In this article, we reflect on a classical physics problem in describing the system of the 3-body motion of celestial objects, originally considered by Newton. At the other end of the scale, in quantum mechanics, the measurement problem was originally addressed by Heisenberg with the introduction of uncertainty relation.
The stochastic resonance synergetics addresses both of these issues in a mathematical description that generalizes the uncertainty relation within the quantum field theory. The mathematical formulation describes the connection between the largest and the smallest structures in a coupled network. In our view, it shows a unifying perspective of looking at motion in the hierarchy of scales and coordinate frames, mathematically described in a 5-dimensional scale-space.
3-body motion problem
Poincare was first to realize that there is no simple solution to the system of 3-body motion in Newtonian dynamics. It has been later described in mathematics as a chaotic motion. The application of this theory is probably the most valued in weather forecasts. In addition, witnessing climate change, possibly applies the chaos theory to the mathematical modeling of a long-term motion in our solar system, as well.
Generalized uncertainty relation in dynamical scaling
Although originally introduced to address the measurement problem in quantum mechanics, it has been shown inherent in wave mechanics. In our work on wave information propagation, a generalized multidimensional uncertainty relation has been derived.
A partition function of the configuration space is decomposed by utilizing two physical principles – the maximum entropy, loosely associated with time, and the least action principle, holding the configuration space in the minimal loop structure. The Hessian of free energy, along with a multidimensional space, scales with the scale parameter β.
Two operators acting on the wave information propagation are represented in the phase-space of conjugate variables, the rotor and divergence. The stability of the decomposition structure is assessed with the generalized uncertainty relation, across the scales.
"Traveling salesman" problem, the elements of prime numbers distribution, and the asymptotic freedom of motion
We have investigated the quantum patterns of the chaotic dynamics. Genotype quantum signatures are computed in 5d, out of generally chaotic dynamics in space-time.
Computation of the minimal circular path is commonly stated as "traveling salesman" problem, in computer science. In our study, we have shown the point in scale at which the asymptotic freedom of motion binds 3 clusters together.
The quantum signatures apply similarly to the distribution of the prime numbers. Stochastic resonance synergies bind 3 scales 6(k+i), i=1,2,3, of natural numbers together in the closed-loop structures. The scale-space wave function preserves the information of the distribution of the prime numbers, across the scales.
We have reflected on the relationship between randomness and chaotic description of motion, in a hierarchy of scales. A mathematical description of generalized uncertainty relation is derived in our study within a quantum field theory. It gives a unifying perspective of looking at the dynamics of complex systems motion.
We have investigated 3-body motion at both ends of the scale-space -- at the largest and at the smallest. Polynomial distribution of a partition function limits the asymptotic freedom of information exchange that binds 3 clusters together.
These limiting signatures make a bridge to the mathematical foundation in our quantum field theory. We have shown results with the "traveling salesman problem" and the distribution of the prime numbers.
We have also proposed, accordingly, a multiscale computational approach to weather and climate change study.
Jovovic, M., Hierarchical scale quantization and coding of motion information in image sequences, Informacione Tehnologije VI, Zabljak, 2002.
Jovovic, M., H. Yahia, and I. Herlin, Hierarchical scale decomposition of images – singular features analysis, INRIA, 2003.
Jovovic, M., and G. Fox, Multi-dimensional data scaling – dynamical cascade approach, Indiana University, 2007.
Jovovic, M., Stochastic Resonance Synergetics – Quantum Information Theory for Multidimensional Scaling, Journal of Quantum Information Science, 5/2:47-57, 2015.