Talk given at Max Planck Institute for the Physics of Complex Systems, Dresden, 9th July, 2014,

1. A brief tour of some fractional
models … and of the physics
they seek to embody
Nick Watkins
(NickWatkins@mykolab.com)
MPIPKS, Dresden, 9th July, 2014

2. Addendum:
These slides were given as a seminar at MPIPKS Dresden on 9th July 2014. I
have corrected errors/typos that I spotted or which were pointed out to
me, and added the remaining references. These are meant to be
indicative, not comprehensive, especially those written as “e.g”
I ran out of time and space in designing the talk, so SOC and turbulence
ended up not being treated in any detail at all – I have added a couple of
suggested book references to their slides in order to rectify this a bit.
I also expect that there are many other areas that could be improved. If
using this material please consult a local expert, especially on topics like
stochastic calculus and fractional derivatives.
Nick

3. Context:
The understanding of Brownian motion has been one of the great
achievements of 20th century physics, mathematics, chemistry and
economics. Its importance spans these fields, as well as biology and
control engineering, at the very least. However, the increasing interest in
processes which are not completely amenable to the now standard
methods has led to new models for "anomalous" time series and
diffusion, most notably the fractional stable family like Mandelbrot's
fractional Brownian motion, and the family of fractional continuous time
random walks. In addition fractals and fractal calculus play a key role in
models for the known physical phenomenon of turbulence, and the
postulated one of self-organised criticality.

4. Aim
• Informally review the “zoo” of fractional models and the physics each
one brings with it …
• Driven in part by my own difficulties over the years in working out
which fractional model corresponds to what physics… have become a
fractal zookeeper in my spare time …
• Main aim is to promote insight via discussion, and build on the
excellent talks at EXEV14; by Rainer & Alexei during their visits; etc.
• WARNING: Haven’t yet reconciled notation.
So if it looks wrong/puzzling please do ask.
Sorry !

5. “A brief tour of some fractional
model … and of the physics
they seek to embody
Nick Watkins
MPIPKS, July, 2014
… and never use “brief tour” in an abstract …
B. Kliban

6. Origins:
"On the motion of small particles suspended in
liquids at rest required by the molecular
kinetic theory of heat.“--A. Einstein,
Ann. D. Physik, 4th ser. 17 549 (1905)
Standard diffusion picture due to Bachelier, Einstein, Smoluchowski, Langevin et al.

7. Three motivating questions
• Q1 stochastics: If we “break” that picture, what models then result for
“anomalous” behaviour, and why ? Do we have more choices ? 2 key examples
are fractional kinetics (e.g. CTRW), and fractional motions (e.g. FBM) …, … How
differ-what’s common ? (*) [Main topic]
• Q2 statistics: How can we reliably distinguish models, measure exponents, etc
when many observables show same or similar behaviour ? …[I will largely
ignore, has been discussed by Barkai, Klages, Chechkin, Froemberg, Klafter,
Sandev and others at PKS over the last year]
• Q3: stat mech: What physical scenarios do these models correspond to … can
that help us design better tests … how do scenarios relate to other observed
physical phenomena such as turbulence, and to postulated ones such as SOC ?
…[I will actually very have little to say about SOC and turbulence]
(* May also briefly mention nonlinear Fokker-Planck and Levy walk cases).

8. Cornerstones
2
Wiener process
( ) ( ) (0,1)t dt
tdX tX t dt t N 
  
2
Diffusion equation
( , )
( , )
( , ) is Gaussian
P y t
D P y t
t
P y t

 


1/
1/
Stability property: Pdf looks the same under
' , ' where =2
( ', ') ( , )
( , ) is Gaussian,
and is an attracting fixed
point for short tailed pdfs.
t y
t y
P y t P y t
P y t



 

 

