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arXiv1994

Symplectic Integration without Roundoff Error

Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious damping or excitation. Methods that do preserve all the Hamiltonian properties, i.e., for which the time-forward map is symplectic, are called symplectic integration algorithms or SIAs. Although such integrators are symplectic in theory, they are not symplectic if implemented using finite-precision arithmetic. This paper explains how to eliminate this problem by using ``Lattice SIAs'' and shows that these methods yield significant advantages when the computational error is dominated by roundoff. Using a lattice SIA is equivalent to evolving the exact solution of a problem with a Hamiltonian that is slightly different from the original. Lattice methods are useful for studies of the long-term evolution of Hamiltonian dynamical systems.

astro-ph↗
arXiv1995

Universality in Turbulence: an Exactly Soluble Model

The present note contains the text of lectures discussing the problem of universality in fully developed turbulence. After a brief description of Kolmogorov's 1941 scaling theory of turbulence and a comparison between the statistical approach to turbulence and field theory, we discuss a simple model of turbulent advection which is exactly soluble but whose exact solution is still difficult to analyze. The model exhibits a restricted universality. Its correlation functions contain terms with universal but anomalous scaling but with non-universal amplitudes typically diverging with the growing size of the system. Strict universality applies only after such terms have been removed leaving renormalized correlators with normal scaling. We expect that the necessity of such an infrared renormalization is a characteristic feature of universality in turbulence.

chao-dyn↗
arXiv1994

The Berry Paradox

Lecture given Wednesday 27 October 1993 at a Physics -- Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa Fe Institute 24--26 May 1994.

chao-dyn↗
arXiv1998

A Way Out of the Quantum Trap

We review Event Enhanced Quantum Theory (EEQT). In Section 1 we address the question "Is Quantum Theory the Last Word". In particular we respond to some of recent challenging staments of H.P. Stapp. We also discuss a possible future of the quantum paradigm - see also Section 5. In Section 2 we give a short sketch of EEQT. Examples are given in Section 3. Section 3.3 discusses a completely new phenomenon - chaos and fractal-like phenomena caused by a simultaneous "measurement" of several non-commuting observables (we include picture of Barnsley's IFS on unit sphere of a Hilbert space). In Section 4 we answer "Frequently Asked Questions" concerning EEQT.

quant-ph↗
arXiv1996

Quantum Fluids and Classical Determinants

A "quasiclassical" approximation to the quantum spectrum of the Schroedinger equation is obtained from the trace of a quasiclassical evolution operator for the "hydrodynamical" version of the theory, in which the dynamical evolution takes place in the extended phase space $[q(t),p(t),M(t)] = [q_i, \partial_i S, \partial_i \partial_j S ]$. The quasiclassical evolution operator is multiplicative along the classical flow, the corresponding quasiclassical zeta function is entire for nice hyperbolic flows, and its eigenvalue spectrum contains the spectrum of the semiclassical zeta function. The advantage of the quasiclassical zeta function is that it has a larger analyticity domain than the original semiclassical zeta function; the disadvantage is that it contains eigenvalues extraneous to the quantum problem. Numerical investigations indicate that the presence of these extraneous eigenvalues renders the original Gutzwiller-Voros semiclassical zeta function preferable in practice to the quasiclassical zeta function presented here. The cumulant expansion of the exact quantum mechanical scattering kernel and the cycle expansion of the corresponding semiclassical zeta function part ways at a threshold given by the topological entropy; beyond this threshold quantum mechanics cannot resolve fine details of the classical chaotic dynamics.

chao-dyn↗
arXiv1997

Investigation of dynamical systems using tools of the theory of invariants and projective geometry

The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form $y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main backbone of this investigation was provided by the theory of invariants developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century and the projective geometry of E. Cartan. In our work two, in some sense supplementary, systems are considered: the Lorenz system $\dot{x}=σ(y-x), \dot{y}=rx-y-zx,\dot{z}=xy-bz $ and the Rößler system $\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary differential equations, which correspond to the systems mentioned abouve, are evaluated. The connection of values of the invariants with characteristics of dynamical systems is established.

solv-int↗
arXiv1999

Bulk Burning Rate in Passive - Reactive Diffusion

We consider a passive scalar that is advected by a prescribed mean zero divergence-free velocity field, diffuses, and reacts according to a KPP-type nonlinear reaction. We introduce a quantity, the bulk burning rate, that makes both mathematical and physical sense in general situations and extends the often ill-defined notion of front speed. We establish rigorous lower bounds for the bulk burning rate that are linear in the amplitude of the advecting velocity for a large class of flows. These "percolating" flows are characterized by the presence of tubes of streamlines connecting distant regions of burned and unburned material and generalize shear flows. The bound contains geometric information on the velocity streamlines and degenerates when these oscillate on scales that are finer than the width of the laminar burning region. We give also examples of very different kind of flows, cellular flows with closed streamlines, and rigorously prove that these can produce only sub-linear enhancement of the bulk burning rate.

