Publications in scientific and technical journals
Number of citations according to ISI - web of Science is listed in curly brackets.
[51] O. Peters, K. Christensen, and J.D. Neelin
Rainfall and Dragon Kings.
Euro. Phys. J. To be publised in Special Topics (2011). {0}
[50] T.O. Richardson, K. Christensen,N.R. Franks, H.J. Jensen, and A.B. Sendova-Franks
Ants in a Labyrinth: A Statistical Mechanics Approach to the Division of Labour.
PLoS One6, e18416 (2011). {0}
[49] T.O. Richardson, K. Christensen,N.R. Franks, H.J. Jensen, and A.B. Sendova-Franks
Group dynamics and record signals in the ant Temnothorax albipennis.
J.R. Soc. Interface8, 518–528 (2011). {1}
[48] T.O. Richardson, E.J.H. Robinson, K. Christensen, H.J. Jensen, N.R. Franks, and A.B. Sendova-Franks
Comment on P. Nouvellet, JP Bacon, D. Waxman, ”Testing the level of ant activity associated with quorum
sensing: An empirical approach leading to the establishment and test of a null-model”.
J. Theor. Biol.269, 356–358 (2011). {2}
[47] A. Espinosa, P.P. Cardoso, E. Arcaute, and K. Christensen
Complexity approaches to self-organisation: a case study from an Irish eco-village.
Kybernetes40, 536–558 (2011). {0}
[46] P. Expert P, R. Lambiotte, D.R. Chialvo, K. Christensen, H.J. Jensen, D.J. Sharp, and F. Turkheimer
Self-similar correlation function in brain resting-state functional magnetic resonance imaging.
J.R. Soc. Interface8, 472–479 (2011). {0}
[45] T.O. Richardson, E.J.H. Robinson, K. Christensen, H.J. Jensen, N.R. Franks, and A.B. Sendova-Franks,
Record dynamics in ants.
PLoS One5, e9621 (2010). {4}
[44] E. Arcaute, K. Christensen, A.B. Sendova-Franks, T. Dahl, A. Espinosa, and H.J. Jensen
Division of labour in ant colonies in terms of attractive fields.
Ecological Complexity6, 396–402 (2009). {2}
[43] V. Pancaldi, P.R. King, and K. Christensen
Hierarchical coarse-graining transform.
Phys. Rev. E79, 036704 (2009). {0}
[42] V. Pancaldi, P.R. King, and K. Christensen
Wavelet-based upscaling of advection equations.
Physics A387, 4760–4770 (2008). {0}
[41] K. Christensen, N. Farid, G. Pruessner, and M. Stapleton
On the scaling of probability density functions with apparent power-law exponent less than unity.
Eur. Phys. J. B 62, 331–336 (2008). {1}
[40] P. Welinder, G. Pruessenr, and K. Christensen
Multiscaling in the Sequence of Areas Enclosed by Coalescing Random Walkers.
New J. Phys. 9, 149-166 (2007). {2}
[39] V. Pancaldi, K. Christensen, and P. King
Permeability Upscaling using Haar Wavelets.
Transp Porous Med 67, No. 3, 395-412 (2007). {2}
[38] N. Farid and K. Christensen,
Evolving networks through deletion and duplication.
New J. Phys. 8, 212-229 (2006). {5}
[37] M. Stapleton and K. Christensen,
One-dimensional Directed Sandpile Models and the Area under a Brownian Curve.
J. Phys. A 39, 9107-9126 (2006). {4}
[36] A. Corral and K. Christensen
Comment on "Earthquakes descaled".
Phys. Rev. Lett. 96, 109801 (2006). {15}
[35] M. Stapleton and K. Christensen,
Universality Class of One-Dimensional Directed Sandpile Models.
Phys. Rev. E 72, 066103, 1-4 (2005). {2}
[34] C. Kamp and K. Christensen,
Spectral analysis of Protein-Protein Interactions in Drosophila melanogaster
Phys. Rev. E 71, 041911, 1-8 (2005). {7}
[33] K. Christensen, N.R. Moloney, O. Peters, and G. Pruessner,
Avalanche Behavior in an Absorbing State Oslo Model
Phys. Rev. E 70, 067101, 1-4 (2004). {8}
[32] M. Stapleton, M. Dingler and K. Christensen,
Sensitivity to initial conditions in self-organised critical systems
J. Stat. Phys. 117, 891-900 (2004). {0}
[31] K. Christensen,
On self-organised criticality in one dimension
Physica A. 340, Issue 4, 527-534 (2004). {5}
[30] O. Peters and K. Christensen,
Rain Viewed as Relaxational Events
Journal of Hydrology 328, Issues 1-2, 46-55 (2006). {12}
[29] S.A. di Collobiano, K. Christensen, and H.J. Jensen,
The Tangled Nature model as an Evolving Quasi-species Model.
