1
I.
CONTEXT
Characteristic Functions.
Gaussian Central Limit Theorem.
Lévy Stable Central Limit Theorem.
Laws of Large Numbers.
Generating Lévy flights on computer.
Appendix 1A: Mathematical Properties of Lévy Stable Distributions.
II.
CHARACTERISTIC FUNCTIONS
Consider the random variable −∞ < X < ∞. The probability of obtaining X in the small
interval (x, xdx) is denoted with PX (x)dx. PX (x) is called a probability density function
(PDF). In this text we assume that it is a smooth function. The PDF must satisfy the
normalization condition
Z
∞
−∞
PX (x)dx = 1,
(1)
and the non-negativity requirement PX (x) ≥ 0. Obviously if these conditioned are not
satisfied PX (x) cannot be used used as a probabilistic measure.
The n th moment of the random variable X is
n
hx i =
Z
∞
−∞
xn PX (x)dx,
(2)
and the variance is defined as
σ 2 = hx2 i − hxi2 .
(3)
2
Not all PDFs have a finite variance, a counter example is the Cauchy or Lorentz PDF
PX (x) =
1
a
.
2
π x + a2
(4)
The characteristic function of the random variable X is
P̃X (k) = he
ikx
i=
Z
∞
−∞
eikx PX (x)dx.
(5)
And P̃X (k) is also called the Fourier transform of PX (x). From normalization condition we
have
P̃X (k = 0) = 1,
(6)
and also |P̃X (k)| ≤ 1.
An important PDF is the Gaussian
1
x2
PX (x) = √
exp − 2 ,
2a
2πa2
!
(7)
and in this case X is called a Gaussian random variable. The characteristic function of X is
a2 k 2
,
P̃X (k) = exp −
2
!
(8)
and its small k expansion, or long wave length behavior, is
P̃X (k) ∼ 1 −
a2 k 2
+ ···.
2
(9)
When X is a Lorentzian random variable, Eq. (4), its characteristic function is
P̃X (k) = e−a|k| ,
(10)
PX = 1 − a|k| · · · .
(11)
its small k expansion is
Unlike the Gaussian case the small k expansion for the Lorentzian PDF is non-analytical.
This is related to the observation that the Gaussian PDF has a finite variance, while the
3
Lorentzian’s variance diverges. More generally, the Taylor expansion of a characteristic
function
∞
X
hxn ik n
(ikx)n
in
PX (x)dx =
n!
n!
−∞ n=0
n=0
√
provided that hxn i is finite, and i = −1. From Eq. (12) we see that
P̃X (k) =
Z
∞
∞
X
hxi = −i
hx2 i =
(12)
d
P̃X (k)|k=0 ,
dk
1 d2
P̃X (k)|k=0
2 dk 2
(13)
etc, hence the characteristic function P̃X (k) serves as a moment generating function. For the
Lorentzian PDF hx2 i = ∞, indeed d2 exp(−a|k|)/dk 2 |k=0 = ∞, and the analytical Taylor
expansion in Eq. (12) is not valid.
1.1 Derive Eqs. (8) and (10).
1.2 Obtain the moments of the Gaussian PDF centered on x0 .
1.3 Show that exp(−|k|µ ) with µ > 2 is not a characteristic function of a non-negative
PDF.
1.4 More generally, show that if P̃X (k) ∼ 1 − ikhxi − a|k|µ , for k → 0, then 0 < µ < 2.
1.5 Let X and Y be independent random variables with PDFs PX (x) and PY (y). Find the
PDF of Z = X/Y and Z = X + Y in terms of integrals of PX (x) and PY (y).