math.AP↗
arXiv2000

On the large time asymptotics of decaying Burgers turbulence

The decay of Burgers turbulence with compactly supported Gaussian "white noise" initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and velocity differences are computed exactly, together with the "time-like" structure functions . The analysis of the answers reveals both well known features of Burgers turbulence, such as the presence of dissipative anomaly, the extreme anomalous scaling of the velocity structure functions and self similarity of the statistics of the velocity field, and new features such as the extreme anomalous scaling of the "time-like" structure functions and the non-existence of a global inertial scale due to multiscaling of the Burgers velocity field. We also observe that all the results can be recovered using the one point probability distribution function of the shock strength and discuss the implications of this fact for Burgers turbulence in general.

chao-dyn↗
arXiv1996

Reversibility, coarse graining and the chaoticity principle

We describe a way of interpreting the chaotic principle of (ref. [GC1]) more extensively than it was meant in the original works. Mathematically the analysis is based on the dynamical notions of Axiom A and Axiom B and on the notion of Axiom C, that we introduce arguing that it is suggested by the results of an experiment (ref. [BGG]) on chaotic motions. Physically we interpret a breakdown of the Anosov property of a time reversible attractor (replaced, as a control parameter changes, by an Axiom A property) as a spontaneous breakdown of the time reversal symmetry: the relation between time reversal and the symmetry that remains after the breakdown is analogous to the breakdown of $T$-invariance while $TCP$ still holds.

chao-dyn↗
arXiv1996

Quasi linear flows on tori: regularity of their linearization

Under suitable conditions a flow on a torus $C^{(p)}$--close, with $p$ large enough, to a quasi periodic diophantine rotation is shown to be conjugated to the quasi periodic rotation by a map that is analytic in the perturbation size. This result is parallel to Moser's theorem stating conjugability in class $C^{(p')}$ for some $p'<p$. The extra conditions restrict the class of perturbations that are allowed.

chao-dyn↗
arXiv1997

Coarsening by Ginzburg-Landau Dynamics

We study slowly moving solutions of the real Ginzburg-Landau equation on the line, by a method due to J. Carr and R.L. Pego. These are functions taking alternately positive or negative values on large intervals. A consequence of our approach is that we can propose a rigorous derivation of a stochastic model of coarsening by successive elimination of the smallest interval, which was described in earlier work by A.J. Bray, B. Derrida and C. Godrèche.

chao-dyn↗
arXiv1998

Distribution of zeros of random and quantum chaotic sections of positive line bundles

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle $L$ over a compact complex manifold $M$. Our first result concerns `random' sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for almost every sequence $\{S^N_j\}$, the associated sequence of zero currents $1/N Z_{S^N_j}$ tends to the curvature form $ω$ of $L$. Thus, the zeros of a sequence of sections $s_N \in H^0(M, L^N)$ chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$ consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

math.CV↗
arXiv1998

Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures

We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of Hörmander used in the study of hypoelliptic differential operators.

chao-dyn↗
arXiv1998

KAM Theorem and Quantum Field Theory

We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are shown to follow from the Ward identities expressing the translation invariance of the corresponding field theory.

chao-dyn↗
arXiv1999

KAM Tori for 1D Nonlinear Wave Equations with Periodic Boundary Conditions

In this paper, one-dimensional (1D) nonlinear wave equations $u_{tt} -u_{xx}+V(x)u =f(u)$, with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u=0. It is proved that for ``most'' potentials V(x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem which allows for multiple normal frequencies.

chao-dyn↗
arXiv1996

Nonadiabatic couplings and incipience of quantum chaos

The quantization of the electronic two site system interacting with a vibration is considered by using as the integrable reference system the decoupled oscillators resulting from the adiabatic approximation. A specific Bloch projection method is applied which demonstrates how besides some regular regions in the fine structure of the spectrum and the associated eigenvectors irregularities appear when passing from the low to the high coupling case. At the same time even for strong coupling some of the regular structure of the spectrum rooted in the adiabatic potentials is kept intact justifying the classification of this situation as incipience of quantum chaos.

chao-dyn↗
arXiv1997

Dry Friction in the Frenkel-Kontorova-Tomlinson Model: Dynamical Properties

Wearless friction is investigated in a simple mechanical model called Frenkel-Kontorova-Tomlinson model. We have introduced this model in [Phys. Rev. B, Vol. 53, 7539 (1996)] where the static friction has already been considered. Here the model is treated for constant sliding speed. The kinetic friction is calculated numerically as well as analytically. As a function of the sliding velocity it shows many structures which can be understood by varies kinds of phonon resonances (normal, superharmonic and parametric) caused by the so-called "washboard wave". For increasing interaction strength the regular motion becomes chaotic (fluid-sliding state). The fluid sliding state is mainly determined by the density of decay channels of m washboard waves into n phonons. We also find strong bistabilities and coherent motions with superimposed dark envelope solitons which interact nondestructively.

cond-mat.stat-mech↗