁgure 7, R right walkers
impinge on site n from the left and L left walkers impinge on n from theright. In steady state it must be that on the left of n there are bR + (1 − b)Lleft walkers moving away, while to the right there are bL + (1 − b)R right14

walkers moving away. Thus the ﬂux to the right of the site is
f = U [(1 − b) R + bL] − U L =δx(1 − b)(R − L) ,
δt

(30)

where we have used U = δx/δt. Because there is a steady state, calculating
the ﬂux to the left of the site gives exactly the same result and so there is a
nondivergent ﬂux of walkers along the line.
Next, we can calculate the concentration diﬀerence across the site inﬁgure 7. To the right of the site the total density of walkers is
c+ = L + bL + (1 − b)R ,(31)

while on the left the density isc− = R + bR + (1 − b)L.

(32)

Combining (31) and (32) we have for the concentration jump across the siteδc = c+ − c− = 2b(L − R)

(33)

Thus, using (30), the ﬂux-gradient relation in steady state is
f = −D

δc
,
δx

D=

(1 − b) (δx)2
.
2b
δt

(34)

Does it seem obvious to you that the diﬀusivity should diverge as b → 0? If
you think of the diﬀusivity as the area under the correlation functions thenthis divergence should be intuitive. It is an instructive exercise to obtain D
in (34b) using Taylor’s formula. (Hint: consider N  1 right walkers which
initially set out together. At t = T δt, after T encounters with scatteringsites, how many of these walkers have changed direction an even number oftimes, and how many odd?)
Comparing the equation above with our earlier expression for the diﬀusivity, D in (13) and (24), we conclude that
αδt =

b
.
1−b

(35)

Thus, with (26) and (35), we can express the parameters of the discretemodel, (δx, δt, b), in terms of the parameters characterizing the continuous
model, U and α.
15

3Age-stratiﬁed populations

The telegraph model from section 2 is Markovian. This means that eachparticle has a constant probability per unit time, α, of switching direction.
Thus, no matter how long a particle has been moving to the right (say),its probability of switching direction in the next dt is always αdt. Consequently an exponentially decreasing number of particles move coherently forlong intervals and the telegraph model in (19) does not exhibit anomalous
diﬀusion.
A satisfactory description of anomalous diﬀusion demands a non-Markovian
model in which particles have some memory of their past motion. To obtainsuperdiﬀusion it is necessary that a right-moving particle is less and lesslikely to change direction as it spends more and more time moving right1 .
Such memory eﬀects are implicit in the models discussed by Weeks et al(1996), and in several of the articles in the conference proceedings edited by
Schlesinger, Zaslavsky & Frisch (1994). The stochastic models discussed inSchlesinger et al. draw heavily on statistical physics. In this lecture we aregoing to develop the theory from scratch using a formalism which is accessible
to people whose background is in ﬂuid mechanics. The climb begins with an
excursion into the theory of age-stratiﬁed populations.
Consider a population of items with a ﬁnite lifetimes and a death rate
which depends on age, a. For example, light bulbs in a large building, or the
population of the United States. At time t the age structure of the populationis characterized by a density function for which f (a, t)da is the number of
items whose age is between a and a + da. In terms of f , the total number of
items in the population, N (t), and the average age, ā(t), are given by
 ∞
 ∞
−1
N (t) =
f (a, t) da,ā(t) ≡ N
af (a, t) da.
(36)
0

0

The density function evolves according toft + fa + αf = 0,
1

(37)

A popular metaphor for the Markovian case is radioactive decay: a molecule has aconstant probability per unit time of decaying. As a metaphor for the non-Markovian
case, imagine entering an enormous maze and then trying to ﬁnd your way back to theentrance. The longer one has wandered, the less the chance of stumbling on the exit in
the next dt.

16where α(a) is the death-rate. The term fa in (37) says that the populationtranslates along the age-axis at a rate one year every year. To completely
specify the problem we must supply an initial condition, and also a boundary condition at a = 0. The boundary condition at a = 0 has an obvious
interpretation:f (0, t) = the birth (or replacement) rate.

