From Phase Transitions to Chaos

The following sections are included:

• PÉTER SZÉPFALUSY: A SHORT BIOGRAPHY

• PROLOGUE

• TABLE OF CONTENTS

Cluster numbers in the 2D Ising model at and above the Curie temperature T_c are observed in computer simulations. While at 1.1 T_c we find all cluster numbers to relax towards zero exponentially, with the same decay time for long times, right at the Curie temperature a different (scaling) behavior sets in. These results are in qualitative agreement with the Becker-Döring equation for the cluster numbers.

Ising strips with an interface and with free boundary conditions are studied applying exact methods and Monte Carlo techniques. Ranges of validity of asymptotic expressions for the correlation lengths and crossover effects are discussed.

We present a phenomenological model for the segregation of a binary alloy in the presence of a gravitational field. Our numerical results indicate that the coarsening domains are characterised by different length scales in the directions along and perpendicular to the gravitational field, with a much faster growth in the direction of the field.

Immiscible binary mixtures containing surfactants, which lower the interfacial tension, show rich phase behaviors. Equilibrium features as well as dynamical aspects of such systems have been extensively investigated and are briefly reviewed in this article emphasizing our own works. Our works described here are concerned with the development of the hybrid model in which a binary mixture is treated as a continuum and surfactants as discrete molecules. The case of block copolymers as surfactants is also discussed from the hybrid point of view.

We review the possible ground states and phase transitions in one-dimensional magnetic chains with special emphasis to the difference between integral and half integral spin values. Both exact and numerical results are presented for the Heisenberg model with nearest neighbour spin exchange and for models with higher order spin couplings. They are compared to conjectures derived from approximate mappings to field theory models. The spin-Peierls transition is also treated.

The isotropic n-component critical dynamics model of Sasvári, Schwabl and Szépfalusy is found to have some general n-independent features. We find that quasi-scaling, important for interpreting experimental thermal conductivity measurements in liquid He⁴, corresponding to n = 2, also occurs at the “natural boundary”, n = 1. In this simpler case, the critical region for the entropy diffusion is again relatively small and the crossover behavior can be described in terms of a quasi-scaling exponent with a value approximately equal to that found for n = 2.

Starting with the description of stochastic processes by path integrals, the methods of renormalized field theory are used to get insight in critical relaxation. It is elucidated that, beginning with a noncritical initial state, even early stages of the relaxation process display universal behaviour governed by a new critical exponent. Using short distance expansions, the form of the correlation function at the late stage and the order parameter relaxation for all times are calculated to one-loop order.

A novel one-dimensional single spin flip kinetic Ising model with checkerboard updating is introduced in which ballistic-type motion of kinks and their random walk behaviour coexist in the form of a crossover phenomenon. The time-dependent structure factor and the dynamical critical exponent are investigated via computer simulation. Dynamic scaling theory with two scaling functions is presented to describe the situation.

The scaling properties of the structure factor in the late stage of a quench from high temperature to zero temperature are obtained analytically by solving the TDGL model, both by integration of the equation of motion and via renormalization group, in the large-N limit. It is found that the usual form of scaling (standard scaling) holds when the order parameter is not conserved, while for a conserved order parameter the structure factor obeys a novel form of scaling (multiscaling) characterized by infinitely many scaling indices.

We present an extension of the Swift–Hohenberg equation to the case of a high Prandtl number Bénard experiment in rotating fluid containers. For the case of circular containers we find complex spatio-temporal behaviour at Taylor numbers smaller than the critical one for the onset of the Küppers-Lortz instability. Furthermore, above the critical Taylor number the experimentally well-known time dependent and spatially disordered patterns in form of local patches of rolls are reproduced.

The thorough understanding of the dynamic behaviour of chemical oscillators makes it necessary to investigate the influence of various perturbing agents. Here we report on our results obtained with chloride perturbed aromatic bromate oscillators. The usual temperature effect which calls forth the transition of a reactive system from a non-oscillatory state to an oscillatory one by raising the temperature only by 1°C has not been reported for any oscillatory system studied so far. We demonstrated some connection between chemical and biological oscillators.

A simple but rigorous approach of sum-rule techniques being applied to the density fluctuations of the centers of cyclotron orbits provides the study of the low-lying collective modes of interacting electrons in a plane perpendicular to high magnetic fields. We derive the dispersion relation of a gapless Goldstone mode within the fractionally filled lowest Landau level, resulting from broken magnetic translational invariance and to be identified with the magnetophonons of this sort of a two-dimensional Wigner-solid. The role of the shear modulus for the stability of the two-dimensional electron solid is investigated in this approach.