Ohmic Langevin
equation and FDT
Kinetic
description
Stability property, Wiener
process & central limit
theorem
( ) ( )Mq q V q f t 
   
e.g. Lemons, 2002; Paul and Baschnagel, 2013
( ) 0 and
( ) ( ') 2 ( ')B
f t
f t f Tt k t t 
 
  

9. Cornerstones of classical stochastics &
equilibrium stat mech [e.g. Lemons, 2002]
• Mathematical BM: Finite variance stable pdf; the central limit
theorem; and embodiment as a random walk. Einstein-Bachelier
Brownian motion, modelled as the Wiener process (WBM).
• Physical BM: embodiment by the Langevin equation (LE) of the
fluctuation-dissipation theorem (and equipartition of energy). LE
follows velocity v, and thus position x, of an individual particle on
timescales comparable with the dissipation time scale.
• An approximation (Smoluchowski’s) that bridges LE physics and WBM
maths. Leads to the Fokker-Planck and diffusion equations which
follow the pdf p(x,t) for position, on timescales long compared to
dissipation time scale.

10. Central Limit Theorem & Mathematical BM
• Adding r.v.’s = convolution of pdfs or multiplication of characteristic functions
[e.g. Mantegna & Stanley, 2000; Bouchaud and Potters, 2003 ]. Can also be
viewed as instance of renormalisation group process [e.g. Sornette, 2004]

11. Langevin equations and physical BM
• “We know that the complementary force … is indifferently positive
and negative and that its magnitude is such as to maintain the
agitation of the particle, which, given the viscous resistance, would
stop without it ”-Langevin, 1908, in Lemons, 2002.
2
Einstein: ( ) ( ) (0,1)t dt
tX t dt X t dtN 
  
2
Langevin: V( ) ( )
(0,1) )( t dt
tV t dt
t dt V t
dtN  
  
 

12. Fluctuation-Dissipation Relation
• “We know that the complementary force … is indifferently positive
and negative and that its magnitude is such as to maintain the
agitation of the particle, which, given the viscous resistance, would
stop without it ”-Langevin, op. cit..
( ) ( ') 2 ( ')
( ) 0
BTf t f t k t t
f t
   
 

13. The Smoluchowski approximation
• Leads to a diffusing X:
2
(dV = 0 = (0,1)) t dt
tV t dt d Nt  

dX = Vdt 2
2
2
0 0 2
dX = (0,1)
whose solution is
( ) (x , )
t dt
t
t
dt
t
N
N
X t







14. So which leg(s) could we break ?
2
Wiener process
( ) ( ) (0,1)t dt
tdX tX t dt t N 
  
2
Diffusion equation
( , )
( , )
( , ) is Gaussian
P y t
D P y t
t
P y t

 


1/
1/
Stability property: Pdf looks the same under
' , ' where =2
( ', ') ( , )
( , ) is Gaussian,
and is an attracting fixed
point for short tailed pdfs.
t y
t y
P y t P y t
P y t



 

 

Ohmic Langevin
equation and FDT
Kinetic
description
Stability property, Wiener
process & central limit
theorem
( ) ( )Mq q V q f t 
   
e.g. Lemons, 2002; Paul and Baschnagel, 2013
( ) 0 and
( ) ( ') 2 ( ')B
f t
f t f Tt k t t 
 
  

15. Symptoms of complex transport: 1
10 July 2014 15
One symptom is existence of very long jumps
(“flights”) compared to the <jump>

16. Extending the CLT -> Lévy flights
• Goes beyond the CLT relatively unambiguously, by dropping
assumption of finite variance.
• Result is Extended Central Limit Theorem and the family of α-stable
distributions, defined by pdf’s characteristic function:
~ exp((k) )LP k
~ ( )( ) e dkx ikx
LP kP



17. Lévy “flight”
10 July 2014 17
Terminology came from Mandelbrot’s “Fractal Geometry of Nature” [1977, p. 289] & his picture of a
rocket traveling between fractally distributed galaxies. Actually a random walk but with non
Gaussian, heavy-tailed jumps. Waiting times still short tailed---temporal memory is short ranged.
Use α-stable distribution,
has asymptotic
power law tail
for its pdf P(x)
with exponent α
(1 )
P(x) x  

