From http://www.numerix.com/ - Kadanoff about the work which Anders Johansen has contributed to:
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This month's column is a vivid illustration of the many connections and similarities between academic physics and practical finance. It is contributed by Nigel Goldenfeld and Leo Kadanoff.
Nigel Goldenfeld is Professor of Physics at the University of Illinois at Urbana-Champaign, and a Managing Partner of NumeriX.
Leo Kadanoff is the John D. and Catherine T. MacArthur Distinguished Service Professor of Physics and Mathematics at the University of Chicago and a member of the NumeriX Advisory Board.
Portions of this article have been adapted from a manuscript which was submitted by one of us (LPK) as a Reference Frames Column for Physics Today.
Can you tell the difference between a stock market crash and a black hole?
We have gotten spoiled in physics. Most of the problems we discuss are treated in terms of differential equations, and most of our fundamental differential equations make sense in a quite global fashion. Schrodinger's equation or Maxwell's equations have the property that if you start out with a physically reasonable situation, the equations' time development gives you another physically reasonable solution and so on to eternity, or at least to the eternity defined by the equation.
But these equations are quite special. They are linear and wonderfully robust. In recent years, physicists, mathematicians, and others have turned their attention to equations which develop mathematical singularities from nothing. You start from a very smooth initial situation and just wait. After a time, an infinity shows up in the solution or in one of its derivatives. Sometimes you can continue the solution past the singularity. Sometimes you need new physics (perhaps an additional boundary condition) to see what happens next. Sometimes you can say nothing at all beyond the singularity time. And sometimes, the equation 'goes bad' in a generic way, the same for all sorts of disparate phenomena.
The simplest example is drawn from the elementary study of ordinary differential equations. Look at the equation
dx/dt = a x,
with a positive and constant, and an x which is positive at time zero. This might be the growth equation for something, perhaps the concentration of some compound in the atmosphere. Note that the solution grows exponentially in time, but x remains positive and well-behaved for all finite times. Perhaps this is environmentally bad, but the worst takes an infinite long time to arise. In contrast, imagine that the growth rate, a, was itself linear in the concentration of the contaminant: a =c x, with c being a constant. A quick calculation shows that the concentration obeys: x(t)=1/c(t*-t), with ct* being an abbreviation for the positive quantity 1/x(0). We notice that the concentration blows up at t* and subsequently has the senseless property of being negative. This example, drawn from the theory of ordinary differential equations, is too trivial for research-style investigation.
However, instances of singularity formation based upon partial differential equations are currently hot topics of research. For example, consider a situation in which bacteria are attracted by something which they produce and which move via a rapid diffusion process. As they produce a more concentrated region of attractant, the bacteria are bound together into a tight little clump in which their density goes to infinity near a singular point as
Note that the blowing up term always has the same basic shape, but its size and extension vary in time. Hence we call it 'scale-invariant'. We use the word 'universal' to suggest that the result will remain the same if the initial situation is varied slightly, or even perhaps if the differential equation is changed a little bit.
Equation (1) describes the simplest way in which a solution to a partial differential equation can 'go bad'. This same sort of mathematical structure reappears time and time again and is sufficiently generic that it can be usefully analyzed in isolation of the physical realization. Indeed, equation (1) is simple in the sense that the exponents in it (namely ½) are rational fractions, and could be obtained by a back-of-the-envelope dimensional analysis of the governing equation. In fact such innocent-looking analyses, although unsurpassed for impressing students, are fraught with hidden assumptions which are frequently unjustified. G.I. Barenblatt and Ya. B. Zeldovich taught us that more often than not, the exponents in a scale-invariant solution are not rational fractions, but transcendental numbers. And, as Nigel Goldenfeld and Yoshi Oono and co-workers have shown, the renormalization group provides a fundamental way to solve such problems, just as in the analogous case of second order phase transitions: equation (1) turns out to be a fixed point of a renormalization group transformation.