(38)

In the case of a population of people, the boundary condition above is a ﬂux
of babies into the system.
The Markovian limit is the special case in which α is independent of a.
This model of α is unrealistic for both light-bulbs and people, though it might
apply to a population of radioactive molecules. The Markovian case is very
simple because one can integrate (37) over a and obtain a closed equation
for N (t):Nt + αN = f (0, t) .

(39)

Thus if α is constant and we need only the total number of functional items
at t then we do not need to solve partial diﬀerential equations and deal withthe age structure of the population.

3.1

The steady-state solution

As a ﬁrst illustrative example, suppose that the replacement rate is adjusted
so that N is constant. (Janitors replace light bulbs as soon as they burn-out.)
In this case the equilibrium solution of (37) isf (a) = N τ −1 Ψ(a),
where

  a



Ψ(a) ≡ exp − α(a ) da ,
0

(40)
 ∞
Ψ(a) da.
τ≡

(41)

0

The function Ψ, and its integral τ , will occur frequently in the sequel. Noticethat the replacement rate is f (0) = N/τ and this suggests that τ should be
the average lifetime of an item. On the other hand, τ will not usually beequal to ā in (36). I suggest brooding on this “paradox” and, as an exercise,see if you can resolve this confusion to your satisfaction by the end of thissection.
17

In (41) we assume that the death rate α(a) is such that as a → ∞,Ψ(a) → 0 fast enough to ensure that τ is ﬁnite. For instance, if α is constant(this is the Markovian case) then Ψ(a) = exp(−αa) and τ = ā = 1/α.
If the death rate α decreases with age then the average liftetime τ might
not be ﬁnite. For example, consider the speciﬁc model

ν
ν
θ
α=
,
⇒
Ψ(a) =
.
(42)
θ+a
θ+a
Provided that ν > 1 then the integral of Ψ(a) converges and τ = θ/(ν − 1).
If ν < 1 then τ = ∞ and there is no steady solution. To understand thiscurious result we must solve an initial value problem (see appendix 5). Here
we just remark that if ν < 1 then the average lifetime of a bulb is inﬁnite.
Detailed solution of the initial value problem in appendix 5 shows that in thiscase the replacement rate is f (0, t) ∝ tν−1 . That is, the total number of new
bulbs which have been installed at time t grows like tν  t. The hypotheticalmanufacturer of lightbulbs with ν < 1 is threatened with bankruptcy: sales
decrease with time, even though every bulb eventually fails.

3.2

A cohort of babies

Imagine a cohort of babies leaving the maternity ward together, or a box of
new lightbulbs shipped fresh from the factory. These items will function forvarying amounts of time, and so we can speak of the PDF of lifetimes. We
denote this PDF by P(a), and our goal is to relate P(a) to the death rateα(a).
Consider a group of N items which all start with a = 0 at t = 0. Whatfraction of this cohort survives at t > 0? The surviving fraction is also thefraction of lifetimes longer than t and so ∞surviving fraction at t ≡ Ψ(t) =P(a) da.
(43)t

To calculate the surviving fraction, we solve (37) with the initial and boundary conditions
f (a, 0) = N δ(a),f (0, t) = 0.

(44)

The solution of (37) and (44) isf (a, t) = N Ψ(t)δ(a − t) ,
18

(45)

where Ψ is deﬁned in (41). Thus Ψ(t) is the fraction of the cohort which isstill alive at time t; we refer to Ψ as the survival function.
It now follows from (43) that the PDF of lifetimes of new items isP(a) = −Ψa = αΨ.

(46)

The average lifetime, τ , is given by the equivalent expressions:
 ∞
 ∞
 ∞
aP(a) da = −aΨa da =Ψ(a) da.
τ=
0

0

(47)

0

Thus, as was suggested in the discussion following (41), to keep a populationin equilibrium the replacement rate is equal to the size of the population, N ,divided by the average lifetime of new items, τ .