Strongly correlated electron systems, such as heavy fermionic materials, show a colourful variety of low-temperature behaviour which arises from a competition between magnetic ordering and the formation of an overall singlet state. The variational method offers a possibility for mapping out the ground state phase diagram of strongly interacting quantum systems. A brief review of recent work on the variational treatment of the Kondo lattice model is given, which led to a phase diagram showing a domain of collective Kondo behaviour lying between the regions of RKKY-type magnetism at weak coupling, and Nagaoka-type ferromagnetism at strong coupling.

We discuss a path integral formula for the partition function, an inequality for the free energy and some bounds on ferromagnetism near half-filling in the above model.

The quantum Hall effect is due to localization-delocalization transitions occurring in two dimensional disordered systems at zero temperature once the Fermi energy coincides with one of the energy eigenvalues of a free electron in a perpendicular magnetic field. In the limit of infinite magnetic field these transitions are manifestly a classical percolation transition from closed to open equipotential orbits in a two dimensional random potential. At finite values of the field quantum corrections become important which change the critical exponent of the localization length to a value near 7/3. Anomalous diffusion and the chaotic structure of the wave function exhibited for systems near the transition point can adequately be described by the concept of multifractality. The aim of the paper is to demonstrate the motivation and advantages of this concept with respect to the quantum Hall effect.

Theories and experiments concerning the dependence of the in-layer resistivity of the cuprate superconductors on the temperature T are reviewed. From the theoretical side, the possible separation between charge and spin of the carriers in the CuO₂ planes is considered as alternative to a phonon mechanism. Both lead to the observed linear dependence in high-T approximation while scatterings between carriers might be the appropriate explanation in low-T approximation. Experimentally, the linear law ρ|∞ ∝ T is regarded as being generic, but exceptions which may be described by a T²-law are considered as possibly significant.

The critical replica number n_c where the uniqueness of the analytic continuation of the moments (Zⁿ) of the partition function from integer to real n's breaks down is calculated to second order in the inverse dimensionality (1/d) expansion and also to first order in the loop expansion for fixed d > 6. Results obtained by both techniques show that n_c, a clear-cut measure of the strength of replica symmetry breaking effects, increases with decreasing dimensionality.

The artificial neural network learning process of Kohonen's Learning Vector Quantization classifying algorithm is treated in the hydrodynamic limit. The hydrodynamic modes are specified. Transport coefficients are calculated and found to obey an Einstein-type relation, offering an explanation to the sharpness of classification. Neurons representing one class, if initialized in an input range dominated by the opposite class, start learning by a process similar to spinodal decomposition.

The one-dimensional random field Ising model (ID RFIM) is related to a nonlinear discrete stochastic mapping for an effective local random field which has for nonzero temperature a multifractal measure which may be thin or fat. By means of symbolic dynamics we distinguish parameter regions where the measure at the boundary of the support diverges or goes to zero with infinite or zero slope, respectively. Within the thermodynamic formalism we calculate generalized fractal dimensions as function of physical parameters.

The properties of the escape probability for a random walk on a one-dimensional lattice with disorder is discussed in terms of random maps. For the case of Sinai disorder, we observe geometric and dynamic behavior similar to that found in the case of intermittent deterministic chaos.

During the past few years a significant progress has been made in our understanding of the dynamics of kinetic roughening due to the convergence of important new results from computer simulations, analytical theories and experiments. This is a good example where these three major approaches of physics can be successfully applied to a class of far-from-equilibrium phenomena. Here we review the associated dynamic scaling formalism and present some of our latest results on the behaviour of theoretically motivated, computer generated and experimentally observed interfaces.

In every drop of water, down at the scale of atoms and molecules, there is a world that can fascinate anyone—ranging from a non-verbal young science student to an ardent science-phobe. The objective of Learning Science through Guided Discovery: Liquid Water and Molecular Networks is to use advanced technology to provide a window into this submicroscopic world, and thereby allow students to discover by themselves a new world. We have developed a coordinated two-fold approach in which a cycle of hands-on activities, games, and experimentation is followed by a cycle of advanced computer simulations employing the full power of computer animation to “ZOOM” into the depths of his or her newly-discovered world, an interactive experience surpassing that of an OMNIMAX theater. Pairing of laboratory experiments with corresponding simulations challenges students to understand multiple representations of concepts. Answers to student questions, resolution of student misconceptions, and eventual personalized student discoveries are all guided by a clear set of “cues” which we build into the computer display. We thereby provide students with the opportunity to work in a fashion analogous to that in which practicing scientists work—e.g., by using advanced technology to “build up” to general principles from specific experiences. Moreover, the ability to visualize “real-time” dynamic motions allows for student-controlled animated graphic simulations on the molecular scale and interactive guided lessons superior to those afforded by even the most artful of existing texts.