1.6 Obtain the characteristic function of X, when PX (x) = 2π/(1 + x2 )2 .
Consider now the random variable 0 < T < ∞, whose PDF is PT (t). Such PDFs are
called one sided. The Laplace transform of PT (t)
P̂T (u) =
Z
∞
0
e−ut PT (t) dt,
(14)
4
is also a moment generating function, since
P̂T (u) =
Z
0
∞
X
(−ut)n
(−u)n hT n i
PT (t) dt =
.
n!
n!
n=0
n=0
(15)
dn
P̂T (u)|u=0 .
dun
(16)
∞
∞X
Hence
hT n i = (−1)n
For the analytical expressions in Eqs. (15,16) to be valid, we require that moments of P T (t)
are finite. An example were the integer moments of PT (t) diverge is the Smirnov PDF
1
1
,
PT (t) = √ t−3/2 exp −
2 π
4t
(17)
whose Laplace transform is
P̂T (u) = e−u
1/2
.
(18)
Similar to the Fourier transform of the Cauchy PDF, the Laplace transform of Smirnov’s
PDF exhibits a non-analytical behavior in the vicinity of u → 0 which has to do with the
divergence of the first moment of the PDF.
1.7 Prove that Eqs. (17) and (18) are Laplace pairs.
1.8 Show that the small u expansion P̂T (u) = A − Buα + O(uα ), must satisfy A = 1 B > 0
and α ≤ 1.
1.9 Construct a one sided PDF PT (t) whose n th moment diverges, while integer moments
of order k < n are finite.
Consider a generic one sided PDF which for long t satisfies
PT (t) ∼ At−(1+α)
(19)
5
and 0 < A, 0 < α < 1. In this case PT (t) is moment-less e.g. hti = ∞. We wish to
investigate the small u behavior of the Laplace transform
P̂T (u) =
Z
+A
Z
t0
0
∞
t0
∞
PT (t)e−ut dt + A
Z
t0
t−(1+α) dt
t−(1+α) e−ut − 1 dt + .
(20)
Here t0 is large meaning that the approximation in Eq. (19) is valid for t > t0 . is small
when t0 is large. If PT (t) = At−(1+α) for t > t0 then = 0 and since we may choose t0 as
being as large as we wish we will take = 0. From the normalization condition the first two
terms on the right hand side (RHS) of Eq. (20) are equal one when u → 0. Also note that
the small u expansion of the first term on the RHS of Eq. (9) is of the form C + Bu and as
we show now the u dependence of this term is not a leading order term in the asymptotic
expansion in the limit of u → 0.
Hence to investigate the small u behavior of PT (t) we must consider the integral
A
Z
∞
t0
t−(1+α) e−ut − 1 dt = Auα
Z
∞
ut0
x−(1+α) e−x − 1 dx
(21)
We consider the limit
lim
Z
∞
ut0 →0 ut0
x−(1+α) e−x − 1 dx = Γ (−α) ,
(22)
where Γ() is the Gamma function. Collecting all the terms we find that if Eq. (19) holds
then
P̂T (u) ∼ 1 + Γ (−α) Auα + · · · ,
(23)
when u → 0. Such relations between small u behavior and large t behavior of Laplace pairs
are called Tauberian and Abelian relations, they are discussed in the mathematical literature
with rigor2,3 . Similar type of relations exists between small k behavior and large x behavior
of Fourier pairs, as stated in the following question.
6
1.10 Consider the symmetric PDF PX (x) = PX (−x). Show that if
PX (x) ∼
A
|x| → ∞
|x|1+µ
(24)
with 0 < µ < 2 then the characteristic function
P̃X (k) ∼ 1 − B|k|µ
|k| → 0.
(25)
Obtain B in terms of A and µ.
1.11 Verify that rules Eqs. (23) and (25) hold for the examples of Smirnov’s and Lorentz’s
PDFs respectively.
III.
GAUSSIAN CENTRAL LIMIT THEOREM 2,4
Let x1 , · · · , xN be a set of N independent identically distributed random variables with
a common PDF PX (x) whose mean is zero and variance is σ < ∞. Consider the scaled sum
Z=
PN
i=1 xi
.
N 1/2
(26)
According to the Central Limit Theorem, in the limit N → ∞
1
z2
√
PZ (z) →
exp − 2 .
2σ
2πσ 2
!