J. Phys. A 36, 883-891 (2003). {27}
[28] O. Peters and K. Christensen,
Rain: Relaxations in the Sky.
Phys. Rev. E 66, 036120, 1-9 (2002). {34}
[27] M. Hall, K. Christensen, S.A. Collobiano, and H.J. Jensen,
Time Dependent Extinction Rate and Species Abundance in a Tangled-Nature Model of ...
Phys. Rev. E 66, 011904, 1-10 (2002). {43}
[26] P. Bak, K. Christensen, L. Danon, and T. Scanlon,
Unified Scaling Law for Earthquakes.
Phys. Rev. Lett. 88, 178501, 1-4 (2002). {217}
Featured in Scientific American news scan article "Scaling the Quakes", June 2002.
[25] K. Christensen, H.J. Jensen, M. Hall, and S.A. Collobiano,
Tangled Nature: A Model of Evolutionary Ecology.
J. Theor. Biol. 216, 73-84 (2002). {50}
[24] K. Christensen, L. Danon, T. Scanlon, and P. Bak,
Unified Scaling Law for Earthquakes.
Proc. Natl. Acad. Sci. USA. 99, 2509-2513 (2002). {43}
[23] S.T. Bramwell, K. Christensen, J.-Y. Fortin, P.C.W. Holdsworth,
H.J. Jensen, S. Lise,
J.M. Lopez, M. Nicodemi, J.-F. Pinton, and M. Sellitto,
Reply to "Comment on 'Universal Fluctuations in Correlated Systems'".
Phys. Rev. Lett. 89, 208902, 1 (2002). {11}
Comment on "Universal Fluctuations in Correlated Systems".
Phys. Rev. Lett. 89, 208901, 1 (2002).
[22] O. Peters, C. Hertlein, and K. Christensen,
A complexity view of rainfall.
Phys. Rev. Lett. 88, 018701, 1-4 (2002). {68}
Featured in United Press International news article "Rainfall and earthquakes may be alike", February 2002.
Featured in New Scientist news article "Rain is 'earthquake in the sky'", May 2002.
Award-winning poster presented at "The Physics Congress", Brighton, UK, April 2002.
[21] S.T. Bramwell, K. Christensen, J.-Y. Fortin, P.C.W. Holdsworth,
H.J. Jensen, S. Lise,
J.M. Lopez, M. Nicodemi, J.-F. Pinton, and M. Sellitto,
Reply to "Comment on 'Universal Fluctuations in Correlated Systems'".
Phys. Rev. Lett. 87, 188902, 1 (2001). {7}
Comment on "Universal Fluctuations in Correlated Systems".
Phys. Rev. Lett. 87, 188901, 1 (2001).
[20] K. Christensen, D. Hamon, H.J. Jensen, and S. Lise,
Comment on "Self-organized Criticality in the Olami-Feder-Christensen Model".
Phys. Rev. Lett. 87, 039801, 1 (2001). {18}
"Self-Organized Criticality in the Olami-Feder-Christensen Model".
Phys. Rev. Lett. 84, 4006-4009, (2000).
[19] K. Christensen and H.J. Jensen,
Mathematical Modelling of Species Formation.
Science Prog. 83, 93-106 (2000). {?}
[18] S.T. Bramwell, K. Christensen, J.-Y. Fortin, P.C.W. Holdsworth,
H.J. Jensen, S. Lise,
J.M. Lopez, M. Nicodemi, J.-F. Pinton, and M. Sellitto,
Universal Fluctuations in Correlated Systems.
Phys. Rev. Lett. 84, 3744-3747 (2000). {152}
[17] A.S. Datta, K. Christensen, and H.J. Jensen,
On the Physical Relevance of Extremal Dynamics.
Europhys. Lett. 50, 162-168 (2000). {9}
[16] A. Malthe-Sørenssen, J. Feder, K. Christensen, V. Frette, and T. Jøssang,
Surface Fluctuations and Correlations in a Pile of Rice.
Phys. Rev. Lett. 83, 764-767 (1999). {12}
[15] K. Christensen, R. Donangelo, B. Koiller, and K. Sneppen,
Evolution of Random Networks.
