Maximum Entropy (Most Likely) Double Helical and Double Logarithmic Spiral Trajectories in Space-Time

M C Parker,C Jeynes

doi:10.1038/s41598-019-46765-w

Abstract

The ubiquity of double helical and logarithmic spirals in nature is well observed, but no explanation is ever offered for their prevalence. DNA and the Milky Way galaxy are examples of such structures, whose geometric entropy we study using an information-theoretic (Shannon entropy) complex-vector analysis to calculate, respectively, the Gibbs free energy difference between B-DNA and P-DNA, and the galactic virial mass. Both of these analytic calculations (without any free parameters) are consistent with observation to within the experimental uncertainties. We define conjugate hyperbolic space and entropic momentum co-ordinates to describe these spiral structures in Minkowski space-time, enabling a consistent and holographic Hamiltonian-Lagrangian system that is completely isomorphic and complementary to that of conventional kinematics. Such double spirals therefore obey a maximum-entropy path-integral variational calculus (“the principle of least exertion”, entirely comparable to the principle of least action), thereby making them the most likely geometry (also with maximal structural stability) to be adopted by any such system in space-time. These simple analytical calculations are quantitative examples of the application of the Second Law of Thermodynamics as expressed in geometric entropy terms. They are underpinned by a comprehensive entropic action (“exertion”) principle based upon Boltzmann’s constant as the quantum of exertion.

Highlights

Formal mathematics establishes tautologies which are frequently very surprising, and we have used well-established formal methods in a properly quantitative treatment of entropy, revealing that measurable quantities from the molecular to the galactic scale can be readily calculated in a simple analytical treatment
We have considered systems of high symmetry which are amenable to our simplified analytical approach, but we expect the method to be readily generalisable to more complex systems
The computational demands of conformational chemistry are very severe; perhaps this approach will stimulate algorithmic advances to speed the calculations for static problems, or even to address dynamic geometrical problems in new ways?

Summary

Double Helical and Double

Logarithmic Spiral Trajectories in Received: 14 November 2018 Accepted: 5 July 2019 Published: xx xx xxxx. The proof that the double-armed logarithmic spiral satisfies the Euler-Lagrange equations in hyperbolic space q (that is, obeys the principle of least exertion) is given in Appendix C (Eq C.47, see Supplementary Information). In Euclidean (x) space, we find that the entropic potential field VS for the logarithmic double spiral is expressed as (see Eq B.42 in Appendix B, Supplementary Information; K0 and K3 are dimensionless): VS(x) imSK0eiκGx 1 − Λx3. The spiral coordinate x3 projected onto the plane in Fig. 2 is associated with an azimuthal angle θ = κGx3 where the appropriate wavelength scale λG for the galaxy is given by the galactic wavenumber κG = 2π/λG This can be calculated from the galactic structural entropy S, well approximated by

Entropic Equivalent

Isomorphism between Mechanics and Entropy

Summary

Findings

Additional Information