From deterministic cellular automata to coupled map lattices
A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit $\kappa \to 0$ of a continuous parameter $\kappa$. Thus, RDCAs generalize CAs in such a way that they constitute CMLs when $\kappa$ is finite and nonvanishing. In the limit $\kappa \to \infty$ all RDCAs are shown to ex…
Semipredictable dynamical systems
A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We prove a general result: The dynamics of any deterministic nonlinear cellular automaton (CA) with $p$ possible dynamical states can be decomposed at each instant of time in a superposition of $N$ layers involving $p_{0}$, $p_{1}$,... $p_{N-1}$ dynamical states each, where the $p_{k\in \mathbb{N}}$, $k \in [0, N-1]$ are divisors of $p$. If the divisors coincide with the prime factors of $p$ this decomposition is unique. Conversely, we also prove that $N$ CA w…
Nonlinear embeddings: Applications to analysis, fractals and polynomial root finding
We introduce $\mathcal{B}_{\kappa}$-embeddings, nonlinear mathematical structures that connect, through smooth paths parameterized by $\kappa$, a finite or denumerable set of objects at $\kappa=0$ (e.g. numbers, functions, vectors, coefficients of a generating function...) to their ordinary sum at $\kappa \to \infty$. We show that $\mathcal{B}_{\kappa}$-embeddings can be used to design nonlinear irreversible processes through this connection. A number of examples of increasing complexity are worked out to illustrate the possibilities uncovered by this concept. These include not only smooth functions but also fractals on the real line and on the complex plane. As an application, we use $\mat…
Cellular automaton for chimera states
A minimalistic model for chimera states is presented. The model is a cellular automaton (CA) which depends on only one adjustable parameter, the range of the nonlocal coupling, and is built from elementary cellular automata and the majority (voting) rule. This suggests the universality of chimera-like behavior from a new point of view: Already simple CA rules based on the majority rule exhibit this behavior. After a short transient, we find chimera states for arbitrary initial conditions, the system spontaneously splitting into stable domains separated by static boundaries, ones synchronously oscillating and the others incoherent. When the coupling range is local, nontrivial coherent struct…
Electrochemical impedance spectroscopy of polyelectrolyte multilayer modified gold electrodes: influence of supporting electrolyte and temperature.
Electrochemical impedance spectroscopy and cyclic voltammetry are employed to characterize poly(styrenesulfonate)/poly(allylamine hydrochloride) multilayers assembled onto cysteamine-modified gold surfaces. The influence of the supporting electrolyte and temperature on the impedance response is studied because of both its practical interest and the need to test further the capillary membrane model recently developed by Barreira et al. [J. Phys. Chem. B 2004, 108, 17973]. The results obtained are interpreted quite satisfactorily in terms of this model, thus providing additional support to its usefulness for the description of ionic transport through polyelectrolyte multilayers. It is observe…
Pore entrance effects on the electrical potential distribution in charged porous membranes and ion channels
Abstract Models for the electrical potential distribution in the interfacial region between a fixed charge membrane and an electrolyte solution have traditionally employed the Donnan equilibrium formalism that assumes discontinuous changes in concentrations and electric potential. In the case of the charged capillary membrane model, we propose to check rigorously the validity of this approach by solving the linearized Poisson–Boltzmann equation for the diffuse electrical double layer at the membrane|solution interface. The comparison of the resulting axial distribution for the electric potential with the Donnan potential drop shows that the discontinuous approach is only valid for membrane …
Unifying vectors and matrices of different dimensions through nonlinear embeddings
Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, derivi…
Preparation of nanostructures composed of dextran sulfate/ruthenium nanoparticles and their interaction with phospholipid monolayers at a liquid–liquid interface