0<α< 2
α-stable
Gaussian

18. Stability
• Stable distribution has property of keeping its shape under convolution
[e.g. Mantegna & Stanley, 2000; Sornette, 2004; Bouchaud and Potters,
2003] but the parameters rescale.
Gaussian
Cauchy

19. A kinetic equation for Lévy flights I
• 1st edition of Paul and Baschnagel (it’s now in Sec 4.2) gives a
heuristic derivation of the kinetic equation that such a pdf must obey.
Note sometimes µ replaces α, in the physics literature. Also watch
out for a µ that differs from α by 1 e.g. classic papers on Levy flights &
walks and DFA.
| |ˆCF for N step walk: ( ) Na k
LP k e



| |ˆCF for one step walk: ( ) a k
LP k e




20. A kinetic equation for Lévy flights II
( / )t| |ˆIf each step takes then ( , ) a t k
t P k t e

 
 
t| |ˆIf each step takes then ( , ) D k
t P k t e


 
0
Effective diffusion coefficient lim
t
a
D
t 


ˆ
ˆA solution of | |
P
D k P
t

 
 Paul & Baschnagel, op cit.

21. Space fractional kinetic equation
Defining fractional derivat
1
| |
2
iveth
ikx
dk e k
x









 
 
fractional kinetic equationWe get
( , ) ( , )P x t D P x t
t x


 

 
Paul & Baschnagel, op cit.

22. Next questions …
• When do the FDT and/or equipartition “break” ? [e.g Klages & Chechkin, various PKS
talks]
• What happens if we use full (generalised) Langevin equation ? [e.g. Sandev, ditto]
• Can we then still make Smoluchowski approximation ? [e.g. Lutz, QMUL seminar, 2006]
• What happens if we break ergodicity – and when might we ? [e.g. Froemberg, EXEV14]
• Why are some processes semi-martingales [*], & others not? [Weron, EXEV14]
• [* and what is a semi-martingale ?]

23. At least two ways to “break” classical diffusion
• One is explicitly non-Markovian,
via generalised Langevin
equation … (fractional motions)
• … Another is semi-martingale route.
Factorising CTRW is of this type,
modifies Brownian diffusion
by using subordination in time
(fractional kinetics).
[Weron and Magdziarz, 2008]

24. Non-Markovian route: Fractional Motions (e.g.
fractional Brownian motion)
• Non-Markovian, and not a semi-martingale
• Keeps a specified (stable) pdf,
• & a (fractional) Langevin equation [e.g. Lutz, 2001;Kupferman, 2004],
• but sacrifice much of the intuition built up about diffusion equations
-except as a formal solution that gives the pdf of fBm/LFSM-as
process no longer has the semi-martingale property.
• Might seem less intuitive of the 2 routes, BUT to understand what we
can say about fractional motions physically, first look back at the
derivation of the Langevin equation …

25. Where does Langevin equation come from ?
• Fundamentally LE is the equation of a preferred degree of freedom
(“system”) interacting with a reservoir made of a set of harmonic
oscillators, …
• … which are usually taken to be a thermalised heat bath-requires us to
impose conditions on the oscillators.
• We are most used to the “ohmic” (linearised resistance) limit of LE:
• But what’s the physical picture behind LE?
( ) ( )Mq V q fq t
   

27. Damped harmonic oscillator [Yurke, 1984]:
• As oscillator moves up and
down it launches waves along string. These
carry away oscillator energy & motion damped.
• Waves propagating along string towards
oscillator will deposit energy and excite it.
• If we give wave modes a thermal spectrum
oscillator is then connected to a heat bath
and the Langevin equation results
Lumped:

