## Periodic Reporting for period 2 - MIXMAX (Development and Implementation of new generation of Pseudo Random Number Generators based on Kolmogorov-Anosov K-systems)

Reporting period: 2017-01-01 to 2018-12-31

Modern powerful computers open a new era for the application of the Monte Carlo Method for the simulation of physical systems with many degrees of freedom and of higher complexity. The Monte Carlo simulations are important computational techniques in many areas of natural sciences and have significant application in particle and nuclear physics, quantum physics, statistical physics, quantum chemistry, material science, among many other multidisciplinary applications. In the heart of the Monte Carlo simulations are Pseudo Random Number Generators (PRNG). The primary objective of the MIXMAX project is a systematic development and implementation of the state of the art new generation of Pseudo Random Number Generators based on Kolmogorov-Anosov C-K systems, which demonstrates excellent statistical properties, into a multidisciplinary usable product. This innovative class of MIXMAX PRNGs was proposed earlier by the members of the network and relies on the fundamental discoveries and results of Ergodic theory. The MIXMAX generator is a fast generator and has exceptional statistical properties. As a fast and high quality generator the MIXMAX will allow to save financial and energy resources in science, research and industry spent for the massive Monte-Carlo simulations in the fields of high energy physics, biophysics, chemistry, Earth and environment, plasma physics, fluid dynamics, molecular structures and development of new materials. A saving of energy consumption by large computing facilities and cooling systems damping the hot water into the river and of the hot air, would be possible with a more efficient Monte Carlo simulations based on MIXMAX generators. The new approach to generate pseudo random numbers can provide a solution for vastly needed high entropy random number generators for communication systems. The overall objectives of the MIXMAX network is to turn these ideas and earlier research on C-K systems generators into a usable product, to develop an efficient program of the C-K system generator with tuneable internal parameters of maximal dimensionality and of the order of the Galois field, embedded into a user friendly environment, to provide statistical data representing internal characteristics of the C-K-system generator, to implement the C-K system generators into the concurrent and distributed software at CERN for applications in LHC and other HEP experiments, to perform large scale simulations in Quantum Gravity and Quantum Field Theory, to disseminate the product at CERN and other research centres.

The theory of the pseudorandom number generators takes a new shape thanks to the result and efforts of the MIXMAX Consortium in applying the cutting edge results of the Ergodic Theory to generate high quality random sequences for Monte-Carlo simulations [1,2,3,4]. A bases of the Consortium new approach is lying on the fundamental work of Anosov on hyperbolic C-systems [5] and on Kolmogorov theory of K-systems of nonzero entropy [6]. The two years of research and development by the Consortium resulted in the production of efficient C and C++ codes of the MIXMAX generator [1,3]. The C and C++ codes of the MIXMAX generator have been published in [1,3] and are publicly avalabel in the depository HEPFORGE.ORG [7] as well as on the webpage of the Consortium [8]. The webpage for the MIXMAX Consortium [8] reflects all aspects of the development of the project. For the needs of the Geant4 and CLHEP software packages at CERN the Consortium has developed the efficient low dimensional MIXMAX generators of dimension N = 8, 17, 40, 60, 96, 120 and 240. These generators have an advantage of having a very high quality sequence for moderate and small values of N. The Complex Statistical Tests of the MIXMAX pseudo random number generators were performed and demonstared that the MIXMAX is a unique 64-bit random number generator which is passing all BigCrushU01 tests [9], the PractRand tests, Dieharder test and many others. It was concluded that the two years (2015-2016) Milestone has been reached and assessment of the results in the execution of the Work packages have been presented at the MTR Meeting [10,11,12].

[1] K.Savvidy The MIXMAX random number generator, Comput.Phys.Commun. 196 (2015) 161-165.

(http://dx.doi.org/10.1016/j.cpc.2015.06.003); arXiv:1404.5355

[2] G. Savvidy, Anosov C-systems and random number generators, Theor.Math.Phys. 188 (2016) 1155-

1171; arXiv:1507.06348 [hep-th].

[3] K. Savvidy and G. Savvidy, Spectrum and Entropy of C-systems. MIXMAX random number generator,

Chaos Solitons Fractals 91 (2016) 33 ( doi:10.1016/j.chaos.2016.05.003); [arXiv:1510.06274

[math.DS]].

