Discrete symmetry analysis of lattice systems

Discrete dynamical systems and mesoscopic lattice models are considered from the point of view of their symmetry groups. Some peculiarities in behavior of discrete systems induced by symmetries are pointed out. We reveal also the group origin of moving soliton-like structures similar to “spaceships” in cellular automata.     

Dynamics of the universe with global rotation

We analyze the dynamics of the FRW models with global rotation in terms of dynamical system methods. We reduce the dynamics of these models to the FRW models with some fictitious fluid which scales like radiation matter. This fluid mimics dynamical effects of global rotation. The significance of the global rotation of the Universe for the resolution of the acceleration and horizon problems in cosmology is investigated. It is found that the dynamics of the Universe can be reduced to the two-dimensional Hamiltonian dynamical system. Then the construction of the Hamiltonian allows for full classification of evolution paths. On the phase portraits we find the domains of cosmic acceleration for the globally rotating universe as well as the trajectories for which the horizon problem is solved. We show that the FRW models with global rotation are structurally stable. This proves that the universe acceleration is due to the global rotation. It is also shown how global rotation gives a natural explanation of the empirical relation between angular momentum for clusters and superclusters of galaxies. The relation J ~ M² is obtained as a consequence of self similarity invariance of the dynamics of the FRW model with global rotation. In derivation of this relation we use the Lie group of symmetry analysis of differential equation.     

Renormalization of one-dimensional avalanche models

We investigate a renormalization group (RG) scheme for avalanche automata introduced recently by Pietroneroet al. to explain universality in self-organized criticality models. Using a modified approach, we construct exact RG equations for a one-dimensional model whose detailed dynamics is exactly solvable. We then investigate in detail the effect of approximations inherent in a practical implementation of the RG transformation where exact dynamical information is unavailable.     

Dynamical dimensional reduction

We propose to call a dynamical dimensional reduction effective if the corresponding dynamical system possesses a single attracting critical point representing expanding physical space-time and static internal space. We show that theBV × T ^D multidimensional cosmological model with a hydrodynamic energy-momentum tensor provides an example of effective dimensional reduction. We also study the dynamics of the multidimensional cosmological model of typeBI × T ^D with an energy-momentum tensor representing low temperature quantum effects, monopole contribution and the cosmological constant. It turns out that anisotropy and the cosmological constant are crucial for the process of dimensional reduction to be effective. We argue that this is the general property of homogeneous multidimensional cosmological models.     

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in ℝ ^k , called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in ℝ ^k . We also discuss a simple signaling pathway related to cancer research, called p53 module.     

Continuum tight binding models and quantum transport in the presence of dynamical disorder

Continuum limits of various tight binding linear chain lattices used in the author's recent study of quantum transport in the presence of dynamical disorder, are analyzed from the point of view of their energy level spectra when disorder is absent. These spectra show a linear dispersion similar to that in the Luttinger model, and describe the energy levels of the corresponding discrete systems in the range of midband wavevectors. Next, the more conventional longwavelength continuum limit, which describes the energy levels of the actual discrete systems near the bottom of the bands is discussed. On the basis of these properties it is argued that the applicability of the continuum models to the study of dynamical properties is restricted to low frequencies in the range of low lying excitations, near the midband Fermi level in half-filled band situations in the case of the midband models, and near the bottom of a nearly empty band for the longwavelength models. Finally, it is shown that in the presence of dynamic disorder the longwavelength continuum limit of a single-band tight-binding model leads to nondiffusive motion, with a mean squared displacement <x ²(t)>t ³, fort.     

Dimension on discrete spaces

In this paper we develop some combinatorial models for continuous spaces. We study the approximations of continuous spaces by graphs, molecular spaces, and coordinate matrices. We define the dimension on a discrete space by means of axioms based on an obvious geometrical background. This work presents some discrete models ofn-dimensional Euclidean spaces,n-dimensional spheres, a torus, and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities. It also analyzes the topology of (3+1)-space-time. We are also discussing the question by R. Sorkin about how to derive the system of simplicial complexes from a system of open coverings of a topological space.     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

Dissipative or conservative cosmology with dark energy?