28. Microscopic model of Brownian motion …
• System interacting with oscillators, often known as Caldeira-Leggett model. Used in study of decoherence in
QM.
• Combined Lagrangian for system, interaction, reservoir (and a counter term):
• System
• See e.g. Caldeira, 2010 from where following slides are taken, and also Paul & Baschagel, 2013, section 3.3.
I R TS CL L L L L  
2
( )
1
2S MqL V q 

29. … Caldeira-Leggett model
• Interaction
• Reservoir
• Counter term
I k k kqqL C 
2 2 21 1
2 2R k k k k k km qL m q  
2
2
2
1
2
k
C k
k k
T
C
qL
m 
  

30. Solve via Euler-Lagrange equations …
• Force on system:
• Force on k-th oscillator of the reservoir:
2
2
( ) k
k k k k
k k
C
M Cq q
m
V q q

   
2
k k k k k kq qm m q C  

31. … and Laplace transforms
2 2
2 2 2
2
2 2 2 2
2
( )
( )
q whic
1 ( )
2
h be
(0) (0)1
co e
2
m s
i
stk k
F kP D
i
stk
i
k k k k
k
k k
k k k k
i
k k k
q s
F
C s q s
e
q
Mq V q
F F V q C e ds
i s s
C
C q
m
ds
m i s











 

 
 
 

   
   
 
 
  





 

Fluctuating force term
Dissipation term

32. The dissipation term and the spectral function
 
 
2
2 2 2
2
2 0
0 0
1 ( )
2
cos ( )
2 ( )
cos ( )
i
stk
D i
k k k k
t
k
k
k k k
t
Cd sq s
F e ds
dt m i s
Cd
t t q t dt
dt m
d J
d t t q t dt
dt

  



 
 
 
 
  

  
 
  
 
 
    
 
      
 
 
 
Above we went from sum over oscillators to an integral by defining a “spectral
function” J. Note J not spectral density, which is effectively J(omega)/omega):
 
2
( )
2
k
k
k k k
C
J
m

   

 

33. Simplify dissipation term: Ohmic ansatz for J(ω)
   
2
2 0
2
cos cos 2 ( )k
k
k k k
C
t t d t t t t
m
   
 

                    
 
Choose a form for the bath’s spectral function and then take the
limit of large cutoff frequency . Brick wall cutoff not necessary, can
also use a smooth exponential cutoff as discussed in Watkins and
Waxman [2004], and Caldeira & Leggett papers cited therein.
( ) if
and 0 if
J   

  
 
0
22 ( ) ( ') ' ( ) (0)
t
DF t t q t
d
qdt t q
dt
    
Linearity of damping in velocity is reason for name “Ohmic” c.f. Ohm’s law.

34. Simplify fluctuating force using equipartition
Assume each oscillator initially in equilibrium about
2
(0)
(0) and (0) (0) (0) 0k
k k k k
k k
C q
q q q q
m 
   
(0) (0) B
k k kk
k
k T
q q
m
   
(0) (0) B
k k kk
k
k T
q q
m
 
2
(0( )0) /k k k kq C q m 
Can use these, and our expression for
fluctuating force to show that :
(0) (0) (0)k k kq q q  
( ) 0 and
( ) ( ') 2 ( ') 0B
f t
f t t k t tf T 

  
Essentially a version of the FDT

35. Ohmic Langevin equation
• Note used hypothesis that environmental oscillators are in
equilibrium to get rid of spurious term
( ) ( )q tV fMq q 
  
2 ( ) (0)t q

36. Beyond the Ohmic case
More generally we can consider other types of spectral function including but not limited
to power laws :
( )
where s 1 is super-Ohmic
and s 1 is sub-Ohmic
s
J  


And in the presence of a memory in the heat bath we have the generalised
Langevin equation of the form:
.. .
0
'(q) ( ) ( ) ( )
t
q V M dt t t t f tM q      
Where memory kernel replaces constant eta [e.g. Kupferman, 2004 and Caldeira, 2010]

