Power law processes and
nonextensive statistical
mechanics: some possible
hydrological interpretations.
Chris Keylock
Talk Outline
(1)Classical and nonextensive statistical
mechanics
(2)Power-law processes in drainage networks
(3)Applying nonextensive statistical mechanics to
historical flood data – the case of the Po river
(4)Hurst exponents and hydrologic time-series
(5)TOPMODEL and spatial distributiveness
Classical statistical mechanics due to Boltzmann
and Gibbs is one of the cornerstones of
modern physics.
Boltzmann (1867) linked macroscopic properties
of a phenomenon (temperature, entropy) to
microscopic probabilities to lay the foundations
for the contemporary subdiscipline of statistical
mechanics.
Subsequently, this has become an extremely
powerful tool for a variety of problems.
Determining the range of validity of this
thermodynamics is not a simple problem.
Krylov (1944) suggested that the property of
mixing was at the heart of the this question.
Quick (exponential) mixing leads to a short
relaxation time. It results in strongly chaotic
behaviour (positive Lyapunov exponent).
However, one can also have slower (power-law)
mixing processes that are weakly chaotic.
The difference emerges because of short (long)
ranged interactions and smooth (complex i.e.
fractal/multifractal) boundary conditions.
At the heart of classical thermodynamics is the
Boltzmann-Gibbs entropy:
n
S   k B  p i ln p i
i 1
This has a maximum when all states pi have
equal probability.
If we apply two simple constraints (normalization
and mean value of the energy):

 p   d  1
0

  p  d  const
0
We obtain the distribution function:
pi 
e i
n
e
i 1
 i
However, in river systems there is a great deal of
evidence for power-law and multifractal
behaviour rather than exponential.
There is a whole book that describes a variety of
aspects of this (Rodriguez-Iturbe and Rinaldo,
1997).
In particular, perhaps the network width function
is important in this context. It gives the number
of links in the basin as a function of distance
from the outlet and has been shown to be
multifractal. The width function has been used
for flood prediction and for defining the
geomorphic instantaneous unit hydrograph.
Rodriguez-Iturbe I., Rinaldo I. 1997. Fractal River Basins: Chance and self-
6
ASI
5
4
3
1750
1800
1850
1900
1950
2000
Year
Data from: Mazzarella A, Rapetti F. 2004. Scale-invariance laws in the recurrence
interval of extreme floods: an application to the upper Po river valley (northern
Italy). J Hydrol 288, 264-271.
2
1.5
log I (Years)
1
0.5
0
-0.5
-1
-1.5
0
0.2
0.4
0.6
0.8
1
log R
Data from: Mazzarella A, Rapetti F. 2004. Events with ASI >= 4
R is the rank of each event and I is the recurrence interval in years
1.2
2
1.5
log I (years)
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
log R
Data from: Mazzarella A, Rapetti F. 2004. Events with ASI >= 3
R is the rank of each event and I is the recurrence interval in years
1.6
An escort distribution for a pdf is used to study
the properties of the pdf in statistical
mechanics. It is defined by:
P
esc
I   
pI 
 pI

q
/ q
dI /
0
We also define averages (q-expectations) rather
differently in this statistical mechanics:

I q   IP  I  dI
q
0
If we take the Tsallis entropy:
1
Sq 
1 q
   p  I  1 dI 

q
0
n
S   k B  p i ln p i
The q-expectation:

I q   IP  I  dI
q
0
And a basic probability constraint:
i 1

  p  d  const
0


0
0
 p   d  1
 p   d  1
and maximise the entropy subject to these
constraints we get:
 i
I / I
e
eq
1/ 1 q 
x
pi  n
popt  I   
eq  1  1  q  x 
 i
I / I
/
e

 eq dI
0
/
0
0
i 1
We can then re-express this
popt  I  
eq I / I0

e
 I / / I0
q
dI /
0
As an escort distribution
P
esc
I   
pI 
 pI

q
/ q
dI /
0
Making use of the q-exponential definition:
1/ 1 q 
e  1  1  q  x 
x
q
To give:
Popt
esc
I   
1  1  q  I / I 0 
/

