Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The Levy walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, bio-physics, and behavioral science demonstrate that this particular type of random walks provides significant insight into complex transport phenomena. This review provides a self-consistent introduction to Levy walks, surveys their existing applications, including latest advances, and outlines further perspectives.

L\'evy walks

The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism—a dynamical Gibbs ensemble construction—to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.

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Thermodynamic Formalism for Systems with Markov Dynamics

With focus on anharmonic chains, we develop a nonlinear version of fluctuating hydrodynamics, in which the Euler currents are kept to second order in the deviations from equilibrium and dissipation plus noise are added. The required model-dependent parameters are written in such a way that they can be computed numerically within seconds, once the interaction potential, pressure, and temperature are given. In principle the theory is applicable to any one-dimensional system with local conservation laws. The resulting nonlinear stochastic field theory is handled in the one-loop approximation. Some of the large scale predictions can still be worked out analytically. For more details one has to rely on numerical simulations of the corresponding mode-coupling equations. In this way we arrive at detailed predictions for the equilibrium time correlations of the locally conserved fields of an anharmonic chain.

Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains

Preface 1. Non-equilibrium statistical mechanics 2. The Boltzmann equation 3. Liouville's equation 4. Poincare recurrence theorem 5. Boltzmann's ergodic hypothesis 6. Gibbs' picture-mixing systems 7. The Green-Kubo formulae 8. The Baker's transformation 9. Lyapunov exponents for a map 10. The Baker's transformation is ergodic 11. Kolmogorov-Sinai entropy 12. The Frobenius-Perron equation 13. Open systems and escape-rates 14. Transport coefficients and chaos 15. SRB and Gibbs measures 16. Fractal forms in Green-Kubo relations 17. Unstable periodic orbits 18. Lorentz lattice gases 19. Dynamical foundations of the Boltzmann equation 20. The Boltzmann equation returns 21. What's next Appendices Bibliography.

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An Introduction to Chaos in Nonequilibrium Statistical Mechanics

Anomalous transport in one-dimensional translation invariant hamiltonian systems with short range interactions is shown to belong in general to the Kardar-Parisi-Zhang universality class. Exact asymptotic forms for density-density and current-current time correlation functions and their Fourier transforms are given in terms of the Prahofer-Spohn scaling functions, obtained from their exact solution for the polynuclear growth model. The exponents of corrections to scaling are found as well, but not so the coefficients. Mode coupling theories developed previously are found to be adequate for weakly nonlinear chains but in need of corrections for strongly anharmonic interparticle potentials. A simple condition is given under which Kardar-Parisi-Zhang behavior does not apply, sound attenuation is only logarithmically superdiffusive, and heat conduction is more strongly superdiffusive than under Kardar-Parisi-Zhang behavior.

Exact results for anomalous transport in one-dimensional hamiltonian systems.

The Lyapunov exponents and the KS entropy for a two dimensional Lorentz gas at low densities are defined for general non-equilibrium states and calculated with the use of a Lorentz-Boltzmann type equation In equilibrium the density dependence of these quantities predicted by Krylov is recovered and explicit expressions are obtained The relationship between the KS entropy, Lyapunov exponents, and escape rate, conjectured by Gaspard and Nicolis for non-equilibrium systems is confirmed and generalized

Lyapunov exponents and Kolmogorov-Sinai entropy for the Lorentz gas at low densities.

In the study of chaotic behavior, Lyapunov exponents play an important part. In this paper, we demonstrate how the Lyapunov exponents close to zero of a system of many hard spheres can be described as Goldstone modes, by using a Boltzmann type of approach. At low densities, the correct form is found for the wave number dependence of the exponents as well as for the corresponding eigenvectors in tangent space. The predicted values for the Lyapunov exponents belonging to the transverse mode are within a few percent of the values found in recent simulations, the propagation velocity for the longitudinal mode is within 1%, but the value for the Lyapunov exponent belonging to the longitudinal mode deviates from the simulations by 30%. For higher densities, the predicted values deviate more from the values calculated in the simulations. These deviations may be due to contributions from ring collisions and similar terms, which, even at low densities, can contribute to the leading order.

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Goldstone modes in Lyapunov spectra of hard sphere systems

We consider the flow of a dilute gas around a macroscopic heavy object. The state of the gas is described by an extended Boltzmann equation where the interactions between the gas molecules and the object are taken into account in computing the rate of change of the distribution function of the gas. We then show that the extended Boltzmann is equivalent to the usual Boltzmann equation, supplemented by boundary conditions imposed on the distribution function at the surface of the object. The remainder of the paper is devoted to a study of the solution of the extended Boltzmann equation in the case that the mean free path of a gas molecule is small compared to some characteristic dimension of the macroscopic object. We show that the Chapman-Enskog normal solution of the ordinary Boltzmann equation is not in general a solution of the extended equation near the surface of the object and must be supplemented by a boundary layer term. We then introduce a projection operator method which allows us to decompose the solution of the extended equation into a normal solution part and a boundary layer part when the gas flow is sufficiently slow. As a specific example of the method we consider the flow around a sphere, and derive the Stokes-Boussinesq form for the frequency-dependent force on the sphere for arbitrary slip coefficient. This derivation is the first one that starts from the Boltzmann equation for a general dilute gas and incorporates the effect of the boundary layer on the drag force.

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Kinetic theory of hydrodynamic flows. I. The generalized normal solution method and its application to the drag on a sphere

We consider the distribution of nucleation times in systems with Brownian type dynamics, as described by classical nucleation theory. This is studied for a prototype system: the two-dimensional Ising model with spin-flip dynamics in an external magnetic field. Direct simulation results for the nucleation times, spanning more than four orders of magnitude, are compared with theoretical predictions. In contrast to usual treatments we determine size-dependent droplet free energies and effective transition rates for growth and shrinkage directly from our simulations. The free energies so determined are well described by the classical BeckerDoring expression, provided one uses an effective surface tension that exceeds the macroscopic surface tension by up to 20%. Within this framework there is good agreement between simulation results and theoretical predictions for the mean nucleation time. In addition we consider the short-time behavior of the nucleation probability after an initial quench into the metastable state. We present theoretical estimates and show that these too agree well with simulation results.

https://dspace.library.uu.nl/bitstream/handle/1874/14016/brendel_05_nucleation.pdf;sequence=1

Henk van Beijeren

Papers

Exact results for anomalous transport in one-dimensional hamiltonian systems.

Lyapunov exponents and Kolmogorov-Sinai entropy for the Lorentz gas at low densities.

Goldstone modes in Lyapunov spectra of hard sphere systems

Kinetic theory of hydrodynamic flows. I. The generalized normal solution method and its application to the drag on a sphere

Nucleation times in the two-dimensional Ising model