arXiv subjects

Vladimir Garcia-Morales

Publications and source records attributed to Vladimir Garcia-Morales.

10 recordsLinked to original sources

arXiv2023

Universality of oscillatory instabilities in fluid mechanical systems

Oscillatory instability (OI) emerges amidst turbulent states in experiments in various turbulent fluid and thermo-fluid systems such as aero-acoustic, thermoacoustic and aeroelastic systems. For the time series of the relevant dynamic variable at the onset of the OI, universal scaling behavior have been discovered in experiments via the Hurst exponent and certain spectral measures. By means of a center manifold reduction, the spatiotemporal dynamics of these real systems can be mapped to a complex Ginzburg-Landau equation with a linear global coupling (GCGLE). In this work, we show that the GCGLE is able to capture the universal behavior of the route to OI, elucidating it as a transition from defect to phase turbulence mediated by the global coupling.

physics.flu-dyn↗

arXiv2016

Fractal surfaces from simple arithmetic operations

Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent $H$ that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.

cs.OH↗

arXiv2015

The $pλn$ fractal decomposition: Nontrivial partitions of conserved physical quantities

A mathematical method for constructing fractal curves and surfaces, termed the $pλn$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal everywhere to the original function. Thus, the method is specially suited for constructing families of fractal objects arising from a conserved physical quantity, the decomposition yielding an exact partition of the quantity in question. Most prominent classes of examples are provided by Hamiltonians and partition functions of statistical ensembles: By using this method, any such function can be decomposed in the ordinary sum of a specified number of terms (generally fractal functions), the decomposition being both exact and valid everywhere on the domain of the function.

cond-mat.stat-mech↗

arXiv2015

Digital calculus: Cellular automata dynamics in closed form

A simple mathematical expression for the universal map for cellular automata is found in closed form with the help of a digit function, whose most basic properties are established. This result is found after proving a theorem on the composition of functions on finite sets. The expression (and the technique used to obtain it) opens the possibility of gaining mathematical insight in any cellular automaton rule since it constitutes at the same time a simple and fast algorithm to implement any such rule.

nlin.CG↗

arXiv2015

Substitution Systems and Nonextensive Statistics

Substitution systems evolve in time by generating sequences of symbols from a finite alphabet: At a certain iteration step, the existing symbols are systematically replaced by blocks of $N_{k}$ symbols also within the alphabet (with $N_{k}$, a natural number, being the length of the $k$-th block of the substitution). The dynamics of these systems leads naturally to fractals and self-similarity. By using $\mathcal{B}$-calculus [V. Garcia-Morales, Phys. Lett. A 376 (2012) 2645] universal maps for deterministic substitution systems both of constant and non-constant length, are formulated in 1D. It is then shown how these systems can be put in direct correspondence with Tsallis entropy. A `Second Law of Thermodynamics' is also proved for these systems in the asymptotic limit of large words.

math-ph↗

arXiv2015

Digital calculus and finite groups in quantum mechanics

By means of a digit function that has been introduced in a recent formulation of classical and quantum mechanics, we provide a new construction of some infinite families of finite groups, both abelian and nonabelian, of importance for theoretical, atomic and molecular physics. Our construction is not based on algebraic relationships satisfied by generators, but in establishing the appropriate law of composition that induces the group structure on a finite set of nonnegative integers (the cardinal of the set being equal to the order of the group) thus making computations with finite groups quite straightforward. We establish the abstract laws of composition for infinite families of finite groups including all cyclic groups (and any direct sums of them), dihedral, dicyclic and other metacyclic groups, the symmetric groups of permutations of $p$ symbols and the alternating groups of even permutations. Specific examples are given to illustrate the expressions for the law of composition obtained in each case.

math-ph↗

arXiv2015

Nonlocal and global dynamics of cellular automata: A theoretical computer arithmetic for real continuous maps

A digit function is presented which provides the $i$th-digit in base $p$ of any real number $x$. By means of this function, formulated within $\mathcal{B}$-calculus, the local, nonlocal and global dynamical behaviors of cellular automata (CAs) are systematically explored and universal maps are derived for the three levels of description. None of the maps contain any freely adjustable parameter and they are valid for any number of symbols in the alphabet $p$ and neighborhood range $ρ$. A discrete general method to approximate any real continuous map in the unit interval by a CA on the rational numbers $\mathbb{Q}$ (Diophantine approximation) is presented. This result leads to establish a correspondence between the qualitative behavior found in bifurcation diagrams of real nonlinear maps and the Wolfram classes of CAs. The method is applied to the logistic map, for which a logistic CA is derived. The period doubling cascade into chaos is interpreted as a sequence of global cellular automata of Wolfram's class 2 leading to Class 3 aperiodic behavior. Class 4 behavior is also found close to the period-3 orbits.

math-ph↗

arXiv2015

Quantum Mechanics and the Principle of Least Radix Economy

A new variational method, the principle of least radix economy, is formulated. The mathematical and physical relevance of the radix economy, also called digit capacity, is established, showing how physical laws can be derived from this concept in a unified way. The principle reinterprets and generalizes the principle of least action yielding two classes of physical solutions: least action paths and quantum wavefunctions. A new physical foundation of the Hilbert space of quantum mechanics is then accomplished and it is used to derive the Schrödinger and Dirac equations and the breaking of the commutativity of spacetime geometry. The formulation provides an explanation of how determinism and random statistical behavior coexist in spacetime and a framework is developed that allows dynamical processes to be formulated in terms of chains of digits. These methods lead to a new (pre-geometrical) foundation for Lorentz transformations and special relativity. The Parker-Rhodes combinatorial hierarchy is encompassed within our approach and this leads to an estimate of the interaction strength of the electromagnetic and gravitational forces that agrees with the experimental values to an error of less than one thousandth. Finally, it is shown how the principle of least-radix economy naturally gives rise to Boltzmann's principle of classical statistical thermodynamics. A new expression for a general (path-dependent) nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics.

physics.gen-ph↗

arXiv2013

Origin of Complexity and Conditional Predictability in Cellular Automata

A simple mechanism for the emergence of complexity in cellular automata out of predictable dynamics is described. This leads to unfold the concept of conditional predictability for systems whose trajectory can only be piecewise known. The mechanism is used to construct a cellular automaton model for discrete chimera-like states, where synchrony and incoherence in an ensemble of identical oscillators coexist. The incoherent region is shown to have a periodicity that is three orders of magnitude longer than the period of the synchronous oscillation.

nlin.CG↗

arXiv2012

Mathematical Physics of Cellular Automata

A universal map is derived for all deterministic 1D cellular automata (CA) containing no freely adjustable parameters. The map can be extended to an arbitrary number of dimensions and topologies and its invariances allow to classify all CA rules into equivalence classes. Complexity in 1D systems is then shown to emerge from the weak symmetry breaking of the addition modulo an integer number p. The latter symmetry is possessed by certain rules that produce Pascal simplices in their time evolution. These results elucidate Wolfram's classification of CA dynamics.

nlin.CG↗