Classical Chaos Described by a Density Matrix

In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states.

Mapping Effective Field Theory to Multifractal Geometry

Fractals and multifractals are well-known trademarks of nonlinear dynamics and classical chaos. The goal of this work is to tentatively uncover the unforeseen path from multifractals and selfsimilarity to the framework of effective field theory (EFT). An intriguing finding is that the partition function of multifractal geometry includes a signature analogous to that of gravitational interaction. Our results also suggest that multifractal geometry may offer insights into the non-renormalizable interactions presumed to develop beyond the Standard Model scale.

Mapping Effective Field Theory to Multifractal Geometry

Fractals and multifractals are well-known trademarks of nonlinear dynamics and classical chaos. The goal of this work is to tentatively uncover the unforeseen path from multifractals and selfsimilarity to the framework of effective field theory (EFT). An intriguing finding is that the partition function of multifractal geometry includes the signature of non-Euclidean metric. Our results also suggest that multifractal geometry may offer insights into the non-renormalizable interactions presumed to develop beyond the Standard Model scale.

Chaotic string dynamics in deformed T1,1

Recently, Arutyunov, Bassi and Lacroix have shown that 2D non-linear sigma model with a deformed T1,1 background is classically integrable [arXiv:2010.05573 [hep-th]]. This background includes a Kalb-Ramond two-form with a critical value. Then the sigma model has been conjectured to be non-integrable when the two-form is off critical. We confirm this conjecure by explicitly presenting classical chaos. With a winding string ansatz, the system is reduced to a dynamical system described by a set of ordinary differential equations. Then we find classical chaos, which indicates non-integrability, by numerically computing Poincaré sections and Lyapunov spectra for some initial conditions.

Classical Chaos Described by a Density Matrix

We work with reference to a well-known semiclassical model, in which quantum degrees of freedom interact with classical ones. We show that, in the classical limit, it is possible to represent classical results (e.g., classical chaos) by means a pure-state density matrix.

Designing Two Secure Keyed Hash Functions Based on Sponge Construction and the Chaotic Neural Network

In this paper, we propose, implement, and analyze the structures of two keyed hash functions using the Chaotic Neural Network (CNN). These structures are based on Sponge construction, and they produce two variants of hash value lengths, i.e., 256 and 512 bits. The first structure is composed of two-layered CNN, while the second one is formed by one-layered CNN and a combination of nonlinear functions. Indeed, the proposed structures employ two strong nonlinear systems, precisely a chaotic system and a neural network system. In addition, the proposed study is a new methodology of combining chaotic neural networks and Sponge construction that is proved secure against known attacks. The performance of the two proposed structures is analyzed in terms of security and speed. For the security measures, the number of hits of the two proposed structures doesn’t exceed 2 for 256-bit hash values and does not exceed 3 for 512-bit hash values. In terms of speed, the average number of cycles to hash one data byte (NCpB) is equal to 50.30 for Structure 1, and 21.21 and 24.56 for Structure 2 with 8 and 24 rounds, respectively. In addition, the performance of the two proposed structures is compared with that of the standard hash functions SHA-3, SHA-2, and with other classical chaos-based hash functions in the literature. The results of cryptanalytic analysis and the statistical tests highlight the robustness of the proposed keyed hash functions. It also shows the suitability of the proposed hash functions for the application such as Message Authentication, Data Integrity, Digital Signature, and Authenticated Encryption with Associated Data.

Correlation Functions of Quantum Artin System

It was conjectured by Maldacena, Shenker and Stanford that the classical chaos can be diagnosed in thermal quantum systems by using an out-of-time-order correlation function. The Artin dynamical system defined on the fundamental region of the modular group SL(2,Z) represents a well defined example of a highly chaotic dynamical system in its classical regime. We investigated the influence of the classical chaotic behaviour on the quantum–mechanical properties of the Artin system calculating the corresponding out-of-time-order thermal quantum–mechanical correlation functions. We demonstrated that the two- and four-point correlation functions of the Liouiville-like operators decay exponentially with temperature dependent exponents and that the square of the commutator of the Liouiville-like operators separated in time grows exponentially.

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.

Hamiltonian Chaos

Nondissipative or Hamiltonian systems are also capable of chaos as phase space volume is twisted and folded in area-preserving maps like the Standard Map. When nonintegrable terms are added to a potential function, Hamiltonian chaos emerges. The Standard Map (also known as the Chirikov map) for a periodically kicked rigid rotator provides a simple model with which to explore the emergence of Hamiltonian chaos as well as the KAM theory of islands of stability. A periodically kicked harmonic oscillator displays extended chaos in the web map. Hamiltonian classical chaos makes a direct connection to quantum chaos, which is illustrated using the chaotic stadium, for which quantum scars are associated with periodic classical orbits in the stadium.

Quantum Chaos and Quantum Randomness—Paradigms of Entropy Production on the Smallest Scales

Quantum chaos is presented as a paradigm of information processing by dynamical systems at the bottom of the range of phase-space scales. Starting with a brief review of classical chaos as entropy flow from micro- to macro-scales, I argue that quantum chaos came as an indispensable rectification, removing inconsistencies related to entropy in classical chaos: bottom-up information currents require an inexhaustible entropy production and a diverging information density in phase-space, reminiscent of Gibbs’ paradox in statistical mechanics. It is shown how a mere discretization of the state space of classical models already entails phenomena similar to hallmarks of quantum chaos and how the unitary time evolution in a closed system directly implies the “quantum death” of classical chaos. As complementary evidence, I discuss quantum chaos under continuous measurement. Here, the two-way exchange of information with a macroscopic apparatus opens an inexhaustible source of entropy and lifts the limitations implied by unitary quantum dynamics in closed systems. The infiltration of fresh entropy restores permanent chaotic dynamics in observed quantum systems. Could other instances of stochasticity in quantum mechanics be interpreted in a similar guise? Where observed quantum systems generate randomness, could it result from an exchange of entropy with the macroscopic meter? This possibility is explored, presenting a model for spin measurement in a unitary setting and some preliminary analytical results based on it.