Power law processes and nonextensive statistical mechanics: some possible hydrological interpretations.

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Power law processes and nonextensive statistical
mechanics some possible hydrological
interpretations.

• Chris Keylock

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• Talk Outline
• Classical and nonextensive statistical mechanics
• Power-law processes in drainage networks
• Applying nonextensive statistical mechanics to
  historical flood data the case of the Po river
• Hurst exponents and hydrologic time-series
• TOPMODEL and spatial distributiveness

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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.
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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. Krylov N. 1944. Nature 153,
709.
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At the heart of classical thermodynamics is the
Boltzmann-Gibbs entropy This has a maximum
when all states pi have equal probability. If we
apply two simple constraints (normalization and
mean value of the energy) We obtain the
distribution function
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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-organization. Cambridge University Press.
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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.
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Data from Mazzarella A, Rapetti F. 2004. Events
with ASI gt 4 R is the rank of each event and I
is the recurrence interval in years
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Data from Mazzarella A, Rapetti F. 2004. Events
with ASI gt 3 R is the rank of each event and I
is the recurrence interval in years
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An escort distribution for a pdf is used to study
the properties of the pdf in statistical
mechanics. It is defined by We also define
averages (q-expectations) rather differently in
this statistical mechanics
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If we take the Tsallis entropy The
q-expectation And a basic probability
constraint and maximise the entropy subject to
these constraints we get
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We can then re-express this As an escort
distribution Making use of the q-exponential
definition To give
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Making use of the following We can integrate
to get the cumulative distribution
function 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)
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From Keylock C.J. 2005. Describing the
recurrence interval of extreme floods using
nonextensive thermodynamics and Tsallis
statistics. Adv. Wat. Res. In press.
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From Keylock C.J. 2005. Describing the
recurrence interval of extreme floods using
nonextensive thermodynamics and Tsallis
statistics. Adv. Wat. Res. In press.
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There is a direct connection between this
entropic form and correlated anomalous diffusion
that is described by nonlinear Fokker-Planck
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 1998576634-6642.
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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.
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Borland (1998) explores the behaviour of a
non-linear Fokker-Planck equation K drift
coefficient, Q is the diffusion constant, ? 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
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The time-dependent solution to this non-linear
Fokker-Planck equation with linear drift
is where Zq normalises the distribution (it is
given as the integral of the expression on the
numerator). Recall, that the Tsallis distribution
is Which is actually the steady-state solution
to the equation.
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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 lt
2 is Hence, H 0.7 in hydrology implies q
1.57.
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An important part of the classic version of
TOPMODEL is the topographic index given by Ln (a
/ tan ß) where a is the upslope area per unit
contour length and tan ß is the surface slope.
(e.g. Lane et al., 2004 for a modification). Rece
ntly, 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.
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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.