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When tea is poured in a cup of hot water, we observe a phenomenon called diffusion: in the end particles of tea spread evenly throughout the mass of water and we enjoy our cup of tea. Diffusion occurs as a result of the second law of thermodynamics (increase of entropy) and can be modeled quantitatively using the diffusion equation (or heat equation). This is a funny equation, since it establishes that the velocity of spreading is infinite while the mean root square fluctuations of the position of the particles grows in time as
Or… Correlation implies causality. This is a logical fallacy. Keep this in mind when analyzing data. There are numerous cases of how people use this logical fallacy nowadays (most typically in newspapers). For example, the following graph
shows a correlation between global warming and the number of remaining pirates. But this does not imply (of course), any causality between them.
By the way, this graph appears in a Bobby Henderson’s clever parody of the type of arguments in Intelligent Design.
I have been on holidays during the last two weeks visiting Argentina. The picture on the left was taken on top of the Perito Moreno glacier, which is amazing. Most glaciers we found where blueish, including the icebergs found floating in the rivers. The reason for that is that the thicker the ice or snow layer is, the better red colors are absorbed by the layer and only the blue colors are reflected (see a more detailed explanation here).
You can’t fold a paper more than seven or eight times. Don’t believe me? Then try it. Thinner paper? Longer paper? It doesn’t matter; you just can’t do it. I used to play this game with my friends which were always amazed and asked for an explanation. Is there any physical or mathematical constrain to do it? Nope: it is simply a matter of scale. If you have a paper of length _L_ and you fold it, the length now is L/2.
In a recent Nature article, Albert-Lászlo Barabási and João Gama Oliveira, have found the perfect excuse for lazy people not answering some emails in their inbox: they analyzed the time response of emails and found that they follow a power law probability distribution of the form P(t) = t-1. In particular this implies that not even the mean response time is finite. Hey! why should you then expect me to answer your emails within my lifetime period!
UCSD physicist Jorge E. Hirsch has propose a quick-and-dirty way to measure quality of academic scientist’s output. His method is explained and studied in a paper to be published in the November 15 issue of PNAS. The idea is very simple and it is called the h-index. This number relies on the number of citations our papers have. In particular the h-index is the maximum number _h_ that verifies the following: at least h of papers of an author have h citations each.
Authors: Esteban Moro and Henri Schurz
Journal: SIAM Journal of Scientific Computing, Volume 29 Issue 4, Pages 1525-1549 (2007). LINK | arXiv
Abstract: Construction of splitting-step methods and properties of related non-negativity andboundary preserving numerical algorithms for solving stochastic differential equations (SDEs) of Ito-type are discussed. We present convergence proofs for a newly designed splitting-step algorithm and simulation studies for numerous numerical examples ranging from stochastic dynamics occurring in asset pricing theory in mathematical finance (SDEs of CIR and CEV models) to measure-valued diffusion and superBrownian motion (SPDEs) as met in biology and physics.
Authors: Esteban Moro
Journal: Physical Review E (Rapid Communication) 70, 045102 (2004). LINK | arXiv
Abstract: We present numerical schemes to integrate stochastic partial differential equations which describe the spatiotemporal dynamics of reaction-diffusion problems under the effect of internal fluctuations. The schemes conserve the non-negativity of the solutions and incorporate the Poissonian nature of internal fluctuations at small densities, their performance being limited by the level of approximation of density fluctuations at small scales.
Authors: Esteban Moro
Book: Advances in Condensed Matter and __Statistical Physics, _E. Korutcheva and R. Cuerno eds. (Nova Science Publishers, New York 2004). LINK | arXiv
Abstract: The Minority Game is a simple model for the collective behavior of agents in an idealized situation where they have to compete through adaptation for a finite resource. This review summarizes the statistical mechanics community efforts to clear up and understand the behavior of this model.
Authors: Esteban Moro
Journal: Physical Review E, Rapid Communication 69, 060101 (2004). LINK | arXiv
Abstract: We study the propagation of pulled fronts in the \(A\leftrightarrow A+A\) microscopic reaction-diffusion process using Monte Carlo simulations. In the mean field approximation the process is described by the deterministic Fisher-Kolmogorov-Petrovsky-Piscounov equation. In particular, we concentrate on the corrections to the deterministic behavior due to the number of particles per correlated volume \(\Omega\). By means of a hybrid simulation scheme, we manage to reach large macroscopic values of \(\Omega\), which allows us to show the importance in the dynamics of microscopic pulled fronts of the interplay of microscopic fluctuations and their macroscopic relaxation.