3.3

Extinction of a population

As a ﬁnal example, suppose that at t = 0 we have the steady-state lightbulb
population in (40). If the janitors then go on strike, so that bulbs burn out
without replacement, then how many bulbs are still operating at t > 0? In
this example we must solve (37) with the initial and boundary conditionsthatf (a, 0) = N τ −1 Ψ(a),f (0, t) = 0 .

(48)

The solution isf (a, t) = H(a − t)N τ −1 Ψ(a) ,(49)

where H(a − t) is the step function. Thus the fraction of surviving bulbs at
t is
 ∞
 ∞
−1
−1
Θ(t) = τΨ(a) da = τ(a − t)P(a) da.
(50)
t

t

Using the speciﬁc model of α in (42), the surviving fraction isΘ(t) = (1 + θ−1 t)1−ν .

(51)

Θ(t) is the most slowly decaying function we have seen so far: as t → ∞,Θ(t)  Ψ(t)  P(t). This model may be relevant to the very slow extinctionof professors once the supply of graduate students is cut-oﬀ.
19Comparing the results in sections 3.1 and 3.2, we see that the steadystate population in section 3.1 contains more long-lived items than are ina cohort of new items section 3.2. This means that the average lifetime of
the light bulbs currently operating in the Empire State building is longer
than the average lifetime of bulbs shipped from the factory. The reason is
obvious: items with brief lifetimes fail quickly, and will likely be replacedwith items whose lifetime is closer to the mean. Thus, fragile individuals areunderrepresented in an operational population.

4

The generalized telegraph model

4.1

Formulation

Using the machinery from the previous section we now construct a generalization of the telegraph model which exhibits anomalous diﬀusion. In thisgeneralization particles switch randomly between moving with u(t) = +U ,u(t) = 0 and u(t) = −U . The transition probabilities between these states
are functions of the time since the last transition. In other words, eachparticle carries an “age”, a, which is the time elapsed since the particle
transitioned into its present state. We denote the density of right moving
particles at (x, t), with age a, by R(a, x, t). For left-moving particles thedensity is L(a, t, x), and for the stationary particles the density is S(a, x, t).
We refer to the left and right–movers collectively as “ﬂying particles” while
the stationary particles are “stickers”.
The ﬂying particles satisfy the conservational laws
Rt + Ra + U Rx + αF R = 0,

Lt + La − U Lx + αF L = 0 ,(52)while the sticking particles have
St + Sa + αS S = 0 .

(53)

The death rates of ﬂying and sticking particles, αF and αS respectively, arefunctions of age a; it is through this device that particles have a memory of
their previous history. The price we pay for this nonMarkovian memory isthat there are now three independent variables, (a, t, x).
Stationary particles are born when left and right-moving particles die.
And, conversely, when a stationary particle dies it is reborn as either a left20

moving particle or a right moving particle with equal probability. Noticethat in order for a right-moving particle to become a left-moving particle itmust pass through the intermediate state with u = 0. These karmic rules are
enforced by boundary conditions at a = 0:
1 ∞L(0, t, x) = R(0, t, x) =αS (a)S(a, t, x) da ,
(54)
2 0
and

 ∞
αF (a) [L(a, t, x) + R(a, t, x)] da .
S(0, t, x) =

(55)