1

1

q
I
/ I 0 



0
 x ; q  
q /(1 q )
q /(1 q )
dI /
Making use of the following:
d x
x q
eq   eq 
dx
We can integrate
Popt
esc
I   
1  1  q  I / I 0 
 1  1  q  I / I 0 
/
q /(1 q )
q /(1 q )
dI /
0
to get the cumulative distribution function
P   I   eq
 I / I0
Which we can fit to the data by maximum-likelihood
methods. Here I fit q but fix I0 as the median of the
distribution (a more difficult fit than 2 parameters)
(a)
0.9
0.8
74 1
0.7
Negative
log-likelihood
Predicted
exceedance
probability
Cumulative exceedance probability
1
(b)
72
0.8
0.6
0.5
70
0.6
0.4
68
0.3
0.2
0.4
66
0.1
0
0
5
10
15
0.2
64
20
25
30
35
40
Observed recurrence interval (years)
0
62 0
1
0.2
1.5
0.4
2
0.6
2.5
0.8
Observed exceedance probability
q
From: Keylock C.J. 2005. Describing the recurrence interval of extreme floods
using nonextensive thermodynamics and Tsallis statistics. Adv. Wat. Res. In
press.
3
1
2
2
1.5
1.5
log I (Years)
1
log I (years)
1
0.5
0.5
0
-0.5
0
-1
-0.5
-1.5
0
0.2
0.4
0.6
0.8
1
log R
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
log R
From: Keylock C.J. 2005. Describing the recurrence interval of extreme floods
using nonextensive thermodynamics and Tsallis statistics. Adv. Wat. Res. In
press.
1.2
There is a direct connection between this
entropic form and correlated anomalous
diffusion that is described by nonlinear FokkerPlanck equations (Borland, 1998).
Consequently, data that can be described by the
q-exponential distribution may be generated by
the relevant underpinning differential equation.
Borland L. Microscopic dynamics of the nonlinear Fokker-Planck equation: A
phenomenological model. Phys Rev E 1998;57:6634-6642.
Correlated diffusive processes have been
previously considered in a hydrological context
when examining the Hurst effect, which leads
to persistence in hydrologic time-series.
Systems with no memory have a value for the
Hurst exponent H of 0.5. However, as noted by
Klemes [1974], typical values for hydrologic
time-series appear to be H  0.7, implying a
fractal structure to the data, memory or a
steadily changing mean.
Kirkby (1987) discussed the implications of this
for the extrapolation of process rates – ‘tricky’.
Kirkby M.J. 1987. The Hurst effect and its implications for extrapolating
process rates. Earth Surface Processes and Landforms 12, 57-67.
Borland (1998) explores the behaviour of a nonlinear Fokker-Planck equation:
df
d
d2
   Kf   Q 2  f  
dt
dx
dx
K = drift coefficient, Q is the diffusion constant, ν
is a real number that introduces the
nonlinearity, f is a probability distribution.
An equation is then derived (Langevin equation)
for the actual trajectories of the system, which
are a function of f.
Borland L. 1998. Microscopic dynamics of the nonlinear Fokker-Planck
equation: A phenomenological model. Phys. Rev. E 57, 6634-442
The time-dependent solution to this non-linear
Fokker-Planck equation with linear drift is:


2

1    t 1  q  x  xM  t  


f  x, t  
Zq
1
1q
where Zq normalises the distribution (it is given
as the integral of the expression on the
numerator).
Recall, that the Tsallis distribution is:
p  x
1  (1  q) x 


1
1q
Zq
H
1
3 q
Which is actually the steady-state solution to the
equation.
Hence, this non-linear Fokker-Planck equation
can be linked to q.
More traditionally, anomalous diffusion is
described by a kind of brownian motion with
memory (fractional brownian motion), which
involves the Hurst exponent.
Both equations are applicable to these types of
problems and the relation between the Hurst
exponent and q for q < 2 is:
1
H
3 q
Hence, H = 0.7 in hydrology implies q = 1.57.
An important part of the classic version of
TOPMODEL is the topographic index given by
Ln (a / tan β) where a is the upslope area per unit
contour length and tan β is the surface slope.
(e.g. Lane et al., 2004 for a modification).
Recently, Ambroise et al. (1996) and others have
proposed generalisations of this to deal with
different catchment characteristics (e.g.
parabolic and linear transmissivity profiles).
Ambroise B., Beven K., Freer J. 1996. Toward a generalization of the
TOPMODEL concepts: Topographic indices of hydrological similarity,
Water Resources Research 32, 2135-2145.
Lane SN, Brookes CJ, Kirkby MJ, Holden J. 2004. A network-index based
version of TOPMODEL for use with high-resolution digital topographic
data. Hydrological Processes 18, 191-201.
Kirkby (1997) argues that a move away from the
exponential treatment of the relation between
discharge and soil moisture may mean that
spatially explicit solutions to the equations for
TOPMODEL become necessary.
Based on the framework presented here, it may
be possible to construct the same argument
from a different starting point. Unless one uses
the logarithmic form of the index, you are
implying a slow mixing of an appropriate
intensive quantity of the hydrological system.
The associated complexity introduced by this
slow mixing implies a spatially explicit solution.
Kirkby M.J. 1997. TOPMODEL: A personal view. Hydrological Processes 11,
1087-97.