(27)
This is a remarkable theorem, of central importance in probability theory, statistical mechanics and beyond, since the PDF of the xi ’s is not necessarily a Gaussian still the scaled
sum Z is Gaussian. Thus we say that the PDF PX (x) belongs to the domain of attraction
of the Gaussian, if σ is finite.
A poor man’s proof of the Gaussian Central Limit Theorem is based on Fourier transforms. Let P̃X (k) be the characteristic function of xi . Since we assume hxi i = 0 we have
7
the small k behavior
P̃X (k) ∼ 1 −
σ2 k2
+ A|k|γ
2
(28)
and 2 < γ. Using problem 1.5 the distribution of a sum of N independent random variable
is a convolution, namely in Fourier space
"
P̃Z (k) = P̃X
k
N 1/2
!#N
.
(29)
Hence using Eq. (28) we find
!N
Ak γ
σ2 k2
+ γ/2 · · ·
P̃Z (k) = 1 −
2 N
N
.
(30)
Taking the limit N → ∞ we obtain
P̃Z (k) → e−
σ 2 k2
2
.
(31)
The inverse Fourier transform of which yields the Central Limit Theorem, Eq. (27).
Two technical remarks are: (i) the choice of the N 1/2 scaling we introduced in Eq. (26)
is justified in Eq. (30), choosing a different type of scaling e.g. Z =
P
xi /N β with β 6= 1/2
will not yield a meaningful limit theorem. (ii) The |k|γ term in Eq. (30) does not contribute
to the N → ∞ limit, hence information on behavior of PX (x), besides its variance is not
important in the N → ∞ limit.
There exists a vast literature on the Gaussian central limit theorem, and more refined
proofs and formulations of it yield insight into questions like how fast is the convergence of
distribution of Z towards the Gaussian (i.e., what is the exact meaning of large N ). An
answer to this question is given in the Berry Eséen theorem, soon to be stated. Another
question is where on the z axis, is the exact PDF PZ (z) well approximate by the Gaussian
PDF. (i.e. when N is large but finite). In next chapter we will check this issue in the
context of random walk theory, and we will show that under certain conditions that PZ (z)
will behave like a Gaussian in its central part (hence the name Central Limit Theorem).
8
A.
Berry Eséen Theorem2
We consider the sum of N random variables Eq. (26), however for convenience use a
dimensionless form by defining ξ = ZN /σ. Denote the PDF of ξ with φN (ξ). The cumulative
distribution function of ξ is
ΦN (ξ) =
Z
ξ
−∞
φN (ξ)dξ.
(32)
We still assume hxi i = 0, hx2i i = σ 2 , and additionally the third moment r3 = h|xi |3 i < ∞ is
assumed finite, and is determined of-course by the precise shape of PX (x).
The Berry–Eséen theorem states that
1
|ΦN (ξ) − √
2π
Z
ξ
−∞
ξ2
e− 2 dξ| ≤
3r3
.
1/2
N (σ 2 )3/2
(33)
This is a very strong bound on ΦN (ξ) since it does not depend on number of summands
N . In the limit N → ∞ we regain the Gaussian Central Limit Theorem. The bound yields
an estimate on how far is the exact solution ΦN (ξ) from the asymptotic Gaussian behavior
when N is finite.
IV.
LÉVY’S GENERALIZED CENTRAL LIMIT THEOREM
Let {x1 , · · · xN } be a set of N independent identically distributed random variables, symmetrically distributed, with zero mean hxi i = 0, and with diverging variance hx2i i = ∞. An
example are random variables distributed according to the Lorentzian PDF Eq. (4). Clearly
for this case the Gaussian Central Limit Theorem does not hold. Still as we show now an
important generalization of the Gaussian Central Limit Theorem is valid.
Let
Zn =
PN
i=1 xi
N 1/γ
(34)
9
and the scaling exponent γ will soon be determined. The PDF of the xi is assumed of the
form
Pxi (x) ∼ A|x|−(1+µ)
when
|x| → ∞
(35)
|k| → 0
(36)
and 0 < µ < 2 hence σ 2 = ∞. In Fourier space
P̃xi (k) ∼ 1 − Ã|k|µ
when
The A and the à are related, see question Eq. 1.12 below. Using an expansion very similar
to that in Eq. (30) we have the characteristic function of the sum Z
!N
|k|µ
P̃Z (k) = 1 − Ã γ/µ · · ·
N
.