0

Trajectories of particles moving with this generalized telegraph process areshown in ﬁgure 8.
The model we have formulated here is a generalization of the telegraphmodel in two ways. First, there are three states: left, right and stationary.
This minor embellishment is motivated by the Texas experiments in whichtrapping in a vortex corresponds to the stationary particles. The nontrivial
generalization is the introduction of the age variable used to capture memory
eﬀects. As an exercise, the student can show that if αF and αS are independent of a then one can easily integrate over a and reduce (52) through (55)to a three-state telegraph model. (This exercise shows how the boundary
condition at a = 0 works.) As a sequel to this exercise, discuss αS → ∞ andshow that in this limit one obtains eﬀectively a two-state telegraph model.
Are you surprised that the diﬀusivity is given by (18)? That is, why do werecover model B, rather than model A, when the sojourns in the intermediate
state u = 0 are very brief?
In order to model slowly fading velocity correlations and anomalous diffusion we useνFνSαF (a) =,αS (a) =.
(56)
θF + aθS + aWith the form above, the transition rates decrease as particles age. Numerical simulations of the three-state model using the transition rates in (56)show that many particles move in the same direction for a long time (seeﬁgure 9).
The main point of (56) is that if a  1 then the transition rates αFand αS are proportional to a−1 . This inverse dependence on age ensuresthat the ﬂying and sticking PDFs, PF and PS in (7), decay algebraically.
21

5

4

t

3

2

1

0

-5

-4

-3

-2

-1

0

1

2

3

4

5

x
Figure 8: Trajectories of particles in the generalized telegraph random process. All particles are released from x = 0.
Thus (56) incorporates important experimental information into the model2 .
One can make a dimensional argument in support of (56): αF and αS havethe dimensions of inverse time. If the only time-scale relevant for long-lived
particles is the particle age, a, then it follows that αF and αS are inversely
proportional to a. We now show that the parameters νF and νS are easily
related to the experimentally measured exponents µF and µS in (9).
As far as scaling exponents are concerned, only the a  1 structure of αF and αSmatter. We use the speciﬁc functional form in (56) for simplicity.
2

22

3

5

4

(a) t=10

800

8

600

6

(b) t=10

x 10

Age

10

Age

1000

400

4

200

2

0
−1000

−500

0
Position

500

0
−1

1000

−0.5

(c) t=103

2.5

2.5
log (Number)3

log10(Number)

0.5

1
5

x 10

(d) t=105

3

2

2
1.5

10

1.5
1

0.5

1
0.5

0
−1000

−500

0
Position

500

0
−4

1000

−2

(e) t=1033

3
log (Number)

3.5

2.5

2

4
4

x 10

2.5
2

10

2

1.5

1.5

1

1

0.5

0.5

0

0
Position(f) t=105

3.5log10(Number)

0
Position

0

200

400

600
Age

800

0

1000

0

200

400

600
Age

800

1000

Figure 9: A simulation with N = 104 particles; αF (a) = αS (a) ∼ 1.35/a. Upper
panels: PDFs as a function of age and position show that there are many particles
that either stick or move at a constant velocity for nearly the whole simualtion.
Center panels: PDFs of the position of particles develop tails larger than Gaussians
as time goes on. Lower panels: PDFs of the age of particles have a spike at large
times, because there is a fraction of particles that never die.

23

4.2

The equilibrium solution

The system (52) through (55) has a solution which is homogeneous (∂x = 0)
and steady (∂t = 0). This equilibrium solution isR(a, x, t) = L(a, t, x) = rΨF (a),S(a, t, x) = 2rΨS (a)(57)

where ΨF,S (a) is
  a



ΨF,S (a) ≡ exp − αF,S (a ) da .

(58)

0

The constant r in (57) is the transition rate between the diﬀerent states; r
is determined by the normalization condition: ∞N = 2r (τS + τF ) ,τF,S ≡
ΨF,S (a) da .
(59)
0

We can use the results from section 3 to interpret τF and τS as the average
lifetimes in the ﬂying and sticking states respectively. Using (46), the PDF
of lifetimes in those states is given by
PF,S = αF,S ΨF,S .

(60)

Using the expression in (56) for αF,S , we see that as a → ∞, the survival
functions decay algebraically with ΨF,S ∼ a−νF,S , and so PF,S ∼ a−νF,S −1 . It
follows that the exponents µF and µS deﬁned in (9) are related to νF and νS
byµF,S = νF,S + 1 .