37. Fractional Langevin equation
(1 2 )
If memory kernel has slowest decay ( ) ~ d
   
.. .
0
(( ) ( )then GLE: M ' ) )(q
t
M dt t t f tV tq q     
..
(1 )
0
2
(1 2 ) 2
becomes FLE: M (q)
( ) 1
where frac. derivative is
(
( )( )
(
)
)
(
F
)
d
d d
t
fM
t
q t
d F
t
tq
t
t



 






   

 
  




V

38. Fractional Brownian motion
• Instead of being defined purely as a stochastic process, the
development in terms of an FLE allows some physical insight into
meaning of fractional Brownian motion.
• It is the noise term in the FLE we have just described [Kupferman,
2004; Lutz, 2001]
• If we want to allow non-Gaussian heavy tailed jumps we can replace
Gaussian steps in fBm by stable ones, to get linear fractional stable
motion (see e.g. refs in Watkins, 2013).

39. At least two ways to “break” classical diffusion
• One is explicitly non-Markovian,
via generalised Langevin
equation … (fractional motions)
• … Another is semi-martingale route.
Factorising CTRW is of this type,
modifies Brownian diffusion
by using subordination in time
(fractional kinetics).
[Weron and Magdziarz, 2008]

40. Semi-Martingale: Fractional Kinetics (FFCTRW)
• Keep a specified pdf (though no longer a stable one), and a (fractional)
diffusion equation [e.g. Klafter and Sokolov, 2011; Brockmann et al, 2006].
• Lose the ergodicity---the reservoir is explicitly nonequilibrium.
• Rather than one Langevin equation now have two coupled ones [e.g
Fogedby, 1994], -> CTRW.
• Need to have a second LE because this is BM subordinated to fractal
time.
• Can retain a lot of stochastic calculus methods & a fractional Taylor
expansion for the pdf.
• Is above true for any CTRW, or just the factorising CTRW ?
• Physical picture: “flights in sticky space and trapping time”.

41. 10/07/2014
One motivation for CTRW is dynamics of
Hamiltonian chaos where the environment is
not “just” hierarchical, like the power law
bath spectral density of the FLE, but also
spatially structured in the KAM sense:
“Chaotic dynamics can be considered as a
physical phenomenon that bridges the regular
evolution of systems with the random [case] …
What kind of kinetics should [there] be for
chaotic dynamics that is intermediate between
completely regular (integrable) and completely
random (noisy) cases ? … These are the subjects
of this paper, where the new concept of
fractional kinetics is reviewed for systems with
Hamiltonian chaos.” – Zaslavsky, Physics
Reports, 2002

42. Continuous Time Random Walk
(CTRW)-simulated discretely
10 July 2014 42
1
( ) i
n
i
X t 

 Impose iid random jumps
/
( )x
 
 
Impose iid random times
1
i
n
n
i
t 

 
Scale factors included
Notation as Fulger et al, PRE (2008)

43. Factorising CTRW
10 July 2014 43
( , ) ( ) ( )i P     
Can in principle study CTRW where the pdfs are
coupled, in practise a factorising ansatz is often made
for the pdf: again use Fulger et al’s notation

44. CTRW = renewal reward process
10 July 2014 44
Here jumps at {J}
become
rewards {W} and
waiting times
become holding
times {S}
CTRW useful as time series model provided one can define & measure events
at arbitrary times ?
Fixed sampling intervals t motivate different class e.g. fractional motions

45. Symptoms of complex transport: 2
10 July 2014 45
... longer waiting times

46. 10 July 2014 46

47. Fractional time process
10 July 2014 47
Jumps still Gaussian but
waiting times now come
from a heavy-tailed,
distribution.
Here use Mittag-Lefler pdf
with parameter β. When β=1
it becomes exponential.
taking β < 1 fattens the tail.
Mittag-Leffler waiting time ccdf, from Fulger et al, PRE (2008)

48. Reality is ambivalent
• Frequently one sees both heavy tailed jumps and waiting times-or at
least non-Gaussian ones ... Brockmann et al (2006) coined apt
phrase “ambivalence” for this property.
• Various models proposed. One is simply to have a decoupled CTRW
with heavy tails in both waiting time and jump size-known as the Fully
Fractional Continuous Time Random Walk (FFCTRW).
10 July 2014 48

49. FFCTRW traces
10 July 2014 49
0 0.5 1 1.5 2 2.5
x 10
8
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
x 10
4
Time
x,ycomponents
Time series
x
y
α=1.5, β=0.7
-14000 -12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
x 10
4
Spatial pattern
x
y
time series
space

50. Dollar bills [Brockmann, 2006]
10 July 2014 50
Data fitted to an FFCTRW with α, β both about 0.6

51. 10/07/2014
Zaslavsky, Physics Reports, 2002 (and book, Chapters 14 & 16) gave
argument for how a fractional Fokker-Planck equation could be obtained for
ambivalent processes. First consider standard F-P route:
3 3 1 1 2 3 3 2 2 2 2 1 1( ; ; ; ) ( ; ; ; ) ( ; ; ; )W x t x t dx W x t x t W x t x t 
( , ; ', ') ( , '; - ')W x t x t W x x t t
0
0 0
( , ; )
( , ; ) ( , ; )
W x x t
W x x t t W x x t t
t

    

0
0 0
0
( , ; )1
lim { ( , ; ) ( , ; )}
t
W x x t
W x x t t W x x t
t t 

  
 
0( , ) ( , ; )P x t W x x t

52. 10/07/2014
Zaslavsky, op. cit. … then makes fractional modification:
0
0 0
0
( , ; )1 ( , )
lim { ( , ; ) ( , ; )}
| |t
W x x t P x t
W x x t t W x x t
t t t
 
   
 
   
  
0
( , ) 1
lim { [ ( , ; ) ( )] ( , )}
( )t
P x t
dy W x y t t x y P y t
t t

 

 

   
  
( )
| |
P
P
t x
 
 
 

 

53. 10/07/2014
From supp. Info. of [Brockmann et al, 2006] comes very useful schematic-
NB they defined (,) opposite way to Zaslavsky 2002
Actually not fBm,
but rather in fact the
fractional time
process (FTP).
Don’t believe
everything you
read in Nature ;-)

54. CTRW vs fixed t models
10 July 2014 54
Table from Watkins et al, PRE, 2009. Here
2 1
2 1
[ ] 2
H
H
tt Ht


 

55. Why name “FF” CTRW
10 July 2014 55
FF CTRW is bottom right example, α, β are the
orders of fractional derivatives--- β= αH

56. Markovian: Nonlinear FP
• Markovian or at least local in time: Can no longer have stable pdf
solution because D is changing in space or time. Keep duality of
diffusion equation and Langevin - > nonlinear Fokker-Planck
equations. Are these in semi-martingale class ?
• Examples:
Wheatcraft and Tyler, 1988; Bassler et al, 2006
Hnat et al, 2003
2
2
( , ) ( ) ( , )P x t D t P x t
t x
 

 
2
2
( , ) (x) ( , )P x t D P x t
t x
 

 

57. Levy walk: couples space & time
10 July 2014 57
Gives a finite velocity by introducing a jump duration τ’ & coupling the jump size to
it – idea known as Lévy walk [Shlesinger & Klafter, PRL (1985)]. Obviously can
be done in many ways, simplest is just to make size proportional to duration.
( , ') ( '| ) ( )
(| | ') ( )
      
    
 
 
Lévy walk
( , ) ( ) ( )      Uncoupled CTRW
In above τ’ means flight duration in Levy walk, and τau waiting time in CTRW.
Deliberately changed notation for these quite different quantities. In
foraging & other literature, walks & flights, durations & waiting times often
treated in a very cavalier fashion – NOT SAFE TO DO SO !!!

