arXiv subjects

Bruno Eckhardt

Publications and source records attributed to Bruno Eckhardt.

99 records · Page 6Linked to original sources

arXiv1999

Transition to turbulence in a shear flow

We analyze the properties of a 19 dimensional Galerkin approximation to a parallel shear flow. The laminar flow with a sinusoidal shape is stable for all Reynolds numbers Re. For sufficiently large Re additional stationary flows occur; they are all unstable. The lifetimes of finite amplitude perturbations show a fractal dependence on amplitude and Reynolds number. These findings are in accord with observations on plane Couette flow and suggest a universality of this transition szenario in shear flows.

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arXiv1997

Fractal Stability Border in Plane Couette Flow

We study the dynamics of localised perturbations in plane Couette flow with periodic lateral boundary conditions. For small Reynolds number and small amplitude of the initial state the perturbation decays on a viscous time scale $t \propto Re$. For Reynolds number larger than about 200, chaotic transients appear with life times longer than the viscous one. Depending on the type of the perturbation isolated initial conditions with infinite life time appear for Reynolds numbers larger than about 270--320. In this third regime, the life time as a function of Reynolds number and amplitude is fractal. These results suggest that in the transition region the turbulent dynamics is characterised by a chaotic repeller rather than an attractor.

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arXiv1995

Approach to ergodicity in quantum wave functions

According to theorems of Shnirelman and followers, in the semiclassical limit the quantum wavefunctions of classically ergodic systems tend to the microcanonical density on the energy shell. We here develop a semiclassical theory that relates the rate of approach to the decay of certain classical fluctuations. For uniformly hyperbolic systems we find that the variance of the quantum matrix elements is proportional to the variance of the integral of the associated classical operator over trajectory segments of length $T_H$, and inversely proportional to $T_H^2$, where $T_H=h\barρ$ is the Heisenberg time, $\barρ$ being the mean density of states. Since for these systems the classical variance increases linearly with $T_H$, the variance of the matrix elements decays like $1/T_H$. For non-hyperbolic systems, like Hamiltonians with a mixed phase space and the stadium billiard, our results predict a slower decay due to sticking in marginally unstable regions. Numerical computations supporting these conclusions are presented for the bakers map and the hydrogen atom in a magnetic field.

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arXiv1995

Semiclassical form factor of matrix element fluctuations

We analyze within a semiclassical approximation the form factor for the fluctuations of quantum matrix elements around their classical average. We find two contributions: one is proportional to the form factor for the density of states, with an amplitude determined by the squared average of the matrix elements. The other is constant and related to the fluctuations of finite time classical trajectory segments around the phase space average. The results are illustrated for an observable in the quadratic Zeeman effect.

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arXiv1993

Symmetry Decomposition of Chaotic Dynamics

Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labor and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. In this paper the general formalism is developed, with the $N$-disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail.

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arXiv1993

Periodic Orbit Theory

[[ RM: A review paper on cycle expansions. I quote the introduction: in section (2) ]] I will summarize Gutzwiller's theory for the spectrum of eigenenergies and extend it to diagonal matrix elements as well. The derivation of the associated zeta function is given (2.2) and the identification of suitable scaling variables discussed (2.3). In section 3 tools necessary for the organization of chaos will be discussed: symbolic dynamics (3.1), the connectivity matrix (3.3), the topological zeta function (3.4) and general transfer matrices and zeta functions (3.5). Although illustrated for the case of hard collisions in a billiard, the symbolic dynamics can be extended to `smooth collisions' in smooth potentials (3.2). In systems with discrete symmetries, zeta functions factorize into zeta functions on invariant subspaces. This symmetry factorization and the associated reduction in symbolics is discussed in section 4. The ideas developed here are illustrated for the example of a free particle reflected elastically off three disks in section 5. Methods to find periodic orbits (5.1), the convergence of the trace formula (5.2), the semiclassical computation of scattering resonances (5.3), the convergence of the cycle expansion (5.4) and methods to obtain eigenvalues of the bounded billiard (5.5) are discussed. The relevant parts of a classical periodic orbit theory are developed in section 6.1, including a discussion of escape rates and the Hannay-Ozorio de Almeida sum rule (6.2). Finally, the issue of semiclassical matrix elements is taken up again and applications to experiments are discussed.

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arXiv1993

Order and chaos in quantum irregular scattering: Wigner's time delay

Recent developments in the semiclassical analysis of chaotic systems are reviewed and illustrated for Wigner's time delay in elastic scattering of a point particle from three disks in the plane. The convergence of the cycle expanded periodic orbit expression for Wigners time delay is demonstrated. Different regimes in form factor (the Fourier transform of the two point correlation function) of the semiclassical time delay are identified and their relation to Berry's semiclassical theory of the spectral rigidity are discussed.

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arXiv1993

Resummation of classical and semiclassical periodic orbit formulas

The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical and the semiclassical Selberg zeta function have poles. Applying a Padé approximation on the expansions of the full Euler products, as well as on the individual dynamical zeta functions in the products, we calculate the leading poles and the zeros of the improved expansions with the first few poles removed. The removal of poles tends to change the simple linear exponential convergence of the Selberg zeta functions to an $\exp\{-n^{3/2}\}$ decay in the classical case and to an $\exp\{-n^2\}$ decay in the semiclassical case. The leading poles of the $j$th dynamical zeta function are found to equal the leading zeros of the $j+1$th one: However, in contrast to the zeros, which are all simple, the poles seem without exception to be {\em double}\/. The poles are therefore in general {\em not}\/ completely cancelled by zeros, which has earlier been suggested. The only complete cancellations occur in the classical Selberg zeta function between the poles (double) of the first and the zeros (squared) of the second dynamical zeta function. Furthermore, we find strong indications that poles are responsible for the presence of spurious zeros in periodic orbit quantized spectra and that these spectra can be greatly improved by removing the leading poles, e.g.\ by using the Padé technique.

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arXiv1993

Continued-fraction expansion of eigenvalues of generalized evolution operators in terms of periodic orbits

A new expansion scheme to evaluate the eigenvalues of the generalized evolution operator (Frobenius-Perron operator) $H_{q}$ relevant to the fluctuation spectrum and poles of the order-$q$ power spectrum is proposed. The ``partition function'' is computed in terms of unstable periodic orbits and then used in a finite pole approximation of the continued fraction expansion for the evolution operator. A solvable example is presented and the approximate and exact results are compared; good agreement is found.

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