"The mind is a compact, multiply connected thought mass with internal connections of the most intimate kind. It grows continuously as new thought masses enter it, and this is the means by which it continues to develop."

Bernhard Riemann On Psychology and Metaphysics ca. 1860

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Wednesday, April 20, 2011

The Trouble with Mathematics Masquerading as Science

Here, dear friends,  in one fell swoop is a sort of preamble to everything that is wrong with today's so called scientific method:

Generalized entropies and the transformation group of superstatistics

  1. Rudolf Hanela,
  2. Stefan Thurnera,b, and
  3. Murray Gell-Mannb,1
+ Author Affiliations
  1. aSection for Science of Complex Systems, Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria; and
  2. bSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501
  1. Contributed by Murray Gell-Mann, March 3, 2011 (sent for review February 17, 2011)

Abstract

Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures β, so that the probability distribution is Graphic, where the “kernel” f(β) is nonnegative and normalized [∫f(β) = 1]. We discuss the relation between this distribution and the generalized entropic form Graphic. The first three Shannon–Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection, we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension.

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