Abstract Nanostructures composed of dextran sulfate (DS)/ruthenium (Ru) nanoparticles (NPs) adsorbed on phospholipid monolayers at a liquid–liquid interface were prepared and characterized electrochemically in relation to their potential use in drug delivery systems. First, positively charged Ru NPs were prepared, and then negatively charged DS was adsorbed on the surface of the NPs, thus forming well-defined and organized structures, as observed under the transmission electron microscope, which are referred to composite nanoclusters. The lipid monolayers were formed by depositing either 1-palmitoyl-2-oleoyl- sn -glycero-3-phosphatidylcholine or 1-palmitoyl-2-oleoyl- sn -glycero-3-phospho- …
A constructive theory of shape
We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of qualitatively similar shapes are constructed giving as input a finite ordered set of characteristic points (landmarks) and the value of a continuous parameter $\kappa \in (0,\infty)$. We prove that all shapes belonging to the same family are located within the convex hull of the landmarks. The theory is constructive in the sense that it provides a systematic means to build a mathematical model for any shape taken from the physical world. We illustrate this with a va…
Ion transport through polyelectrolyte multilayers under steady-state conditions
Abstract The permeability of a self-assembled polyelectrolyte multilayer to small ions under the influence of an applied potential difference is studied as a function of the number of layers and the nature of the supporting electrolyte. The multilayer is described as a series of homogeneously charged membranes with alternating sign of their fixed charge. Ion transport is described on the basis of the diffusion equation and the assumption of (Donnan) electrochemical equilibrium at the boundaries between layers. The calculated steady-state current–voltage curves are found to be in good agreement with experimental linear sweep voltammograms (at low sweep rate). The permeability of polystyrenes…
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.
Estimation of pKa shifts in weak polyacids using a simple molecular model: effects of strong polybases, hydrogen bonding and divalent counterion binding
Abstract The pKa values of ionizable groups in macromolecules can be significantly different than those of the isolated groups in solution. We have estimated theoretically the changes in the dissociation constant of a weak acid (a) in the vicinity of another ionizable group (b) on the basis of the theoretical approach by Hill (J. Am. Chem. Soc. 78 (1956) 3330) for matching pairs of interacting sites on two large molecules. Three cases are considered for group b: the strong base, the same weak acid as group a with hydrogen bonding between them, and the same weak acid as group a with divalent counterion binding. The pKa shifts are evaluated in each case as a function of the interaction energy…
Electrochemical Impedance Spectroscopy of Polyelectrolyte Multilayer Modified Electrodes
Electrochemical impedance spectroscopy, Fourier transform infrared reflection−absorption spectroscopy, and cyclic voltammetry were employed to characterize polyelectrolyte multilayers (PEMs) fabricated with poly(styrenesulfonate) as the polyanion and the polypeptides poly-l-histidine, poly-l-lysine, and poly-l-arginine as polycations. The layer-by-layer electrostatic assembly was produced onto alkanethiol-modified gold surfaces. The frequency response reveals that the effect of the number of layers seems to be related to a progressive reduction in the active area of the PEM-modified electrodes. The active area after the deposition of seven layers can be lower than 10% of its original value.…
Electrochemical Characterization of Polyelectrolyte/Gold Nanoparticle Multilayers Self-Assembled on Gold Electrodes
Polyelectrolyte/gold nanoparticle multilayers composed of poly(l-lysine) (pLys) and mercaptosuccinic acid (MSA) stabilized gold nanoparticles (Au NPs) were built up using the electrostatic layer-by-layer self-assembly technique upon a gold electrode modified with a first layer of MSA. The assemblies were characterized using UV-vis absorption spectroscopy, cyclic and square-wave voltammetry, electrochemical impedance spectroscopy, and atomic force microscopy. Charge transport through the multilayer was studied experimentally as well as theoretically by using two different redox pairs [Fe(CN)(6)](3-/4-) and [Ru(NH(3))(6)](3+/2+). This paper reports a large sensitivity to the charge of the out…
Hybrid Circuits with Nanofluidic Diodes and Load Capacitors
