Fractional Calculus

1

• Fractional Calculus
• Modeling
• Physiologic Networks
• 6th Symposium on Complex Systems
• Department of Physics
• University of Illinois at Urbana-Champaign
• May 15, 2006
• Bruce J. West
• Mathematical Information
• Sciences Directorate
• Army Research Office
• Research Triangle Park, NC

2006
2006
Collaborators Mirek Latka Nicola Scafetta
2
Outline of Talk

• Set the stage
• some history
• Physics and statistics
• where it began
• when the mean is not enough
• inverse power laws
• Modeling complexity
• fractional operators
• evolving fractal processes
• fractional Langevin equations
• Applications of the fractional calculus
• multifractality
• headaches
• walking

Gauss
Asymmetric Lévy
Inverse power law
(Pareto)
3
Modeling complexity in physics (history)

• Mandelbrot
• Fractional Random walks 81
• Anomalous diffusion 75
• Fractional Langevin equation 63
• Fractional diffusion equation 83
• Lévy statistics
• Anomalous diffusion

• Newton
• Random walks 05
• Classical diffusion 05
• Langevin equation 16
• Fokker-Planck equation 17
• Gaussian statistics
• Normal diffusion

4

• The simplest statistical models
• importance of scaling
• random walks
• Langevin equation
• Fokker-Planck equation

5

• The scaling analysis is based on the joint use
  of the Diffusion Entropy Analysis (DEA) and the
  Standard Deviation Analysis (SDA) of the
  diffusion process with Probability Density
  Function p(x,t).

There are two scaling exponents H d
SDA
DEA
Simple phenomena
Complex phenomena
6
Simple Random Walks

• Simple many particle systems are understood in
  terms of statistical properties and in physics
  this starts with a simple random walk, circa
  1905.
• no memory
• delta correlated, Gaussian jumps
• random-walk variable
• second-moment properties (ordinary diffusion)

Gaussian Noise
Sub-trajectories
Brownian trajectory
Scaling H d 0.5
time
7
Continuum Limit of Simple Random Walk

• The continuum form of the random walk has two
  limits, the dynamic equation and the phase space
  equation.
• Langevin equation (1908)
• Phase space (diffusion) equation
• Gaussian statistics

dissipation
random force
Brownian motion
Exponential relaxation
FPE
Scaling H d 0.5
Log-log spectrum
8

• Complexity one definition

9
One spectrum of complexity
Renormalization groups Scaling Fractals Partial
differential equations Random
Computational complexity Chaos Nonlinear
dynamics Hamiltonians Operator dynamics Control
theory
10

• Complexity and inverse power laws in the social
  and life sciences

11
Paretos Children

• Paretos
  Auerbachs
• Law (1890)
  Law (1913)
• money
  
  cities
• Willis
  Zipfs
• Law (1912)
  Law (1949)

12
Paretos Children

• physiology
  science
• BJW AG
  publications
• (J. Appl. Physiol. 1986)
  D. Price (1963)
• (Am. Sci. 1987)
• lungs
  
  of citations
  
  
• internet
  trial
• M. Buchanan
  
  error
• Nexus (2002)
  A. Newell
• 
  
  Unified Theories
• of Cognition
• (1990)
  

13
Pareto inverse power law (circa 1900)
Gauss bell curve (circa 1800)

• Simple scientific world view
• linear output is proportional to input
• additive
• simple rules yield simple results
• stable
• predictable
• quantitative
• normal distribution

• Complex scientific world view
• nonlinear small changes may diverge
• multiplicative
• simple rules yield complex results
• unstable
• limited predictability
• qualitative plus quantitative
• inverse power law distributions

14
Chance and change simple inverse power law
1985
additive fluctuations
1990
multiplicative fluctuations
Steady-state distributions
Pareto inverse power-law distribution
Gauss distribution
15
1990
1985
1995
1994
In modern jargon the inverse power law
distribution is explained in terms of fractals.
But how do we handle the dynamics of fractal
processes?
2006
2006
1999
2003
2004
16
Fractional Random Walks