(37)
where · · · are terms of order higher than |k|µ . We now determine the scaling exponent, and
choose
γ = µ,
(38)
and find using Eq. (37) in the limit of large N
µ
P̃Z (k) → e−Ã|k| .
(39)
The characteristic function of Z has a form of a stretched exponential. Inverse Fourier
transform of Eq. (39) yields the PDF of Z in the limit of N → ∞, such PDFs are called
stable PDFs, or sometimes Lévy’s PDFs and are denoted here with lµ,Ã,0 (z). The PDF
lµ,Ã,0 (z) is symmetric which is obviously related to the fact that we assumed that the xi ’s
is symmetrically distributed random variable. The expenent µ is called the characteristic
exponent, and à is a scale factor. In the special case µ = 2 we recover from Eq. (39) the
Gaussian PDF. For µ = 1 we obtain the Lorentzian PDF. For other choices of µ Schneider
showed that lµ,Ã,0 (z) can be expressed in terms of Fox functions, though generally they are
10
not tabulated. Hopefully this will change in the near future, and programs like Mathematica
will yield stable PDFs.. Obtaining stable PDFs is in principle easy, using numerical inverse
Fourier transform. We investigate some of the Mathematical properties of lµ,Ã,0 (z) in the
Appendix.
1.12 Show that
A = Ã
sin µπ
Γ (µ) µ
2
.
π
(40)
1.13 van-Kampen writes “The unfortunate name “stable” is used for distributions having
the property that the sum of two variables so distributed has again the same distribution (possibly shifted and/or rescaled)”. Show that the symmetric Lévy PDFs Eq.
(39) are stable. Is the term stable PDFs unfortunate or is van-Kampen’s “unfortunate”
remark unfortunate.
V.
LAWS OF LARGE NUMBER’S
Consider the sum of N random independent identically distributed random variables
tN =
PN
i=1 ti
,
Nβ
(41)
and ti > 0 have a common PDF Pt (t). Let P̂t (u) be the Laplace transform of Pt (t). When
β = 1, tN is a random variable, however we expect that in the limit
lim tN |β=1 = hti =
N →∞
Z
∞
0
tPt (t)dt,
(42)
provided that hti is finite. The law of large numbers considers the mathematical meaning
of the identity in (42). Here we will consider this law briefly, using Laplace transforms
P̂tN (u) = P̂t
u
Nβ
N
.
(43)
11
Using the small u expansion we have two classes of PDFs, those with infinite hti
P̂t (u) = 1 − Auα + · · · 0 < α < 1,
(44)
and those with a finite first moment
P̂t (u) = 1 − htiu + · · · u → 0.
(45)
Using Eq. (43,45) we have for the case when the underlying PDF of {ti } is moment-less
P̂tN
N
Auα
(u) = 1 − αβ + · · ·
N
,
(46)
we choose β = 1/α and find
α
lim P̂tN (u) = e−Au .
N →∞
(47)
For the second class of PDFs, those with finite first moments Eq. (44), we obtain
lim P̂tN (u) = e−htiu .
(48)
lim PtN (t) = δ(tN − hti).
(49)
lim PtN (t) = lα,A,1 (t).
(50)
N →∞
and hence as expected
N →∞
For α < 1 we find that
N →∞
lα,A,1 (t) is the one sided Lévy stable PDF which is the inverse Laplace transform of lα,A,1 (u) =
exp(−Auα ). We see that when α < 1 the “average” tN remains random even when N → ∞.
Some mathematical properties of one sided Lévy PDFs are discusses in Appendix A.
VI.
GENERATING LÉVY FLIGHTS ON A COMPUTER
Further insights into power law statistics is gained through simple numerical experiments.
It is easy to generate on a computer a random number, call it x, uniformly distributed in
12
the interval [0, 1]. Starting with such a random number we seek a transformation which will
yield a random variable which is described by power law statistics. One such transformation
is
t = (1 − x)1/α .