An example of the weakly nonlinear resonance is considered. The unbounded resonance structure is described in detail, and is shown to be unstable against weak perturbations. Peculiarities of diffusive motion within the intricate chaotic component are discussed.

Statistical structures of widespread chaos in Hamiltonian systems are shown to be characterized by the coexisting islands of tori and the homoclinic tangencies of unstable and stable manifolds. The time correlations of local quantities obey power-law decay C_t ∞ t(−(β−1) with 1 < β < 2 due to the intermittent sticking to islands. The spectrum of the coarse-grained expansion rates of nearby chaotic orbits has two linear parts with slopes zero and −2. The slope zero is caused by the intermittent sticking to islands, whereas the slope −2 is produced by the dynamics of the tangency points. The probability density of the coarse-grained expansion rate Λ_n over n (> 1) iterates is given by P(Λ:_n)∞ nexp[2nΛ] for 0 > Λ > Λmin and the scaling law P(β:n) = n⁶p(n⁶Λ), (Λ ≡ Λ − Λ^∞) with δ = (β − l)/β for 0 < Λ < Λ^∞, (Λ^∞ ≡ Liapunov exponent). For 0 < Λ < Λ^∞, p(x) ∞ |x|^−(1+β) for |x| ≫ 1, whereas p(x) ∞ expl[−ax^1/δ], (a > 0) for 0 < Λ < Λ^∞.

The problem of normal and anomalous transport of a particle in the phase space of a 4D map is examined. The map arises from the problem of the motion of a charged particle in a constant magnetic field and an electrostatic wave packet. The case of weak chaos is considered. Due to the finite observation time, the particle diffusion possesses strongly inhomogeneous properties. We introduce new objects into the analysis of dynamical chaos — we called them “chaotic jets” These are long living bundles of orbits with coherent propagation properties. The existence of chaotic jets depends in a complex manner on the topology of the phase space and strongly influences the asymptotic law of transport. The results of wavelet analysis applied to the trajectories are also shown.

We briefly review several recent results in the theory of diffusion in the light of the last decade advances on dynamical systems. We show how the diffusion coefficient can be calculated from the classical Ruelle resonances of the dynamical system, i.e., from complex zeros of the dynamical zeta function. The diffusion coefficient is also related to the characteristic properties of chaos on the fractal repeller underlying the diffusion process in open deterministic systems. Finally, we present a thermodynamic formalism based on the generating function of the diffusion coefficient and higher moments of diffusion and we show how anomalous diffusion is related to dynamical phase transitions in this formalism.

The linear response of a completely chaotic map to a particularly chosen constant perturbation has been calculated analytically. In the infinite-time limit, it is compared with the full nonlinear response. All physically relevant observables relax to the asymptotic value, some of them after a single time step; one of the explicitly calculated observables shows exponential relaxation with a relaxation time related to the Lyapunov exponent of the unperturbed map.

A systematic method for casting chaos into a probabilistic process obeying to a Master reqation is developed and illustrated on simple examples of chaotic dynamics such as the tent map with slope less than 2 and a piecewise linear map showing intermittent behavior.

This is a brief account of various aspects of the evolutionary pattern of a driven nonlinear oscillator. The model was proposed a century ago by Helmholtz in order to explain the combinational tones in the drumskin, and more recently, by J. M. T. Thompson, to study ship stability against waves in windy situations.

We review linear and nonlinear properties of the response of the discrete Ricker system x_n+1 = x_nexp[r(l − x_n)] against modulating the control parameter r with periodic, aperiodic and stochastic perturbations ξ_n via r(l + Δξ_n) in the range where the unmodulated map undergoes its first pitchfork bifurcation from the fixed point x^* = 1 into a period-2 limit cycle. The stability threshold of the fixed point is determined for periodic and stochastic modulation types as a function of the modulation amplitude Δ. The time-dependent nonlinear solutions which bifurcate out of the fixed point and which depend sensitively on details of the modulation dynamics are investigated analytically in comparison with numerical simulations.