(61)

We can summarize our arguments to this point by observing that the experiments provide the ﬂying velocity, U , the average lifetime in the ﬂying andsticking states, τF,S , and the exponents µF,S . These ﬁve experimental data
determine the ﬁve parameters in the generalized telegraph model, namely(U, νF,S , θF,S ).

4.3

Formulation of the initial value problem

Now that we have determined the model parameters using experimental constraints it is time to do some mathematics and use the model to predict the24

exponent γ in (6). The simplest intial value problem we can consider is (52)through (55) with[R(a, 0, x), S(a, 0, x), L(a, 0, x)] = r [ΨF (a), 2ΨS (a), ΨF (a)] δ(x).

(62)

The constant r is given in (57). Thus, the initial population has an equilibrium distribution of ages and is released at x = 0. Because of the symmetry
between left and right moving particles
R(a, t, x) = L(a, t, −x),S(a, t, x) = S(a, t, −x).

(63)

Equation (63) greatly simpliﬁes subsequent algebra.
One technical point (which I confess confuses me) is using the equilibrium
age distribution as the initial condition in (62). This choice leads to simple
calculations below. And perhaps the gross details of the dispersion process,such the exponent γ, are independent of the initial distribution of ages? As
an excercise I suggest solving the initial value problem using other initialconditions e.g., R(a, 0, x) = δ(a)δ(t) etcetera. Are there any signiﬁcant differences in the t → ∞ structure of the solution?
Our strategy will be to obtain a closed hierarchy of spatial moments bymultiplying the conservation laws (52) and (53) by xn and integrating overx. It is possible to solve the ﬁrst few members of the hierarchy and showthat if αF,S and has the form in (56) with 1 < ν < 2 then as t → ∞ ∞ ∞
2x  =
x2 [R(a, t, x) + S(a, t, x) + L(a, t, x)] dxda ∝ t3−νF . (64)0−∞Before entering this calculation, we give a simple argument which suggests
how the anomalous exponent 3 − νF > 1 arises in (64).
The variance x2  in (64) can alternatively be written as1  2x  =x .
N n=1 nN

2

(65)

At time t > 0 some of the N particles will have moved coherently withunchanging velocity (either +U or −U ) ever since t = 0; half of these particles
will be at x = U t and the other half at x = −U t. These “coherent particles”
each contribute a term U 2 t2 to the sum on the right hand side of (65). The
number of coherent ﬂying particles is just Θ(t)N where Θ(t) is given by25

(51) with ν replaced by νF . Thus, because every term in the sum in (65) is
positive, one hasx2  > Θ(t)U 2 t2 ∼ U 2 θFν−1 t3−νF(66)

The inequality (66) has teeth only if 3 − νF > 1: then we learn that thecoherent particles alone produce a superdiﬀusive contribution to the variance.
The argument above may suggest to you that superdiﬀusion is due solely
to the few extreme particles which move without changes in direction. Thisis an overstatement: the lower bound in (66) is generously less than theexact result for x2  which we obtain in the next section. Thus “nearlycoherent” particles, meaning particles which change direction only once or
twice, also make a large contribution to the sum in (65). This is an essential
point, because in their analysis of the experiments Solomon et al. discarded
all coherent particles from the data set3 . Thus the exponent measured by
Solomon et al reﬂects only the contribution of nearly coherent particles.

4.4

Solution of the initial value problem

This is a dry section which contains the details of the analytic calculationof x2 . The main point of interest here is that a lot of the algebra can be
avoided by proving (75) below. (I suggest this as an exercise.)
The spatial moments are deﬁned by ∞[Rn (a, t) , Sn (a, t), Ln (a, t)] ≡xn [R(a, t, x), S(a, t, x)L(a, t, x)] dx,−∞(67)
Because of the symmetry in (63)Rn (a, t) = (−1)n Ln (a, t),

Sn (a, t) = 0 if n is odd.