[EN] The chemical and physical input signals characteristic of micro- and nanofluidic devices operating in ionic solutions should eventually be translated into output electric currents and potentials that are monitored with solid-state components. This crucial step requires the design of hybrid circuits showing robust electrical coupling between ionic solutions and electronic elements. We study experimentally and theoretically the connectivity of the nanofluidic diodes in single-pore and multipore membranes with conventional capacitor systems for the cases of constant, periodic, and white-noise input potentials. The experiments demonstrate the reliable operation of these hybrid circuits ove…
A new approach to fuzzy sets: Application to the design of nonlinear time-series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillations
It is shown that characteristic functions of sets can be made fuzzy by means of the $\mathcal{B}_{\kappa}$-function, recently introduced by the author, where the fuzziness parameter $\kappa \in \mathbb{R}$ controls how much a fuzzy set deviates from the crisp set obtained in the limit $\kappa \to 0$. As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time-series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter…
Substitution systems and nonextensive statistics
Abstract 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 B -calculus (Garcia-Morales, 2012) 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 prove…
The $p\lambda n$ fractal decomposition: Nontrivial partitions of conserved physical quantities
A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ 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 (g…
Calculation of the wetting parameter from a cluster model in the framework of nanothermodynamics
The critical wetting parameter ${\ensuremath{\omega}}_{c}$ determines the strength of interfacial fluctuations in critical wetting transitions. In this Brief Report, we calculate ${\ensuremath{\omega}}_{c}$ from considerations on critical liquid clusters inside a vapor phase. The starting point is a cluster model developed by Hill and Chamberlin in the framework of nanothermodynamics [Proc. Natl. Acad. Sci. USA 95, 12779 (1998)]. Our calculations yield results for ${\ensuremath{\omega}}_{c}$ between 0.52 and 1.00, depending on the degrees of freedom considered. The findings are in agreement with previous experimental results and give an idea of the universal dynamical behavior of the cluste…
Effect of cationic polyamidoamine dendrimers on ionic transport through nanochannels
Abstract The effect of polyamidoamine (PAMAM) dendrimers (generations G0–G3) on the ion transport properties of nanochannels (conical and cylindrical) is studied either by surface functionalization or by addition to the electrolyte solution. Surface functionalization with cationic dendrimers lead to inversion of the ion current rectification, indicating the anion selectivity of the modified nanochannels. This anion selectivity increases by immobilizing higher-generation dendrimers, as expected for an increase in the surface density of amino groups. However, compared to PAMAM G2, functionalization with PAMAM G3 results in higher cation and lower anion fluxes. Diffusion experiments of charged…
Weakly coupled map lattice models for multicellular patterning and collective normalization of abnormal single-cell states
We present a weakly coupled map lattice model for patterning that explores the effects exerted by weakening the local dynamic rules on model biological and artificial networks composed of two-state building blocks (cells). To this end, we use two cellular automata models based on: (i) a smooth majority rule (model I) and (ii) a set of rules similar to those of Conway's Game of Life (model II). The normal and abnormal cell states evolve according with local rules that are modulated by a parameter $\kappa$. This parameter quantifies the effective weakening of the prescribed rules due to the limited coupling of each cell to its neighborhood and can be experimentally controlled by appropriate e…
Microcanonical foundation of nonextensivity and generalized thermostatistics based on the fractality of the phase space
We develop a generalized theory of (meta)equilibrium statistical mechanics in the thermodynamic limit valid for both smooth and fractal phase spaces. In the former case, our approach leads naturally to Boltzmann-Gibbs standard thermostatistics while, in the latter, Tsallis thermostatistics is straightforwardly obtained as the most appropriate formalism. We first focus on the microcanonical ensemble stressing the importance of the limit $t \to \infty$ on the form of the microcanonical measure. Interestingly, this approach leads to interpret the entropic index $q$ as the box-counting dimension of the (microcanonical) phase space when fractality is considered.