Schröder's ‘generating function’ z = Φ(t), f(z) = Φ(ωt) for the logistic map f(z) = ωz(1 + z) is investigated. For |ω| > 1, Φ is an entire function. Its basic mathematical properties: Taylor series, asymptotic behaviour, product representation and inverses are discussed. Scaling laws are described. A periodic function S is introduced, providing the connection between Φ and the F-function of Douady and Hubbard. The region of interest for applications to the iterated map f is the negative t axis. There, the behaviour of Φ is intimately tied to the Julia set of f. Computer graphs of Φ for several characteristic values of ω are shown.

Application of a symbolic dynamics to a particular control-variable-tuned fully developed chaotic iterated map leads to an exactly solvable model of a phase transition in one dimension.

The fluctuation spectrum approach aiming at a long-time characterization of dynamical fluctuations of chaotic systems and a new approach to their infinite time-correlations are briefly reviewed. These are complementary to the multifractal theory of strange sets in the sense that they deal with explicit dynamical fluctuations. We will discuss fundamental approximations, periodic cycle determination, and scaling laws near transition points for quantities relevant to the present approaches.

For dynamical systems perturbed by weak noise, satisfying a large deviation condition, “non-equilibrium potentials” can be defined, whose properties generalize well-known ones of thermodynamic potentials in thermodynamic equilibrium: they are related to the invariant probability density in a familiar way and can serve as Lyapunov functions for the unperturbed dynamical system. In the present article we review recent progress in the study of non-equilibrium potentials for dynamical systems described by one- and two-dimensional maps, including examples with fractal attractors and repellers.

The valoric interpretation developed already by Clausius, Helmholtz and Ostwald is taken as the basis for a reinterpretation of the various entropy concepts developed by Clausius, Boltzmann, Gibbs, Shannon and Kolmogorov. As the key points, we consider the value of energy with respect to work and the value of entropy with respect to information-processing. The last part is devoted to an entropic analysis of strings of letters with special emphasis to long range correlations.

We show that for piecewise continuous piecewise monotone interval maps to define topological entropy one can use the standard Bowen's method, as well as the ideas of symbolic dynamics. As for continuous interval maps, positive entropy is due to the existence of horseshoes. This allows us to prove lower semi-continuity of topological entropy as a function of a map on some natural spaces of piecewise continuous piecewise monotone maps.

We discuss some recent developments in the study of diffusion in different versions of the Lorentz gas with the special emphasis on rigorous results and open problems.

First we briefly review a recent breakthrough in the verification of the Boltzmann-Sinai ergodic hypothesis formulated for systems of elastic hard balls known to be isomorphic to semi-dispersing billiards. A basic tool behind these results: the Fundamental Theorem for Semi-Dispersing Billiards is, however, also applicable to establishing the ergodicity of planar billiards with convex boundary components. As an application, we elucidate the idea of the proof of ergodicity of Wojtkowski's planar billiards with convex-scattering boundary components generalizing Bunimovich's famous stadium and, moreover, survey related results and questions.

It is shown that certain aspects of the quantum mechanics of a free particle on the modular surface are encoded in spectral properties of a generalized transfer operator of a conveniently chosen Poincaré map for the classical Hamiltonian flow for this particle.

We study noncommutative generalizations of the Bernoulli shift. With the appropriately extended definitions of dynamical and entropic entropy we calculate how the noncommutativity decreases these quantities which depend sensitively on the details of the model.

A random domain with smooth boundary is considered in the plane. Let us magnify this domain by some positive number λ. We are interested in the difference between the area of this magnified domain and the number of lattice points contained in it. We show that if the domain is chosen by a probability measure with some nice properties, then the expected value of this difference is bounded by a constant, independent of λ, and, if λ > 1, its variance is bounded by constant times the expected value of the length of the boundary. Problems like this arise in the investigation of quantum chaos, as well.

Some mathematical problems of Nonequilibrium Statistical Mechanics are exposed in the case of coupled anharmonic oscillators. The macroscopic behaviour of such systems is described in terms of hydrodynamic scaling. The crucial step of the argument is the Ergodic Hypothesis, which can be verified only in the presence of some artificial random effects.

We describe a nonlinear partial differential equation on a compact domain with solutions regular but with rough behaviour down to a small scale.