Summary

The book Entropy and infinite noise deals in chapters 2, 4 and 5 with the historical emergence of the physical concept of entropy and the criticism of the Boltzmannian understanding and definition of entropy as it was raised in his time, for example in the form of the Loschmidt paradox. Central concepts of these chapters are expansion and contraction, in relation to which entropy behaves like an expansion measure, i.e. a measure for the propagation of particles in the position space and of energy in momentum space. However, in my experience, this relation is only hinted at in the public presentation of entropy, which is probably due to the fact that expansion is commonly reduced to its extensive side and therefore cannot be understood. In fact, expansion also has an intensive side, and entropy is a measure of both, extensive and intensive expansion. Thus, the increase of the entropy of motion, as I call it to distinguish it from the density entropy, in a closed system is first connected with the expansion of the momentums in the intensive momentum space, because the conservation of energy prevents the extensive expansion - averaged.

The introduction and chapters 3 and 7 focus on the Gibbs and the Shannon entropy on the one hand and the density entropy derived from the ideal gas in the sixth chapter on the other hand. It is shown that the first two are not extensive, but the density entropy is when the absolute number of particles is considered as a factor. It turns out that the density entropy - without absolute particle number - coincides with the differential entropy, which is a special case of the relative entropy. Finally, based on the density entropy, the U-entropy is defined, which, unlike all others, is a contraction measure and increases monotonically with increasing refinement of the partitioning of space, like the Gibbs and Shannon entropies, raising the question of its convergence in the limit to infinite resolution. In this question, the U-entropy is consistently compared with the Shannon entropy, and their differences are highlighted. It is shown that, unlike the latter, U-entropy can converge - even under the tightened assumption that the particle density is the foundation of the spatial order, thus defining it, which is only possible if there are distinguished points in space with infinite particle density. These points are the rational points of the continuous space. The irrational points are derived from the rational points. Both together form the space to the hierarchically ordered address space with distinguishable points, for which the infinite noise of the particle density is a condition.

Another pair of terms are self-reference and double-reference, in the context here the double-reference between the particle continuum on the one hand and the space continuum on the other hand, by which they define each other. The infinite noise is in contrast to the general physical understanding that physical quantities are continuous or even differentiable functions.

In the course of the research of the above mentioned connections also a volume about the mathematical continuum has been written, which will probably appear in a few months also at this place.