• Another way to incorporate complexity is
  using memory in random walks through
  fractional differences (Fractional
  Gaussian Noise H¹0.5).
• fractional random walk Hosking (1983)
• solution of discrete equation is (infinitely
  long coupling)
• correlation function is inverse power law
• H measures long-range memory statistics remain
  Gaussian

1999
Persistent noise H0.75
Fractional Brownian trajectory
DEA d0.75
Scaling H d 0.75
SDA H0.75
time
17
Continuum Limit of Fractional RWM

• The continuum limit of fractional differences
  are fractional differentials and fractional
  integrals.
• Continuum limit of fractional random walk is a
  fractional Langevin equation (BJW, MB and PG,
  Physics of Fractal Operators, 2003)
• mean-square displacement
• sub-diffusive process

2003
fractional integral
random force
fractional derivative
Initial condition
Hgt0.5
H0.5
Hlt0.5
anomalous diffusion
18
Derivatives of fractal functions

• Generalized Weierstrass function
• Renormalization group relation implies complex
  scaling index
• Fractal dimension
• Derivative diverges
• Riemann-Liouville operator

2003
new fractal dimensions
Rocco BJW Physica A 265 (1999)
19
Fractional Brownian motion

• Wiener Process
• with autocorrelation function
• Fractionally integrated process
• with autocorrelation function
• compared with the Hurst exponent
• Thus, the equation of motion is a fractional
  Langevin equation

1999
2003
20
General observations

• Many complex systems can be represented by
  fractal functions.
• If is a fractal function with fractal
  dimension D then the fractional derivative of the
  function has a new fractal dimension
  .
• The dynamics of a fractal function can be
  described by a fractional equation of evolution.
• If the statistics of a process are non-local in
  space and/or in time then the evolution of the
  probability density obeys a fractional diffusion
  equation.

21

• Fractal time series

22
Arterial blood pressure variability
Body Temperature variability
Heart rate variability HRV
Breath rate Variability BRV
Stride rate variability SRV
23
Fractal Heart Beats
Heart beat time series

• Fractional diffusion equation (no FPE) with
  Seshadri (1983)
• Lévy distribution (symmetric solution)
• scaling
• fractal dimension
• asymptotic form (inverse power law)

Heart beat histograms
24
Pathological Breakdown of fractal dynamics

• Healthy heart rate
• multiple scales
• long-range order
• fractal time series
• (A.L. Goldberger,
• Lancet 347, 1312, 1996)
• Correlation index

Healthy dynamics
Decreased correlation
Increased correlation
Uncorrelated randomness atrial fibrillation
Single scale heart failure
25

• Multifractal time series

26
Multifractality of Cerebral Blood Flow
Fractional Langevin equation describes the blood
flow through the middle cerebral artery BJW,
Latka, Latka Latka (2003) scaling q-mome
nts, when is a Gaussian random
variable multifractal distribution of Hölder
exponents h Similar results are obtained for
heartbeat variability after heart attacks.
normals
migrainures
27
Multifractality and human gait
Fractional Langevin equation describes
variability in the stride interval in walking BJW
Latka, J. NeuroEng. Rehab. 2 2
(2006) scaling q-moments, when is a
Lévy random variable multifractal distribution
of q-moments .
b 0.03 ? 1.5
28
Conclusions
2006
2006
2004

• Complex phenomena (regular random) cannot be
  described by differential equations of motion for
  either the dynamical variable or the probability
  density.
• The scaling properties of many physiologic
  systems indicate an underlying fractal structure,
  either in the geometrical structure or in the
  statistical behavior.
• The evolution of fractal statistical processes
  in time can be described by fractional Langevin
  equations and/or fractional diffusion equations.
• The fractional calculus can describe the
  dynamics of many of the complex stochastic
  phenomena observed in medicine for which
  traditional methods are inadequate.

2003
1999