Phys. Rev. Lett. 81, 2380-2383 (1998). {32}
[14] K. Christensen, A. Corral, V. Frette, J. Feder, and T. Jøssang,
Tracer Dispersion in a Self-Organized Critical System.
Phys. Rev. Lett. 77, 107-110 (1996). {135}
[13] T. Rage, V. Frette, G. Wagner, T. Walmann, K. Christensen, and T. Sun,
Construction of a DLA Cluster Model.
Eur. J. Phys. 17, 110-115 (1996). {?}
[12] C.J. Perez, A. Corral, A.D.-Guilera, K. Christensen, and A. Arenas,
On Self-Organized Criticality and Synchronization in Lattice Models of Coupled ...
Int. J. Mod. Phys. B 10, Nos. 8 & 9, 1-41 (1996). {27}
[11] V. Frette, K. Christensen, A. Malthe-Sørenssen, J. Feder, T. Jøssang, and P. Meakin,
Avalanche Dynamics in a Pile of Rice.
Nature 379, 49-52 (1996). {288}
Featured in
Nature 379, 22 (1996) news and views "Avalanche theory in rice" by M. Kardar.
Physikalishe Blätter 52, 203 (1996) "Langkorn- oder Kurzkornreis" by S. Luding.
[10] K. Christensen, H. Flyvbjerg, and Z. Olami,
Self-Organized Critical Forest-Fire Model: Mean-field Theory and Simulation Results ...
Phys. Rev. Lett. 71, 2737-2740 (1993). {74}
[09] K. Christensen,
Reply to "Comment on 'Self-Organized Criticality in a Continuous, Nonconservative ...'".
Phys. Rev. Lett. 71, 1289 (1993). {9}
Comment on "Self-Organized Criticality in a Continuous, Nonconservative ...".
Phys. Rev. Lett. 71, 1288, (1993).
[08] K. Christensen and Z. Olami,
Sandpile Models with and without an Underlying Spatial Structure.
Phys. Rev. E 48, 3361-3372 (1993). {65}
[07] K. Christensen and Z. Olami,
Scaling, Phase Transitions, and Nonuniversality in a Self-Organized Critical Cellular ...
Phys. Rev. A 46, 1829-1838 (1992). {142}
[06] Z. Olami and K. Christensen,
Temporal Correrations, Universality, and Multifractality in Spring-Block Model ...
Phys. Rev. A 46, R1720-R1723 (1992). {28}
[05] K. Christensen, Z. Olami, and P. Bak,
Deterministic 1/f Noise in Nonconservative Models of Self-Organized Criticality.
Phys. Rev. Lett. 68, 2417-2420 (1992). {73}
[04] K. Christensen and Z. Olami,
Variation of the Gutenberg-Richter b Values and Nontrivial Temporal Correlations in ...
J. Geophys. Res. 97, 8729-8735 (1992). {72}
[03] Z. Olami, H.J.S. Feder, and K. Christensen,
Self-Organized Criticality in a Continuous, Nonconservative Cellular Automaton ...
Phys. Rev. Lett. 68, 1244-1247 (1992). {548}
[02] K. Christensen, H.C. Fogedby, and H.J. Jensen,
Dynamical and Spatial Aspects of Sandpile Cellular Automata.
J. Stat. Phys. 63, 653-684 (1991). {53}
[01] H.J. Jensen, H.C. Fogedby, and K. Christensen,
1/f Noise, Distribution of Lifetimes, and a Pile of Sand.
Phys. Rev. B 40, R7425-R7427 (1989). {135}
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Book
K. Christensen and Nicholas R. Moloney,
Complexity and Criticality
List of Content
Published October 2005 by Imperial College Press
Summary:
The book provides a challenging and stimulating introduction to the
contemporary topics of complexity and criticality, and
explores their common basis of scale invariance, a central unifying theme of the book.
The word `complexity' takes on a variety of meanings depending on the context, and its official definition
is continuously being revised. This is because the
study of complexity is in its infancy and is a rapidly developing field
at the forefront of many areas of science including mathematics, physics, geophysics,
economics and biology, to name just a few.
Institutes and departments have been
formed, conferences and workshops organised,
books and countless articles written, all in the name of
complexity. And yet, nobody agrees on a clear and concise theoretical
formalism with which to study complexity. The danger is therefore that
complexity research may become unstructured or even misleading.
For our purposes, complexity refers to the repeated application
of simple rules in systems with many degrees of freedom
that gives rise to emergent behaviour
not encoded in the rules themselves.