(68)

The result above allows us to work exclusively with Rn and Sn while retaining
full information about the distribution. Using the symmetry, the variancecan be written as ∞2
x  =
2R2 + S2 da .
(69)
0
3

This drastic procedure is necessary because some fraction of the experime ntal particles
are in integrable regions and will ﬂy forever. Retaining all these particles will ultimately
lead to the trival ballistic exponent γ = 2.

26

The zeroth moment of (52) through (55), with the initial condition in(62) is[R0 (a, t), S0 (a, t), L0 (a, t)] = r [ΨF (a), 2ΨS (a), ΨF (a)] .

(70)

That is, the zeroth moment is just the equilibrium solution. (This is why using the equilibrium age distribution as the initial condition is so convenient.)
Using (68), the ﬁrst spatial moment is S1 = 0, L1 (a, t) = −R1 (a, t) andR1t + R1a + αF R1 = U rΨF ,R1 (0, t) = 0,

R1 (a, 0) = 0.

(71)

The solution of the initial value problem (71) is
R1 (a, t) = U rΨF (a) min(a, t).

(72)

The second moment equations are L2 = R2 andR2t + R2a + αF R2 = 2U R1 ,S2t + S2a + αS S2 = 0,(73)

with the a = 0 boundary condition that ∞ ∞αS (a)S2 (a, t) da,
S2 (0, t) = 2
αF (a)R2 (a, t) da. (74)
2R2 (0, t) =00

To obtain the variance in (69) we do not need the complete solution of (73)and (74). Instead, after some judicious integration over a, one ﬁnds that ∞d 2
R1 (a, t) da.
(75)
x  = 4U
dt
0
Substituting (72) into the result above we obtain
 t

 ∞
d 2
2
x  = 4U raΨF (a) da + t
ΨF (a) da .
dt
0
t

(76)

If the right hand side of (76) approaches a constant as t → ∞ then thevariance grows diﬀusively. Otherwise there is anomalous diﬀusion.
With (76) in hand, one can easily determine if particular models of αFand ΨF lead to anomalous diﬀusion. For example, with the model in (56),evaluating the integrals in (42) gives a pleasant exact solution(1 + t̃)2−νF
d 2
1
2 2
x  = 4U rθF
+
,
(77)
dt
(2 − νF )(νF − 1) (νF − 1)(νF − 2)27

where t̃ ≡ t/θ.
The asymptotic behaviour at large time depends crucially on νF . If νF > 2
then there is normal diﬀusion:d 2
4U 2 rθF2
x  ≈+ O(t2−νF ) .
dt(νF − 1)(νF − 2)

(78)

If 1 < νF < 2, there is superdiﬀusion
4U 2 rθF2 t̃2−νFd 2x  ≈+ O(1) .
dt(2 − νF )(νF − 1)

(79)(At νF = 2 there is a logarithmic term.)
Notice the minor role of αS (a) in the solution above: if νS > 1, so thatthe mean sticking time is ﬁnite, then the parameters νS and θS occur only inr.

4.5

An exercise for the diligent student

Consider an asymmetric two-state model
Lt + La + UL Lx + αL (a)L = 0,

Rt + Rx + UR Rx + αR (a)R = 0, (80)with the boundary conditions ∞αR (a)R(a, t, x) da,
L(0, t, x) = ∞R(0, t, x) =αL (a)L(a, t, x) da.

0

0

(81)
Show that the average velocity isτL UL + τR UR
Ū =,τL + τ R

 ∞τL,R ≡
ΨL,R (a) da,

(82)

0

where ΨL and ΨR are deﬁned by analogy with (41). Show that the Laplacetransform of the velocity autocorrelation function is given by2
corr(s)ˆ= URM
S

1 τL + τR (1 − ψ̂L )(1 − ψ̂R )
−
,
s
τL τR
s2 (1 − ψ̂L ψ̂R )

28

(83)

whereτL + τ R
2
URM
UR UL .
S ≡ √
τL τR

(84)

(If you use the moment method, you will need Laplace transforms to solve
the integral equation which arises at n = 1.) Using the model
αR,L (a) =νR,L,θR,L + a(85)

perform an asymptotic analysis of (83) to identify the anomalous diﬀusion
exponents which occur if either or both of νL and νR are less than 2.