(51)
Using the chain rule
Pt (t) =
dx Px (x) ,
dt (52)
and the uniform PDF Px (x) = 1 when 0 < x < 1 we find
Pt (t) = αt−(1+α)
(53)
and 1 < t < ∞. This is only one example of how we can generate a random number
whose PDF decays like t−(1+α) for long times. In some cases we may wish to obtain random
variables t described by a specific PDF Pt (t) and for which simple transformations like Eq.
(51) are difficult to obtain. In that case a method called the Accept-Reject method can be
used, the latter is discussed in1 .
Figures: Devil Staircase, Exp Stair case, Lévy Flight.
1.14 Invent a transformation that maps a uniformly distributed random variable, onto a
power law random variable whose PDF behaves like Pt (t) ∝ t−(1+α) , and which is
different than Eq. (51).
1.15 Consider the sum TN =
PN
i=1 ti
and tmax = maximum of {ti }. Obtain TN and tmax M
times. Plot TN versus tmax for two cases: (i) ti uniformly distributed between [0, 1]
and (ii) for a one sided power law PDF with α < 1 e.g. use Eq. (51). Use large M
and N and explain your observations.
13
1.16 Consider a sum of N random variables uniformly distributed in [−1/2, 1/2], use numerical simulation to observe Gaussian Central Limit behavior. For what values of N
do you expect a reasonable convergence to Gaussian behavior. Note that you may also
obtain exact solution to the problem in Fourier space, and then invert the solution
numerically.
1.17 Consider a sum of N random variables X, with Px (X) ∝ |x|−2 when |X| → ∞, and
PX (x) = PX (−x). Use numerical simulations to observe Lévy Central Limit behavior.
VII.
LONG RANGE INTERACTIONS AND LÉVY STATISTICS
Consider a system of particles interacting via a two body central field. Select one of the
particles called the tracer particle, and place it on the origin. The density of particles is
ρ = N/V , where N is the number of particles and V is the volume. In the thermodynamic
limit both N and V are large, while ρ is finite. The force the particles exert on the test
particle is
F=
N
X
i=1
F(|ri |).
(54)
For example for a point masses m embedded in a sea of other point masses M
F(|ri |) =
GmM r̂
|r|3
(55)
and all other symbols have their usual meaning. In some physical situations the particles
are uniformly distributed in space. Then we may think of an ensemble of tracer particles,
each embedded in a sea of bath particles, and ask what is the distribution of forces on the
tracer particle. The problem is a problem of summation of random variables.
This type of problem appears in many brunches of physics, with long range interaction.
Chandrasekar considered it in the context of the calculation of the distribution of forces
14
acting on a gravitational object. Stoneham’s theory of inhomogeneous line broadening of
defected crystals, can be interpreted in terms of Lévy statistics. Klauder and Anderson 18
showed Lévy statistics is important in the context of spectral diffusion in spin systems governed by dipolar interaction (note that these authors do not use the term Lévy statistics
to describe their results). Such Lévy statistics describe statistics of single molecule spectroscopy in low temperature glasses (14 for theory, and15 for experiment) and also certain
aspects of turbulence. It was Holtsmark who considered this problem first in the context of
spectroscopy.
Let us consider a specific example and let us calculate the distribution of forces along the
z direction for the Newtonian attraction in three dimensions
N
X
Fz =
i=1
GmM cos(θi )
.
|ri |2
(56)
The factor cos(θi ) yields the projection of the force on the z axis. We now consider the
distribution of Fz , assuming a uniform distribution of the other objects (a rather strong
assumption, which we discuss later). We find the characteristic function of Fz , making use
of the fact that the bath of particles are statistically uncorrelated (Physically we neglect the
interaction of bath particles) we find
he
ikFz
"
i=h 1+
R
2
V
dV (eiGmM cos(θ)/|r| − 1) N
i
V
#
(57)
Here I used the fact that a bath star is uniformly distributed in space, hence its density is
1/V . In the thermodynamic limit N → ∞ we have
he
ikFz
i = exp −ρ
Z
V
dV (1 − e
ikGmM cos(θ)/|r|2
) .
