![]() ![]() In this paper we are studying the properties of their spectrum and of the entropy. These systems have exponential instability of all their trajectories and as such have mixing of all orders, nonzero Kolmogorov entropy and a countable set of everywhere dense periodic trajectories. The uniformly hyperbolic Anosov C-systems defined on a torus have very strong instability of their trajectories, as strong as it can be in principle. ![]() The corresponding MIXMAX generator has the best combination of speed, size of the state and is currently available generator. Using analytical and computer calculations, we are studying a distribution function of periodic trajectories and their deviation from asymptotic behavior. The asymptotic distribution of chaotic trajectories of C-K systems with periods less than a given number is well known in mathematical literature, but a deviation from its asymptotic behavior is unknown. The C-K systems on a torus have countable set of everywhere dense periodic trajectories and their distribution play a crucial role in coding and implementation of the pseudorandom number generator. All trajectories of the C-K systems are exponentially unstable and pseudorandom numbers are represented in terms of coordinates of very long chaotic trajectories. We are considering the hyperbolic C-K systems of Anosov–Kolmogorov which are defined on high dimensional tori and are used to generate pseudorandom numbers for Monte-Carlo simulations. This result allows to define decorrelation and relaxation times in terms of entropy and characterise the statistical properties of the MIXMAX generator. We have found that the upper bound on the rate of the exponential decay of the correlation functions universally depends on the value of the system entropy. It is important to specify the parameters of a dynamical C-system which quantify the exponential decay. The correlation functions of the physical observables which are defined on a torus phase space are tend to zero and become uncorrelated exponentially fast. The C-systems on a torus are perfect candidates to be used for Monte-Carlo simulations. Of special interest are C-systems that are defined on a high dimensional torus. The hyperbolic Anosov C-systems have exponential instability of their trajectories and as such have mixing of all orders and nonzero Kolmogorov entropy. We are developing further our earlier suggestion to use high entropy Anosov C-systems for the Monte-Carlo simulations. We also provide the alternative parameters for the generators, $N=8$ and $N=240$ with $m$ in this optimised range. When $d > N$ the vectors of the generated numbers fall into the parallel hyperplanes and the distances between them can be larger than the genuine "resolution" of the MIXMAX generators, which is $ 2^$, a range which is inclusive of the value of the $N=17$ generator. ![]() These tests have been performed by L'Ecuyer and collaborators. Therefore the spectral test is important to perform in dimensions $d > 8$ for $N=8$ generator, $d> 17$ for $N=17$ and $d> 240$ for $N=240$ generator. In particular, the default MIXMAX generators have various dimensions: $N=8,17,240$ and higher. The test is aimed at answering the question of distribution of the generated pseudo-random vectors in dimensions $d$ that are larger than the genuine dimension of a generator $N$. An important statistical test on the pseudo-random number generators is called the spectral test. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |