Chapter 5 Chance and necessity

The ordinary probability

When the party is over

A rare dissemination

Potentially abnormal

Growing

Tumours

Enormous molecules

Referencias y comentarios:

· La ley de los grandes números se desarrolla con detalle en en.wikipedia.org, mathworld.wolfram.com y www.stat.berkeley.edu, el teorema del límite central en video.google.es (vídeo explicativo de éste y otros conceptos en la teoría de la probabilidad) y en.wikipedia.org, y la construcción de histogramas se ilustra en w3.cnice.mec.es, o bien pueden consultarse estos conceptos en un libro o manual sobre probabilidad como, por ejemplo, el de

® William Feller, An introduction to probability theory and its applications, dos volúmenes, Wiley, NY 1968 y 1971.

En línea puede verse www.dartmouth.edu. También: www.mathcs.carleton.edu.

Nótese que este capítulo no puede considerarse, en absoluto, como una introducción a la teoría de probabilidades, pues no se hace una descripción sistemática del tema, de modo que ha de completarse en este sentido con alguna de estas referencias.

· The importance of the mesoscopic world to understand the mystery of life was highlighted by

® Mark Haw in Middle World: The Restless Heart of Matter and Life (Palgrave Macmillan, 2006).

· For Brownian motion, visit the Center for Polymer Studies which illustrates random walks in one and two dimensions in the site polymer.bu.edu/java/; see also galileo.phys.virginia.edu/classes/109N/more_stuff/applets/brownian/brownian.hml.

The original experiment is described in

® “Einstein, Perrin, and the reality of atoms: 1905 revisited”, by Ronald Newburgh, Joseph Peidle and Wolfgang Rueckner, American Journal of Physics 74, 478 (2006).

The mathematical theory is in

® Dynamical theories of Brownian motion, by Edgard Nelson (Princeton University Press, 1967) whose second edition is posted at www.math.princeton.edu/ ~nelson/books.html.

Further details on diffusion:

® “The dichotomous history of diffusion”, by T.N. Narasimhan, Physics Today (July 2009), page 48.

· The classical book for stochastic processes in science is

® Stochastic Processes in Physics and Chemistry by Nicolaas G. Van Kampen (Elsevier, 2007).

· For experimental reports and theory on anomalous diffusion, see

® “Lévy flight search patterns of wandering albatrosses”, Gandhimohan M. Viswanathan et al., Nature 381, 413 (1996);

® “Revisiting Lévy flight search patterns of wandering albatrosses, bumblebeees and deer”, Nature 449, 1044 (2007), also by Gandhimohan M. Viswanathan et al.;

® “Scale-free dynamics in the movement patterns of jackals”, R.P.D. Atkinson et al., Oikos 98, 134 (2002);

® “Anomalous diffusion spreads its wings”, Joseph Klafter and Igor M. Sokolov, Physics World, page 29, August 2005;

® Diffusion and reactions in fractals and disordered systems, Daniel ben-Avraham and Shlomo Havlin (Cambridge University Press 2000);

® “Strange kinetics of single molecules in living cells”, Eli Barkai, Yuval Garini, and Ralf Metzler, Physics Today 65, 29 (August 2012)

® “Einstein relation in superdiffusive systems”, Giacomo Gradenigo et al., Journal of Statistical Mechanics: Theory and Experiments L06001 (2012).

Also, it is interesting to note here that Nassim N. Taleb has developed on the impact of rare events in the financial markets —with possible extension to many fields—, and on our tendency to rationalize them too simply a posteriori, in

® The black swan — The impact of the highly improbable, by Nassim N. Taleb (Random House 2010).

· For a description of power laws and their properties:

® “A brief history of generative models for power law and lognormal distributions”, Michael Mitzenmacher, Internet Mathematics 1, 226 (2003), and

® “Power laws, Pareto distributions and Zipf’s law”, Mark E. J. Newman, Contemporary Physics 46, 323 (2005).

® “The scaling laws of human travel”, Dirk Brockmann et al., Nature 439, 462 (2006).

· On the processes of growth, see the classic

® D'Arcy W. Thompson, On Growth and Form (Cambridge University Press 1917; revised edition, Dover 1992).