Analysis of adsorption of phospholipids at the 1,2-dichloroethane/water interface by electrochemical impedance spectroscopy: A study of the effect of the saturated alkyl chain
Abstract The adsorption behaviour of a series of phosphatidylcholines (PCs) with saturated carbon chains of different length (DLPC, DPPC, DSPC, DAPC, and DBPC) at the electrified 1,2-dichloroethane/water interface was studied by measuring electrochemical impedance spectroscopy at the polarized interface. Two different trends in the interfacial capacitance were observed for any of the PCs the capacity dependence on the applied potential: strong adsorption occurs at negative potential with a marked decrease of C ( E ); increase of capacity is observed at positive potentials. It is demonstrated that the interfacial lipid adsorption was dependent on phospholipid concentration, applied potential…
Ion size effects on the electrokinetic flow in nanoporous membranes caused by concentration gradients
The space charge model (SCM) relies on the Poisson−Boltzmann (PB) equation, and hence on the assumption of negligible ion size effects, to calculate the radial distribution of the electrical potent...
Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators
We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several e…
Digit replacement: A generic map for nonlinear dynamical systems
A simple discontinuous map is proposed as a generic model for nonlinear dynamical systems. The orbit of the map admits exact solutions for wide regions in parameter space and the method employed (digit manipulation) allows the mathematical design of useful signals, such as regular or aperiodic oscillations with specific waveforms, the construction of complex attractors with nontrivial properties as well as the coexistence of different basins of attraction in phase space with different qualitative properties. A detailed analysis of the dynamical behavior of the map suggests how the latter can be used in the modeling of complex nonlinear dynamics including, e.g., aperiodic nonchaotic attracto…
Correct thermodynamic forces in Tsallis Thermodynamics: connection with Hill Nanothermodynamics
The equivalence between Tsallis Thermodynamics and Hill Nanothermodynamics is established. The correct thermodynamic forces in Tsallis thermodynamics are pointed out. Through this connection we also find a general expression for the entropic index $q$ which we illustrate with two physical examples, allowing in both cases to relate $q$ to the underlying dynamics of the Hamiltonian systems.
Fluoride-Induced Negative Differential Resistance in Nanopores: Experimental and Theoretical Characterization
We describe experimentally and theoretically the fluoride-induced negative differential resistance (NDR) phenomena observed in conical nanopores operating in aqueous electrolyte solutions. The threshold voltage switching occurs around 1 V and leads to sharp current drops in the nA range with a peak-to-valley ratio close to 10. The experimental characterization of the NDR effect with single pore and multipore samples concern different pore radii, charge concentrations, scan rates, salt concentrations, solvents, and cations. The experimental fact that the effective radius of the pore tip zone is of the same order of magnitude as the Debye length for the low salt concentrations used here is su…
Diagrammatic approach to cellular automata and the emergence of form with inner structure
We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allow CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time, arbitrary neighborhood ranges and topology. We provide several examples of the method, illustrating how it can be applied to the mathematical modeling of the emergence of order out of disorder. Specif…
A new approach to fuzzy sets: Application to the design of nonlinear time series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillations
Abstract It is shown that characteristic functions of sets can be made fuzzy by means of the B κ -function, recently introduced by the author, where the fuzziness parameter κ ∈ R controls how much a fuzzy set deviates from the crisp set obtained in the limit κ → 0. As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter κ plays an analogous role to temper…
Coupling theory for counterion distributions based in Tsallis statistics
It is well known that the Poisson-Boltzmann (PB) equation yields the exact counterion density around charged objects in the weak coupling limit. In this paper we generalize the PB approach to account for coupling of arbitrary strength by making use of Tsallis q-exponential distributions. Both the weak coupling and the strong coupling limits are reproduced. For arbitrary coupling we also provide simple analytical expressions which are compared to recent Monte Carlo simulations by A. G. Moreira and R. R. Netz [Europhys. Lett. 52 (2000) 705]. Excellent agreement with these is obtained.