When equations go bad, it is frequently a sign that the phenomena are becoming more interesting. For example, scientists have long studied situations in which a mass of fluid forms a thin neck, which then breaks so that the fluid separates into two pieces. A group of mathematicians and physicists at the University of Chicago looked for similarity solutions in these situations. Using simulations, they found a whole zoo of such solutions, one of which was, in parallel, realized experimentally. In this solution, the derivative of the pressure blew up at the breaking point. This situation of broken necks is part of a broader problem, discussed by de Gennes and others, in which one must understand fluid flow on a surface which is partially wet and partially dry. The first time a dry spot appears on an initially wet surface, there is a mathematical singularity. At that point, one must somehow deal with the new boundary conditions which are required to describe the motion of the interface. The number and type of those boundary conditions, can in principle, define another kind of similarity solution, one which describes the wet-dry edge.
There is a classical and unsolved problem related to these singularity issues. The most fundamental equations for fluid flow are the Navier-Stokes and Euler equations. Nobody knows whether or when or how these equations develop singularities. One suspects that near-singular behavior of these equations bedevil the construction of accurate simulations of their results and complicate practical things like weather prediction and the design of aircraft.
Not all equations go bad the same way, however. In the last couple of years, it has been realized that true scale-invariant behavior is sometimes replaced by a discrete scale-invariance, in which the size and extension all vary periodically, but on a scale that gets more and more compressed as the singularity is approached. This behavior is a limit-cycle in renormalization group language, and although not usually encountered in quantum field theory, is a generic possibility for systems far from equilibrium.
One example is provided by black hole formation in general relativity. Black holes are singularities. In an elegant numerical study, Matthew Choptuik showed that in contrast to the usual situation with stellar collapse, certain situations allow black holes of mass smaller than the Chandrasekhar limit to be formed and that, in the process of formation, the solutions oscillate periodically in the logarithm of (t*-t). Many more papers followed which agreed with and extended this conclusion.
Even more remarkable are recent claims that this sort of behavior has been observed in far more complex systems, where we do not have an inkling of how to construct a differential equation description, or even if such a local description is appropriate. For instance, Didier Sornette and co-workers have observed that seismic precursors to several recent earthquakes in California, Japan and elsewhere can be fitted to the mathematical form
where t* is the singularity formation time. Finally, D. Sornette, A. Johansen and J.-P. Bouchaud, and simultaneously and independently, J.A. Feigenbaum and P. Freund have argued from time series data that major international stock indices follow Equation (2) in the period leading up to a stock market crash. They have analyzed data from crashes in 1929, 1962, 1990 and 1987. Feigenbaum and Freund also found that no evidence for Equation (2) could be found in the data during years when no crash occurred.
What should we conclude? Homogeneous systems can exhibit log-periodic behavior, only if there is an instability which can break the scale invariance and then sufficient non-linearity to couple together various length- and time- scales. Hence, Choptuik's results on the highly symmetric Einstein equations are presumably the manifestation of the well-known black hole instability, together with a considerable non-linear coupling among space-time scales. Indeed, depending upon the nature of the matter field coupled to gravity, both continuous and discrete self-similar behavior have been found. Earthquakes and stock market crashes also occur in systems plausibly unstable and containing many different scales. The relationships between tectonic plates, fault zones, and local crack and rupture points in one case, and the relationships between individual investors, traders, corporations, mutual funds, pension funds, major banks and governments in the other are so interconnected that perhaps we should not be surprised that this system divides into a Russian doll structure when the system is at its weakest and most stressed condition. The log-periodic variations show a hierarchy in the stressed system, but one still one cannot know whether there are a continuum of scales or a discrete hierarchy in the unstressed case.
For econometricians, there is a potentially useful conclusion to be drawn: if the presence of log-periodic scaling is confirmed as a precursor to market crashes, then we must take seriously not only the potentially hierarchical nature of the world financial system, but also its heterogeneity under stress. Recent attempts to model with agents the heterogeneity introduced by market participants with different positions will become more germane if the log-periodic scaling is confirmed.
Breaking things is how physicists like to explore nature. Modelling non-extreme events tests only the most superficial analytic properties of our equations. Only when our equations go bad in the same way as the piece of the real world they describe, can we be confident that we have caricatured all the right features.