All evolutional paths for all admissible initial conditions of FRW cosmological models with dissipative dust fluid (described by dark matter, baryonic matter and dark energy) are analyzed using dynamical system approach. With that approach, one is able to see how generic the class of solutions leading to the desired property—acceleration—is. The theory of dynamical systems also offers a possibility of investigating all possible solutions and their stability with tools of Newtonian mechanics of a particle moving in a one-dimensional potential which is parameterized by the cosmological scale factor. We demonstrate that flat cosmology with bulk viscosity can be treated as a conservative system with a potential function of the Chaplygin gas type. We characterize the class of dark energy models that admit late time de Sitter attractor solution in terms of the potential function of corresponding conservative system. We argue that inclusion of dissipation effects makes the model more realistic because of its structural stability. We also confront viscous models with SNIa observations. The best fitted models are obtained by minimizing the χ² function which is illustrated by residuals and χ² levels in the space of model independent parameters. The general conclusion is that SNIa data supports the viscous model without the cosmological constant. The obtained values of χ² statistic are comparable for both the viscous model and ΛCDM model. The Bayesian information criteria are used to compare the models with different power-law parameterization of viscous effects. Our result of this analysis shows that SNIa data supports viscous cosmology more than the ΛCDM model if the coefficient in viscosity parameterization is fixed. The Bayes factor is also used to obtain the posterior probability of the model.     

Discrete symmetry groups of vertex models in statistical mechanics

We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq ⁴-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.     

Solutions of several coupled discrete models in terms of Lamé polynomials of arbitrary order

Coupled discrete models are ubiquitous in a variety of physical contexts. We provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lamé polynomials of arbitrary order. The models discussed are: (i) coupled Salerno model, (ii) coupled Ablowitz?CLadik model, (iii) coupled ? ⁴ model and (iv) coupled ? ⁶ model. In all these cases we show that the coefficients of the Lamé polynomials are such that the Lamé polynomials can be re-expressed in terms of Chebyshev polynomials of the relevant Jacobi elliptic function.     

Fano resonance and wave transmission through a chain structure with an isolated ring composed of defects

We consider a discrete model that describes a linear chain of particles coupled to an isolated ring composed of N defects. This simple system can be regarded as a generalization of the familiar Fano-Anderson model. It can be used to model discrete networks of coupled defect modes in photonic crystals and simple waveguide arrays in two-dimensional lattices. The analytical result of the transmission coefficient is obtained, along with the conditions for perfect reflections and transmissions due to either destructive or constructive interferences. Using a simple example, we further investigate the relationship between the resonant frequencies and the number of defects N, and study how to affect the numbers of perfect reflections and transmissions. In addition, we demonstrate how these resonance transmissions and refections can be tuned by one nonlinear defect of the network that possesses a nonlinear Kerr-like response.     

Coherent State Path Integral for the Harmonic Oscillator and a Spin Particle in a Constant Magnetic field

Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP¹ model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.     

A New Class of Cellular Automata with a Discontinuous Glass Transition

We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density ρ _c for convergence to a completely empty configuration is non trivial, 0<ρ _c <1, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, ρ<ρ _c , emptying always occurs exponentially fast and that ρ _c coincides with the critical density for two-dimensional oriented site percolation on ℤ². This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is discontinuous and at the same time the crossover length diverges faster than any power law. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques. Finally, we will discuss why, due to the peculiar mixed critical/first order character of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D.S. Fisher.     

