B. V. Chirikov

Linear and Nonlinear Dynamical Chaos

Interrelations between dynamical and statistical laws in physics on the one hand, and between the classical and quantum mechanics on the other hand, are discussed with the emphasis on the new phenomenon of dynamical chaos.The principal results of the studies into chaos in classical mechanics are presented in some detail within the general picture of chaos as a specific case of dynamical behavior. These results include the strong local instability and robustness of motion, continuity of both the phase space as well as the motion spectrum, and time reversibility but nonrecurrency of statistical evolution.The analysis of apparently very deep and challenging contradictions of this picture with the quantum principles is given. The quantum view of dynamical chaos, as an attempt to resolve these contradictions guided by the correspondence principle and based upon the characteristic time scales of quantum evolution, is explained. The picture of the quantum chaos as a new generic dynamical phenomenon is outlined together with a few other examples of such a chaos, including linear (classical) waves and a digital computer.I conclude with the discussion of two fundamental physical problems: the quantum measurement (ψ-collapse) and the causality principle, which both appear to be related to the phenomenon of dynamical chaos.

Arnold diffusion in large systems

A new regime of Arnold diffusion in which the diffusion rate has a power-law dependence on the perturbation strength is studied theoretically and in numerical experiments. The theory developed predicts this new regime to be universal in the perturbation intermediate asymptotics, the width of the latter increasing with the dimensionality of the perturbation frequency space, particularly in large systems with many degrees of freedom. The results of numerical experiments agree satisfactorily with the theoretical estimates.

Adiabatic invariance and separatrix: Single separatrix crossing

Detailed numerical experiments on the dynamics and statistics of a single crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described. The results are compared with a simple approximate theory first developed by Timofeev and further improved and generalized by Tennyson and coworkers. The main attention is paid to a new, ballistic, regime of separatrix crossing in which the violation of adiabaticity is maximal. Some unsolved problems and open questions are also discussed.

Multiple Separatrix Crossing: A Chaos Structure ¶

Numerical experiments on the structure of the chaotic component of motion under multiple-crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described with the emphasis on the ergodicity problem. The results clearly demonstrate nonergodicity of this motion due to the presence of a regular component of a relatively small measure with a very complicated structure. A simple 2D-map per crossing is constructed that qualitatively describes the main properties of both chaotic and regular components of the motion. An empirical relation for the correlation-affected diffusion rate is found including a close vicinity of the chaos border where evidence of the critical structure is observed. Some unsolved problems and open questions are also discussed.

The problem of quantum chaos

The new phenomenon of quantum chaos has revealed the intrinsic complexity and richness of the dynamical motion with discrete spectrum which had been always considered as most simple and regular one. The mechanism of this complexity as well as the conditions for, and the statistical properties of, the quantum chaos are explained in detail using a number of simple models for illustration. Basic ideas of a new ergodic theory of the finite-time statistical properties for the motion with discrete spectrum are discussed.

Diffusion in Smooth Hamiltonian Systems

A family of models determined by a smooth canonical 2D-map that depends on two parameters is studied. Preliminary results of numerical experiments are reported; they are evidence of substantial suppression of global diffusion in a wide range of perturbation values. This effect is caused by the little-known phenomenon of the conservation of resonance separatrices and other invariant curves under the conditions of strong local dynamic chaos. Such a total suppression of diffusion occurs although invariant curves are only conserved for a countable zero-measure set of parameter values. Simple refined estimates of diffusion rates in smooth systems without invariant curves were obtained and numerically substantiated. The principal boundary of diffusion suppression in a family of models with invariant curves was described by a semiempirical equation in dimensionless variables. The results were subjected to a statistical analysis, and an integral distribution for diffusion suppression probability was obtained.

