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> He began by rewriting the complicated Boltzmann equation as the sum of a series of decreasing terms. Theoretically, this chunky decomposition of the equation

> He began by rewriting the complicated Boltzmann equation as the sum of a series of decreasing terms. Theoretically, this chunky decomposition of the equation would be more easily recognizable as a different, but axiomatically equivalent, physical description of a gas — perhaps, a fluid description. The terms in the series quickly become unruly, however; energy, instead of diminishing at shorter and shorter distances in the gas, seems to amplify. This prevented Hilbert and others from summing up the series and interpreting it. Nonetheless, there was reason for optimism: The leading terms of the series looked like the Navier-Stokes equations when a gas becomes denser and more fluidlike. “So the physicists were happy, sort of," said Ilya Karlin, a physicist at ETH Zurich in Switzerland. “It’s in all the textbooks.” This reminds me a lot of perturbation theory, a method used to solve the complicated equations of quantum field theory. The technique basically involves summing up a bunch of Feynman diagrams (of decreasing significance), and it has been used to calculate the value of the gyromagnetic ratio of an isolated electron to within 10 decimal places of its experimentally measured value (which is absolutely amazing, both from a theoretical and experimental standpoint). However, what's peculiar about this summation is that it fails to converge. You would think that by adding up smaller and smaller terms, the series would eventually reach some limiting value, but that doesn't occur. So the most predictive theory that mankind has ever created (quantum electrodynamics) works only as long as you don't keep adding up more terms. (*Technically speaking, this isn't a failure of QED, but of the method used to solve its equations. There are other solution techniques that don't have this problem.)

plus 11y ago
The issue isn't that an infinite sum of tiny terms don't converge -- the issue is that individual terms of perturbation theory diverge. An example can be found in J. Chem. Phys. 112, 2000, 9736-9748 "Divergence in Moller--Plesset Theory: A Simple Explanation Based on a Two-State Model" DOI 10.1063/1.481611 (Note that this is specifically in reference to Moller--Plesset Perturbation Theory, but the divergence is a general phenomenon) I'm not saying that all perturbation theories diverge. Moller--Plesset perturbation theory doesn't even always diverge. But the divergence behaviour is not in the form of an infinite sum of tiny terms being infinite, but rather the individual terms of the perturbation theory increasing without bound (and oscillating sign). Also note that it is possible for truncations of perturbation theory to diverge with increasing order, but for the infinite sum of all (divergent) PT terms to converge and be finite.

plus 11y ago

The issue isn't that an infinite sum of tiny terms don't converge -- the issue is that individual terms of perturbation theory diverge. An example can be found in J. Chem. Phys. 112, 2000, 9736-9748 "Divergence in Moller--Plesset Theory: A Simple Explanation Based on a Two-State Model" DOI 10.1063/1.481611 (Note that this is specifically in reference to Moller--Plesset Perturbation Theory, but the divergence is a general phenomenon) I'm not saying that all perturbation theories diverge. Moller--Plesset perturbation theory doesn't even always diverge. But the divergence behaviour is not in the form of an infinite sum of tiny terms being infinite, but rather the individual terms of the perturbation theory increasing without bound (and oscillating sign). Also note that it is possible for truncations of perturbation theory to diverge with increasing order, but for the infinite sum of all (divergent) PT terms to converge and be finite.