This arises from issues I encountered in 2007. From September 2005 to the end of May 2007 I was looking for the base of a logarithm that would connect the rate of English lexical growth, about 3.39% per decade to the rate of divergence measured by Morris Swadesh for related Indo-European languages, which he found to be less than 14% per thousand years. I was looking for a number, but after many spreadsheets and failed guesses found that a network’s mean path length (mu, say) worked as the base of the logarithm. (I wrote a paper in early 2008 about this, on arxiv.)
I noticed this peculiarity about Clog (n) where n is the number of nodes in the network, and C is the clustering coefficient. From one perspective, if there are n nodes, then log (n) = k is the number of degrees of freedom in the network (or ensemble) relative to the mean path length mu. But at the same time, log (n) = k could also represent k time periods occurring for a set of nodes equal in number to the mean path length mu. So which is it? Is k the degrees of freedom for an ensemble? Or does k represent the length of time for a single cluster of mu nodes? Is this an irrelevant mathematical curiosity? Or, is there some connection between k as degrees of freedom for n nodes relative to mu and the way that time works? Why are the number of degrees of freedom for n nodes equivalent to the degrees of freedom relative to a single cluster of mu nodes over k periods of time. For example, does this imply that an event that occurs for an entire ensemble can in some way be emulated by an event repeated over k time periods for mu nodes? Perhaps it is nothing But it seems that J. Willard Gibbs in his Elementary Principles of Statistical Mechanics used a kind of variation of this when he calculated the statistical distribution of multiple copies of the same ensemble. Do these mathematical aspects indicates something about the nature of time, that a period of time for mu nodes can correspond to an event at a point of time for an entire ensemble? Does that seem to echo Minkowski’s notion of space time?
1. Logic of the Second Law of Thermodynamics: Subjectivism, Logical Jump, Interdisciplinary Argumentation.
2. New thermodynamics pursues universality, two theoretical cornerstones:
2.1 Boltzmann formula: ro=A*exp(-Mgh/RT) - Isotope centrifugal separation experiments show that it is suitable for gases and liquids.
2.2. Hydrostatic equilibrium: applicable to gases and liquids.
3. The second and third sonic virial coefficients of R143a derived from the new thermodynamics are in agreement with the experimental results.
3.1. The third velocity Virial coefficient derived is in agreement with the experimental data, which shows that the theory is still correct when the critical density is reached.
4. See Appendix Pictures and Documents for details.
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