Fractal diffusion in smooth dynamical systems with virtual invariant curves

Preliminary results of extensive numerical experiments with a family of simple models specified by the smooth canonical strongly chaotic 2D map with global virtual invariant curves are presented. We focus on the statistics of the diffusion rate D of individual trajectories for various fixed values of the model perturbation parameters K and d. Our previous conjecture on the fractal statistics determined by the critical structure of both the phase space and the motion is confirmed and studied in some detail. In particular, we find additional characteristics of what we earlier termed the virtual invariant curve diffusion suppression, which is related to a new very specific type of critical structure. A surprising example of ergodic motion with a “hidden” critical structure strongly affecting the diffusion rate was also encountered. At a weak perturbation (K ≪ 1), we discovered a very peculiar diffusion regime with the diffusion rate D=K2/3 as in the opposite limit of a strong (K ≫ 1) uncorrelated perturbation, but in contrast to the latter, the new regime involves strong correlations and exists for a very short time only. We have no definite explanation of such a controversial behavior.

Separatrix conservation mechanism for nonlinear resonance in strong chaos

We propose a simple, approximate theory of the fairly general mechanism for the separatrix conservation of nonlinear resonance, which leads to the complete suppression of global diffusion despite the strong local chaos of motion. This theory allows the separatrix splitting angle to be plotted against system parameters and, in particular, yields their values at which the separatrix remains unsplit. We present the results of our numerical experiments confirming theoretical conclusions for a certain class of dynamical Hamiltonian systems. New features of chaos suppression have been found in such systems. In conclusion, we discuss the range of application of the proposed theory.

Big Entropy Fluctuations in Statistical Equilibrium: the Macroscopic Kinetics ¶

Large entropy fluctuations in the equilibrium steady state of classical mechanics are studied in extensive numerical experiments in a simple strongly chaotic Hamiltonian model with two degrees of freedom described by the modified Arnold cat map. The rise and fall of a large separated fluctuation is shown to be described by the (regular and stable) “macroscopic” kinetics, both fast (ballistic) and slow (diffusive). We abandon a vague problem of the “appropriate” initial conditions by observing (in a long run) a spontaneous birth and death of arbitrarily big fluctuations for any initial state of our dynamical model. Statistics of the infinite chain of fluctuations similar to the Poincaré recurrences is shown to be Poissonian. A simple empirical relationship for the mean period between the fluctuations (the Poincaré “cycle”) is found and confirmed in numerical experiments. We propose a new representation of the entropy via the variance of only a few trajectories (“particles”) that greatly facilitates the computation and at the same time is sufficiently accurate for big fluctuations. The relation of our results to long-standing debates over the statistical “irreversibility” and the “time arrow” is briefly discussed.

Big entropy fluctuations in the nonequilibrium steady state: A simple model with the gauss heat bath

Large entropy fluctuations in a nonequilibrium steady state of classical mechanics are studied in extensive numerical experiments on a simple two-freedom model with the so-called Gauss time-reversible thermostat. The local fluctuations (on a set of fixed trajectory segments) from the average heat entropy absorbed in the thermostat are found to be non-Gaussian. The fluctuations can be approximately described by a two-Gaussian distribution with a crossover independent of the segment length and the number of trajectories (“particles”). The distribution itself does depend on both, approaching the single standard Gaussian distribution as any of those parameters increases. The global time-dependent fluctuations are qualitatively different in that they have a strict upper bound much less than the average entropy production. Thus, unlike the equilibrium steady state, the recovery of the initial low entropy becomes impossible after a sufficiently long time, even in the largest fluctuations. However, preliminary numerical experiments and the theoretical estimates in the special case of the critical dynamics with superdiffusion suggest the existence of infinitely many Poincaré recurrences to the initial state and beyond. This is a new interesting phenomenon to be further studied together with some other open questions. The relation of this particular example of a nonequilibrium steady state to the long-standing persistent controversy over statistical “irreversibility”, or the notorious “time arrow”, is also discussed. In conclusion, the unsolved problem of the origin of the causality “principle” is considered.