5

Solution of an initial value problem

In this appendix we discuss the issue raised at the end of section 3.1 andanalyze a problem in which the death rate of old items is so small that the
average lifetime τ is inﬁnite. For example, this is the case ν < 1 in (42).
Speciﬁcally, consider the initial value problem posed by (37) with the initialand boundary conditions
f (a, 0) = N δ(a) ,f (0, t) = r(t) .

In (86) the replacement rate r(t) is determined by requiring that ∞N=f (a, t) da .

(86)

(87)

0

The solution of (37) and (86) is
f = N Ψ(t)δ(a − t) + Ψ(a)r(t − a).

(88)

The ﬁrst term on the right hand side of the equation above is the cohort
of initial items aging and dying. The second term is inﬂux of new items.
Imposing (87) on (88), we obtain an integral equation for r:
 t
N = N Ψ(t) + Ψ(a)r(t − a) da.
(89)
0

The integral relation above is known as the renewal equation
29

Because of the convolution in (89), the Laplace transform ∞e−sa [Ψ(a), r(a)] da ,[Ψ̂(s), r̂(s)] ≡

(90)

0

is gratifying. In this way we ﬁnd from (89) thatr̂ = N

1 − sΨ̂
sΨ̂

.

(91)

The large-time behaviour of r(t) can be obtained from (91) using standard
asymptotic methods.
If α(a) ∝ 1/a as a → ∞, then the rightmost singularity of Ψ̂(s)in thecomplex s-plane is the branch-point at s = 0. We show below in (94) through(97) that the structure of Ψ̂ at this branch-point isΨ(s) = (sν−1 + τ + · · ·(92)

If ν < 1 then the singular term involving sν−1 dominates the constant τ ass → 0. In this case, from(91),r̂(s) ∼

N
,
(sν

⇒

r(t) ∼

N tν−1
,
(Γ(ν)

as

t→∞

(93)

Because ν < 1 the replacement rate vanishes as t → ∞.
To explain the small-s expansion in (92), we use the model death-rate in(42), which produces the survival functionνθΨ(a) =.
(94)
θ+a
The Laplace transform in (90) is thenΨ̂(s) = θν sν−1 eθs Γ(1 − ν, θs),(95)

where Γ(a, x) is the incomplete Γ-function deﬁned by Abramowitz & Stegunin their article 6.5.3. This Laplace transform can be rewritten ase Γ(1 − ν) − θΓ(1 − ν)

ν ν−1 θsΨ̂(s) = θ s

30

∞


(θs)n
.
Γ(2 − ν + n)
n=0

(96)

The form above is convenient because the singular terms containing sν−1
are localized in the ﬁrst function on the right hand side. When s  1 theexpansion of (96) isΨ̂(s) = θν Γ(1 − ν)sν−1 +θ+ O(s, sν ) ,
ν−1

(97)

which is the form assumed in (92).

References[1] D.D. Joseph and L. Preziosi. Heat waves. Rev. Mod. Phys., 61:41–73,
1989.
[2] M. Schlesinger, G.M. Zaslavsky, and U. Frisch. Lévy Flights and Related
Topics in Physics. Springer, Berlin, 1994.
[3] T.H. Solomon, E. Weeks, and H.L. Swinney. Chaotic advection in a
two-dimensional ﬂow: Lévy ﬂights and anomlaous diﬀusion. Physica D,
76:70–84, 1994.
[4] G.I. Taylor. Diﬀusion by continuous movements. Proc. London Math.
Soc., 20:196–212, 1921.
[5] E.R. Weeks, J.S. Urbach, and H.L. Swinney. Anomalous diﬀusion inasymmetric random walks with a quasi-geostrophic example. Physica D,
97:291–310, 1996.

31