The integration of the imaginary part in the exp, vanishes because of symmetry
Z
V
dV sin[kGmM cos(θ)/|r|2 ] = 0,
(58)
15
hint using spherical coordinates
he
ikFz
R1
−1
i = exp −ρ
sin(cos(θ))d cos(θ) = 0. Hence the integral to solve is
Z
V
h
2
dV 1 − cos(GmM cos(θ)/|r| )
i
.
(59)
The fact that the integral is real means that the PDF of Fz is symmetric, and we are equally
likely to find Fz > 0 or Fz < 0. We can also consider only only k > 0 since heikFz i must be
an even function of k. After a change of variables and integration over angles
heikFz i = exp −
Z ∞
8πρ
dyy 2 (1 − cos 1/y 2 ) .
(GmM )3/2 |k|3/2
5
0
(60)
The integral is solved according to
Z
∞
0
dyy 2(1 − cos 1/y 2 ) =
√
2π
.
3
Eq. (60) shows that the PDF of Fz is a symmetric Lévy stable law, with a characteristic
exponent 3/2 (because of the |k|3/2 ). The variance of the force in z direction is infinite. The
distribution was found by Holtsmark in a different context, a long time ago.
Why did we get Lévy statistics for this problem. Since we assumed that particles are
uniformly distributed in space, and independent of each other the problem is similar to the
mathematical problem of summation of independent random variables. Some of the terms in
the sum Eq. (56) may become very big, when two gravitational objects i.e. stars, are closely
situated. On the other hand many other terms are typically very small, due to the stars in
the background. This causes a wide distribution of forces and leads to Lévy statistics.
We assumed that the system of point masses is uniformly distributed in space, while
Chandrasekar seems to claim that this is a reasonable assumption it clearly must be doubted.
Interaction among the bath particles will lead to complex correlations among their position
in space. However, in other disordered condensed matter systems uniformly distributed
defects leading Lévy statistics are indeed quite common. For example low density quenched
16
defects in crystals, for example dislocations, are to an excellent approximation randomly
distributed in space. And these defects when interacting with an optical marker, create the
so called inhomogeneous line broadening effect. The physics is as follows. An atom in a
gas phase has a certain absorption frequency ω0 . This atom is embedded in a crystal, and
it interacts with the randomly and uniformly distributed defects. These defects will cause
a frequency shift, namely the atom’s absorption frequency is ω0 +
P
ωi . The ωi are due to
interactions of the atom with defects, they are usually calculated using quantum mechanics
in particular first order perturbation theory13 . The details of these interaction are an issue
for a condensed matter course. Here we are interested only in certain statistical aspects of
the problem, and we claim without proof that in many cases, the frequency shifts decay
like ωi ∝ ri−δ where ri is the distance of the defect to the optical marker. δ will depend
on details of the interaction of the atom and defect, however for simple interaction like
Coloumb interactions, interactions with certain strain defects, or dipoles one finds δ = 1, or
δ = 2, or δ = 3. The important point is that the defects are uniformly distributed in space
with a density ρ, and hence:
1.18 Show that the PDF of
ω = ω0 +
N
X
A cos θi
|ri |δ
i=1
(61)
in d dimensions is a symmetric Lévy stable law with a characteristic exponent d/δ.
Show that Holtsmark PDF is a special case. Explain why the choice of angular
dependence (i.e. the artificial cos θ term) does not change your result in qualititative
way (universality) provided that on average negative and positive frequency shifts are
equally likely (symmetry).
17
Since the crystal will have in it many markers (called chromophores), these optical centers
will have a very wide Lévy spectrum of frequencies. Namely the absorption line shape of
a single atom in gas phase is very narrow, the width being the inverse natural life time of
the excited state. In contrast the absorption line of an ensemble of such atoms embedded
in a crystal, is in many cases very broad its shape being a symmetric Lévy function. The
particular type of L’evy function will depend on the type of defects in the sample (see Table
1 for some details). This topic is very nicely reviewed by Stoneham, who unfortunately did
not recognize the relation of this problem with Lévy statistics.