[4] A. Görlich, M. Kalomenopoulos, K. Savvidy and G. Savvidy, Distribution of periodic trajectories

of Anosov C-system, Int. J. Mod. Phys. C 28 (2017) 1750032 (doi: 10.1142/S0129183117500322)

[5] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Trudy Mat.

Inst. Steklov., Vol. 90 (1967) 3 - 210

[6] A.N. Kolmogorov, New metrical invariant of transitive dynamical systems and automorphisms of

Lebesgue spaces, Dokl. Acad. Nauk SSSR, 119 (1958) 861-865

[7] HEPFORGE.ORG http://mixmax.hepforge.org; http://www.inp.demokritos.gr/˜savvidy/mixmax.php

[8]http://www.inp.demokritos.gr/˜savvidy/mixmax.php

[9] P. L’Ecuyer and R. Simard, TestU01: A C Library for Empirical Testing of Random Number

Generators, ACM Transactions on Mathematical Software, 33 (2007) 1-40.

[10] First Consortium Meeting and Workshop at CERN, July 2015: https://indico.cern.ch/event/404547

[11] The Second Consortium Meeting and Workshop at CERN, July 2016:https://indico.cern.ch/event/544108/

[12]Conference and MTR Meeting in Athens, September 2016:https://indico.cern.ch/event/558996/

[1] K.Savvidy The MIXMAX random number generator, Comput.Phys.Commun. 196 (2015) 161-165.

(http://dx.doi.org/10.1016/j.cpc.2015.06.003); arXiv:1404.5355

[2] G. Savvidy, Anosov C-systems and random number generators, Theor.Math.Phys. 188 (2016) 1155-

1171; arXiv:1507.06348 [hep-th].

[3] K. Savvidy and G. Savvidy, Spectrum and Entropy of C-systems. MIXMAX random number generator,

Chaos Solitons Fractals 91 (2016) 33 ( doi:10.1016/j.chaos.2016.05.003); [arXiv:1510.06274

[math.DS]].

[4] A. Görlich, M. Kalomenopoulos, K. Savvidy and G. Savvidy, Distribution of periodic trajectories

of Anosov C-system, Int. J. Mod. Phys. C 28 (2017) 1750032 (doi: 10.1142/S0129183117500322)

[5] D. V. Anosov, Geodesic flows on closed Riemannian manifolds with negative curvature, Trudy Mat.

Inst. Steklov., Vol. 90 (1967) 3 - 210

[6] A.N. Kolmogorov, New metrical invariant of transitive dynamical systems and automorphisms of

Lebesgue spaces, Dokl. Acad. Nauk SSSR, 119 (1958) 861-865

[7] HEPFORGE.ORG http://mixmax.hepforge.org; http://www.inp.demokritos.gr/˜savvidy/mixmax.php

[8]http://www.inp.demokritos.gr/˜savvidy/mixmax.php

[9] P. L’Ecuyer and R. Simard, TestU01: A C Library for Empirical Testing of Random Number

Generators, ACM Transactions on Mathematical Software, 33 (2007) 1-40.

[10] First Consortium Meeting and Workshop at CERN, July 2015: https://indico.cern.ch/event/404547

[11] The Second Consortium Meeting and Workshop at CERN, July 2016:https://indico.cern.ch/event/544108/

[12]Conference and MTR Meeting in Athens, September 2016:https://indico.cern.ch/event/558996/

The main advantages of the MIXMAX generator are based on its exceptional characteristics: a)the MIXMAX generator is one of fastest generators producing 64-bit pseudorandom number in 5 nanoseconds b) it has very large Kolmogorov entropy of 0.9 per/bit c) it has long periods of order 10 to the power five d) a new skipping algorithm allows generate independent seeds and not overlapping streams e) it is a unique 64bit generator. Thus MIXMAX generator is a fast generator and has exceptional statistical properties. As a fast and high quality generator the MIXMAX will allow to save financial and energy resources in science, research and industry spent for the massive Monte-Carlo simulations in the fields of high energy physics, biophysics, chemistry, Earth and environment, plasma physics, fluid dynamics, molecular structures and development of new materials. A saving of energy consumption by large computing facilities and cooling systems damping the hot water into the rivers and releasing the hot air, would be possible with a more efficient Monte Carlo simulations based on MIXMAX generators. A further development of the MIXMAX system of generators can provide a solution for vastly needed high entropy random number generator for communication systems.