Unification of the Nature's Complexities via a Matrix Permanent-Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity

Vitaly Kocharovsky; Vladimir Kocharovsky; Sergey Tarasov

doi:10.3390/e22030322

Unification of the Nature's Complexities via a Matrix Permanent-Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity

Entropy (Basel). 2020 Mar 12;22(3):322. doi: 10.3390/e22030322.

Authors

Vitaly Kocharovsky^{1

2}, Vladimir Kocharovsky², Sergey Tarasov²

Affiliations

¹ Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA.
² Institute of Applied Physics, Russian Academy of Science, Nizhny Novgorod 603950, Russia.

Abstract

We reveal the analytic relations between a matrix permanent and major nature's complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathematics, and ♯P-complete problems in the theory of computational complexity. They follow from a reduction of the Ising model of critical phenomena to the permanent and four integral representations of the permanent based on (i) the fractal Weierstrass-like functions, (ii) polynomials of complex variables, (iii) Laplace integral, and (iv) MacMahon master theorem.

Keywords: MacMahon master theorem; NP-complexity; Toeplitz determinant; critical phenomena; fractals; matrix permanent; quantum computing; ♯P-complexity.

Abstract

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