The word `criticality', on the other hand, is well defined
among statistical physicists. Criticality refers to the behaviour
of extended systems at a phase transition
where observables are scale free, that is, no characteristic
scales exist for these observables.
At a phase transition, the many constituent microscopic `parts' give
rise to macroscopic phenomena that cannot be understood by considering
the laws obeyed by a single part alone. Criticality
is therefore a cooperative feature emerging from the
repeated application of the microscopic laws of a system of
interacting `parts'.
The phenomenology of phase transitions is well developed and there
exists a sound theoretical formalism for its description.
The book is divided into three chapters. In the first two chapters,
we carefully introduce the reader to the concepts of critical phenomena
using percolation and the Ising model as paradigmatic
examples of isolated equilibrium systems.
These systems undergo a phase transition only if an external agent
finely tunes certain external parameters to particular values.
The underlying theoretical formalism of criticality is carefully explained
through the concept of scale invariance, a central unifying theme of
the book.
However, there are many examples in Nature of complexity, that is,
the spontaneous emergence of criticality in slowly-driven non-equilibrium
systems: earthquakes in seismic systems, avalanches in granular media
and rainfall in the atmosphere. Key models of self-organised criticality
illustrate how such systems may naturally evolve into a stationary state
displaying scale invariance, and analogies are drawn between complexity
and criticality.
Although mathematical methods have been developed to describe
complexity and criticality, it is our experience that these methods
are unfamiliar to scientists outside the field.
Therefore, throughout the book we
emphasise the mathematical quantitative techniques available.
Our hope is that this
book will help students and researchers to treat complexity and
criticality more quantitatively.
The book is based on the lecture notes developed for the Statistical Mechanics course.
The target audiences are undergraduate and graduate students and researches in various fields.
The book will be self-contained and accessible to readers not familiar with the concepts of
complexity and criticality. The text can form the basis for advanced undergraduate or graduate
courses, and serve as an introductory reference
for researches in various fields.
The book includes a generous number of figures, and has an
associated website containing solutions to exercises and animations of
the models considered.
Each chapter is accompanied by exercises, full solutions to which can be
obtained by contacting the authors via the book's associated
website, http://www.worldscibooks.com/physics/p365.html.
On this site, readers will also find animation codes to visualise the behaviour
of the models considered.
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Publications in conference proceedings
[06] V. Pancaldi, K. Christensen, and P. King,
Two-phase Flow Upscaling using Haar Wavelets.
V IMA Conference on Modelling Permeable Rocks, University of Edinburgh, 26-29 March 2007.
[05] V. Pancaldi, K. Christensen, and P. King,
Upscaling using Haar Wavelets.
Computational Methods in Water Resources XVI, Copenhagen, Denmark, 18-22 June 2006.
[04] O. Peters and K. Christensen,
Micro rain radar measurements reveal evidence of self-organised
criticality in precipitation processes.
Sixth International Symposium on Tropospheric Profiling, Leipzig, Germany,
pp. 312-314 (2003).
[03] K. Christensen,
Physics in Dry Granular Media.
Editors: H.J. Hermann, J.-P. Hovi, and S. Luding.
Avalanches in Piles of Rice.
NATO ASI Series E, Vol. 350, pp. 475-480, Kluwer Academic Publishers (1998).
[02] P. Bak, K. Christensen, and Z. Olami,
Nonlinear dynamics and predictability of geophysical phenomena.
Editors: W.I. Newman, A. Gabrielov, and D.L. Turcotte.
Self-Organized Criticality: Consequences for Statistics and Predictability of Earthquakes.
Geophysical Monograph 83, pp. 69-74, American Geophysical Union, Washington D.C. (1994).
[01] K. Christensen,
Spontaneous formation of space-time structures and criticality.
Editors: T. Riste and D. Sherrington.
Dynamical Aspects of Sandpile Cellular Automata.
NATO ASI Series C, Vol. 349, pp. 33-36, Kluwer Academic Publishers (1991).
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Editorial work
[1] P. Alstrøm, T. Bohr, K. Christensen, H. Flyvbjerg, M.H. Jensen, B. Lautrup, and K. Sneppen,
Complexity and Criticality
Physica A, Vol. 340, Issue 4 (2004).
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Ph.D. thesis
[1] K. Christensen,
Self-organization in models of sandpiles, earthquakes, and fireflies.
Ph.D. thesis, 1-131, University of Aarhus, Denmark (1992).
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