Table Summary of Stoneham’s theory on inhomogeneous line shape broadening,
for different types of interactions, and dimensions d.
The results show that normal-
ized inhomogeneous lines are symmetrical stable laws ld/δ,0 (ω).
L10 is the Lorentzian,
L3/2,0 is Holtsmarks function, and L∗2,0 (ω) is a Gaussian with logarithmic corrections.
18
Physical problem
d
δ
Line Shape
Strain Broadening due to dislocations
2
1
L∗2,0 (ω)
Strain due to dislocation dipoles
2
2
L1,0 (ω)
Strain broadening due to point defects
3
3
L1,0 (ω)
Magnetic and electric dipole broadening
3
3
L1,0 (ω)
Broadening due to random electric fields
3
2
L3/2,0 (ω)
An important issue are cutoffs. We derived the Lévy distribution assuming that the
point defects (or masses) are uniformly distributed in space. However Physically two objects
cannot come into close contact with each other, for example a star has a finite size. Or the
1/r δ interaction is not valid for short distances since particles repel each other on short
distances. We also assumed that the size of our system is infinite, while in practice it is
finite. These lower and upper bounds can be included in the theory and then non-universal
deviations from Lévy behavior can be observed. These deviations depend ofcourse on the
details of the problem, for example what exactly are the cutoffs etc. Without going into
details if δ >> d usually a small cutoff is sufficient to destroy the Lévy behavior. Thus in
Physics we expect Lévy behavior only when the interactions are long range, namely δ ≤ d.
The reason is clear only if the interaction is long ranged, then we have a summation of many
19
random variables (i.e. the forces or frequency shifts), and a possibility for a central limit
theorem argument to hold.
1
Press Numerical recipes
2
Feller
3
G. Weiss
4
N. G. van-Kampen Stochastic Processes in Physics and Chemistry (North Holland Amsterdam
1981).
5
Abramovitz
6
G. Samorodnitsky and M. S. Taqqu Stable Non-Gaussian Random Processes (Chapman and
Hall/CRC New York 2000).
7
B. V. Gnedenko, and A. N. Kolmogorov Limit distributions for sums of independent random
variables (Addison-Wesley, Reading MA 1954).
8
W. R. Schneider in Stochastic Processes in Classical and Quantum Systems edited by S. Albeverio, G. Casatti, and D. Merlini (Springer, Berlin 1986).
9
H. Scher, and E. Montroll Phys. Rev. B 12 2455 (1975).
10
E. Barkai, Phys. Rev. E 63 046118 (2001).
11
E. W. Montroll, and J. T. Bendler, J. Stat. Phys. 34 129 (1984).
12
E. W. Montroll, and B. J. West, On an Enriched Collection of Stochastic Processes in Fluctuation Phenomena Chapter 2, North Holland Publishing Company (1979) Montroll and Lebowitz
Editors.
13
A. M. Stoneham Rev. Mod. Phys. 41 82 (1969).
14
E. Barkai, R. Silbey, and G. Zumofen Phys. Rev. Lett. 84 5339 (2000).
20
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E. Barkai, A. V. Naumov, Yu. G. Vainer, M. Bauer, and L. Kador Phys. Rev. Lett. 91 075502
(2003).
16
Holtsmark
17
I. A. Min, I. Mezik, and A. Leonard Phys. Fluid 8 1169 (1996).
18
J. R. Klauder and P. W. Anderson Phys. Rev. 125 912 (1962).
VIII.
APPENDIX A: MATHEMATICAL PROPERTIES OF STABLE PDFS
A.
Symmetric Stable Laws
We have found that an important class of stable PDFs are symmetric stable laws whose
Fourier transform is exp(−A|k|µ ). The inversion of this characteristic function is now considered. Besides the case µ = 1 (Lorentzian) and µ = 2 (Gaussian) the inversion is not
straight forward. Schneider8 expresses the symmetric Lévy stable laws in terms of certain
Fox functions. These Fox functions are generally not tabulated, though one can use known
asymtotic behavior of Fox functions to derive asymptotic behaviors of Lévy stable laws. We
will consider the case A = 1 since the generalization to A 6= 1 is straight forward.