58. Turbulent cascade processes
Many systems have
aggregation, but not by an
additive route. Classic
example is turbulence.

59. SOC

60. Data-inspired question:
• How much of any given complex system’s time series properties can be
predicted from just a few parameters e.g. from its pdf and power spectral
exponents ? Examples include wind power paper, solar wind, AE etc.
• One motivation for why this might be possible is the effective collapse onto
a few degrees of freedom used not only in LD chaos, but also in the
synergetics idea.
• In particular, how much does this hopefully simple parameterisation
then govern the extremes ?
• Begs a “physics” question: are these parameters actually physical in origin
? Or more to do with the aggregation and measurement processes ?

61. Sources Used I:
• Bassler, K., et al, “Markov processes, Hurst exponents, and nonlinear diffusion equations”, Physica A 369, 343,
2006 [and subsequent exchanges of comment(s) with Frank].
• Bouchaud, J.-P., and M. Potters, “Theory of financial risk and derivative pricing”, 2nd edition,CUP, 2003.
• Brockmann, D., et al, “The scaling laws of human travel”, Nature, 439, 462, 2006.
• Caldeira, A. O., “Caldeira-Leggett model”, Scholarpedia, 2010.
• Fogedby, H. C., “Langevin equations for continuous time Levy flights”, PRE, 50, 1657, 1994.
• Fulger, D., et al, “Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic
solution of the space-time fractional diffusion equation”, PRE, 77, 021122, 2008
• Hnat, B., et al, “Intermittency, scaling, and the Fokker-Planck approach to fluctuations of the solar wind bulk
plasma parameters as seen by the WIND spacecraft”, PRE, 67, 056404, 2003
• Lemons, D., “An introduction to stochastic processes in physics”, Johns Hopkins, 2002.
• Mantegna, R., and H. E. Stanley, ”Introduction to Econophysics”, CUP, 2000.
• Kupferman, R., “Fractional kinetics in Kac-Zwanzig heat bath models”, J. Stat. Phys., 114, 291, 2004.
• Lutz, E., “Fractional Langevin equation”, Phys. Rev. E, 64, 051106, 2001.

62. Sources Used II:
• Paul, W., and J. Baschnagel, “Stochastic Processes from Physics to Finance, 2nd Edition”, Springer , 2013.
• Klafter, J., and I. M. Sokolov, “First Steps in Random Walks”, OUP, 2011.
• Shlesinger, M. F. and J. Klafter, Phys. Rev. Lett., 54, 2551, 1985.
• Sornette, D., “Critical phenomena in natural sciences”, 2nd edition, Springer, 2004.
• Watkins, N. W. and Waxman, D., “Path integral derivation of Bloch-Redfield equations for a qubit weakly
coupled to a heat bath: application to nonadiabatic transitions”, arXiv:cond-mat/0411443v1, 2004.
• Watkins, N. W., et al, “Kinetic equation of linear fractional stable motion and applications to modelling the
scaling of intermittent bursts”, PRE, 79, 041124, 2009.
• Watkins, N. W., “Bunched black (and grouped grey) swans: dissipative and non-dissipative models of correlated
extreme fluctuations in complex systems”, GRL , 40, 1-9, doi:10.1002/GRL.50103, 2013.
• Weron, A., and M. Magdziarz, “Anomalous diffusion and semi-martingales”, EPL, 86, 60010, 2009
• Wheatcraft, S. W. and S. W. Tyler, “An Explanation of Scale-Dependent Dispersivity in Heterogeneous Aquifers
Using Concepts of Fractal Geometry”, Water Resour. Res., 24, 566, 1988
• Yurke, B., “Conservative model for the damped harmonic oscillator”, Am. J. Phys., 52(12), 1099, 1984
• Zaslavsky, G., “Chaos, fractional kinetics, and anomalous transport” , Physics Reports, 371, 461, 2002 (and his
book, OUP, 2005)