Correlation and fluctuations in relativistic nucleus-nucleus collisions

Correlation and fluctuations are now well accepted analysis techniques in heavy-ion collisions at relativistic energies. At the current stage of RHIC exploration, matter in bulk and many of the physics questions about the final stage of collisions are addressed with the help of correlation techniques. In the present work after a general introduction to the underlying formalism to the exotic phenomena of correlation and fluctuations, discussion on various parameters disentangling dynamical fluctuations is presented. Analysis to investigate dynamical fluctuations and correlation is carried out in terms of F _q - and G _q -moments. A study of various other parameters involving multiplicity and pseudorapidity of relativistic charged particles produced in high energy nuclear interactions reveals the presence of correlation and fluctuations in particle production in these collisions. The experimental data on 14.5A GeV/c ²⁸Si-nucleus interactions has been analyzed. A parallel analysis of correlation free data generated using MC-RAND Monte Carlo code, UrQMD data and for the HIJING generated events has also been carried out.     

Spectral Representation and the Averaging Problem in Cosmology1

We investigate the averaging problem in cosmology as the problem of introducing a distance between spaces. We first introduce the spectral distance, which is a measure of closeness between spaces defined in terms of the spectra of the Laplacian. Then we define S _N, a space of all spaces equipped with the spectral distance. We argue that S _N can be regarded as a metric space and that it also possesses other desirable properties. These facts make S _N a fundamental arena for spacetime physics. Then, we apply the spectral framework to the averaging problem: We describe the model-fitting procedure in terms of the spectral representation, and also discuss how to analyze the dynamical aspects of the averaging procedure with this scheme. In these analyses, we are naturally led to the concept of the apparatus-dependent and the scale-dependent effective evolution of the universe. These observations suggest that the spectral scheme seems to be suitable for the quantitative analysis of the averaging problem in cosmology.     

Molecular and Functional Aspects of Bacterial Chemotaxis

We consider the dynamics of chemotaxis in the model bacterium Escherichia coli. We analyze both its molecular mechanisms and the functional causes governing the evolution of the observed behaviors. We review molecular models of the transduction network controlling the bacterial chemotaxis in response to chemoattractant binding to the receptors. In particular, recent progress stimulated by FRET experiments is presented for statistical physics allosteric models. The response function to a pulse of chemoattractant is expressed in terms of microscopic parameters of the allosteric models. The functional causes for the shape of the response function, as measured in experimental tethering assay, are then investigated. A hydrodynamic equation, valid for space-time scales larger than the microscopic running length and time, is derived for the position of a swimming bacterium. It is then shown how optimization over the microscopic parameters of the response function yields the curve observed experimentally. In particular, the observed property of adaptation to the background level of aspartate emerges as being produced by fluctuations in the space-time chemoattractant profiles sensed by bacteria along their trajectories. This functional cause is distinct from arguments based on the extension of the dynamical range. Future directions and experiments to probe the adaptation of E. coli chemotaxis to the environmental conditions and its response to realistic space-time chemoattractant stimuli are finally discussed.     

Boolean delay equations: A simple way of looking at complex systems

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time; such BDEs can be seen therefore as metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal solution sets, such as Devil’s staircases and “fractal sunbursts.” All known solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades of loading and failure in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid-earth problems. The former have used small systems of BDEs, while the latter have used large hierarchical networks of BDEs. We moreover introduce BDEs with an infinite number of variables distributed in space (“partial BDEs”) and discuss connections with other types of discrete dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.     

Self-gravitating darkon fluid with anisotropic scaling

The fluid model for the dark sector of the universe (darkon fluid), introduced previously in Phys. Rev. D 80, 083513, 2009, is reformulated as a modified model involving only variables from the physical phase space. The Lagrangian of the model does not possess a free particle limit and hence the particles it describes, darkons, exist only as a self-gravitating fluid. This darkon fluid presents a dynamical realization of the zero-mass Galilean algebra extended by anisotropic dilational symmetry with dynamical exponent z=\frac53z=\frac{5}{3}. The model possesses cosmologically relevant solutions which are identical to those in the previous paper. We derive also the equations for the cosmological perturbations at early times and determine their solutions. In addition, we discuss also some implications of adding higher spatial-derivative terms.