Consider the symmetric Lévy PDF
1
l1,µ,0 (x) =
2π
Z
∞
−∞
e
−ikx−|k|µ
1
dk =
π
Z
∞
0
µ
cos kxe−k dk.
(62)
Integrating by parts and changing the integration variable we have
µ Z ∞ µ−1
µ Z ∞ −( v )µ µ−1
−k µ
l1,µ,0 (x) =
k
sin kxe dk =
sin vdv.
e x v
πx 0
πxµ+1 0
(63)
For large x
l1,µ,0 (x) ∼
µ
πxµ+1
Z
∞
0
v µ−1 sin vdv
(64)
21
and hence if 0 < µ < 1 then
l1,µ,0 (x) ∼
µ
µπ
Γ(µ)
sin
.
πx1+µ
2
(65)
It is easy to see that this large x behavior is valid also for the Lorentzian case µ = 1. If
1 < µ the integral Eq. (64) is not well defined and a different approach must be used.
The proof that Eq. (65) is still valid for 1 < µ < 2 is slightly more complicated
1Z∞
µ
lµ,1,0 (x) =
cos(kx)e−k dk
π 0
change variable ω = kx and let η = x−µ and then
1
lµ,1,0 (x) =
πx
Z
∞
0
e−ηω (cos ω)e−η(ω
µ −ω)
dω.
For large x hence small η
lµ,1,0 (x) ∼
1 Z ∞ −ηω
e
cos ω [1 − η (ω µ − ω) + · · ·] dω.
πx 0
(66)
The integrals are solved using a formula from a Table (e.g. Mathematica)
Z
∞
0
ω µ cos ωe−ωη dω = 1 + η −2
−(1+µ)/2
η −1−µ cos [(1 + µ)ArcTan(1/η)] Γ (1 + µ) .
Using Eq. (66)
1
lµ,1,0 (x) ∼
πx
(
η
η2 − 1
1
+
η
− η(1 + η −2 )−(1+µ)/2 η −(1+µ) cos (1 + µ)ArcTan( ) Γ(1 + µ) .
2
2
2
1+η
(1 + η )
η
"
#
In the limit of x → ∞ corresponding to η → 0 we obtain Eq. (65).
B.
One Sided Stable Laws
One sided Lévy stable laws, lα,1,1 (t) 0 < t < ∞ defined via their Laplace transform
ˆlα,1,1 (s) = exp(−sα ), are now investigated. Generalization to the case lα,A,1 (t) with A 6= 1
is straightforward. Recall also that 0 < α < 1.
)
22
We write the stable density in terms of Laplace inversion integral
1
lα,1,1 (t) =
2πi
Z
c+i∞
c−i∞
α
dsest e−s ,
(67)
a change of variables −st = z yields
1
tlα,1,1 (t) =
2πi
Z
c+i∞
c−i∞
α
dze−z−(−z/t) .
(68)
We consider 1/tα << 1, hence expand exp(−z α /tα ) in Taylor series
∞
X
1 1
1
− α
tlα,1,1 (t) =
t
n=0 2πi n!
n Z
C
dz(−z)nα e−z .
(69)
The integral in this equation is the Hankel representation of the Γ function5 . Thus we have
1 (−1)n
1/Γ (−αn).
lα,1,1 (t) =
nα+1
n=0 n! t
∞
X
(70)
Using the reflection formula for Γ functions5
Γ (z) Γ (1 − z) = π csc πz,
(71)
we find
∞
1X
Γ(1 + nα)
lα,1,1 (t) =
(−1)n−1 sin(πnα)t−(αn+1) .
π n=1
n!
(72)
The opposite limit of small t can be treated using steepest descent method (Scher-Montroll).
A summary of some properties of one sided stable PDFs is given in Appendix A of10 .