Periodic orbit action correlations in the Baker map

Tanner, Gregor

doi:10.1088/0305-4470/32/27/307

Cited by 17 publications

(27 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Second, to test the general semiclassical arguments on action correlations for a paradigm chaotic dynamical system -the Baker map. This system was investigated previously by a number of groups, [2,3,6,8], who demonstrated numerically the existence of the expected correlations. Here, we develop another approach for the analysis of the action spectrum, where we try to systematically asses the way the periodic orbits and their actions can be partitioned to families which are dynamically related.…”

Section: Introductionmentioning

confidence: 97%

Action correlations and random matrix theory

Smilansky

Verdene

2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The correlations in the spectra of quantum systems are intimately related to correlations which are of genuine classical origin, and which appear in the spectra of actions of the classical periodic orbits of the corresponding classical systems. We review this duality and the semiclassical theory which brings it about. The conjecture that the quantum spectral statistics are described in terms of Random Matrix Theory, leads to the proposition that the classical two-point correlation function is given also in terms of a universal function.We study in detail the spectrum of actions of the Baker map, and use it to illustrate the steps needed to reveal the classical correlations, their origin and their relation to symbolic dynamics.

show abstract

Section: Introductionmentioning

confidence: 97%

Action correlations and random matrix theory

Smilansky

Verdene

2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

show abstract

“…Indeed, the overall features of the transfer-matrix spectrum described above can also be found in the transferoperator spectra discussed in Refs. [6,24]. Looking for an explanation for this similarity, we recall that the transfer operator is infinite-dimensional and independent of L. Thus, one may be tempted to compare Eq.…”

Section: Spectral Propertiesmentioning

confidence: 99%

“…Agreement with the exact results is met, within the specified statistical errors, for r ≥ 12, corresponding to f > ∼ 4. In rows (fi) we consider (N, r) fixed and large values of L which can in no way be reached by direct computation of the periodic orbit summation, so that exact results are not known for such L. In these cases we compare with the approximate results obtained using the transfer-operator method [6,24].…”

Section: Truncation Schemesmentioning

confidence: 99%

Transfer matrices and circuit representation for the semiclassical traces of the baker map

Carlo

Vallejos²,

Abreu³

2010

Phys. Rev. E

View full text Add to dashboard Cite

Because of a formal equivalence with the partition function of an Ising chain, the semiclassical traces of the quantum baker map can be calculated using the transfer-matrix method. We analyze the transfer matrices associated with the baker map and the symmetry-reflected baker map (the latter happens to be unitary but the former is not). In both cases simple quantum-circuit representations are obtained, which exhibit the typical structure of qubit quantum bakers. In the case of the baker map it is shown that nonunitarity is restricted to a one-qubit operator (close to a Hadamard gate for some parameter values). In a suitable continuum limit we recover the already known infinite-dimensional transfer-operator. We devise truncation schemes allowing the calculation of long-time traces in regimes where the direct summation of Gutzwiller's formula is impossible. Some aspects of the long-time divergence of the semiclassical traces are also discussed.

show abstract

“…As examples they considered the deformed cat-map and the baker-map. This has been elaborated further in a number of studies by Dittes et al [29], Aurich and Sieber [30], Cohen et al [31], Tanner [32], Sano [33], Primack and Smilansky [34], Sieber and Richter [17] and Smilansky and Verdene [35]. Argaman et al started from the assumption that spectral fluctuations of chaotic quantum systems follow the predictions of RMT and they derived a universal expression for classical correlation functions of periodic orbits via Gutzwiller's semi-classical trace formula.…”

Section: Introductionmentioning

confidence: 98%

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

Laprise

Hosseinizadeh

Kröger

2015

View full text Add to dashboard Cite

For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity, we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evidence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems.

show abstract

Periodic orbit action correlations in the Baker map

Cited by 17 publications

References 40 publications

Action correlations and random matrix theory

Action correlations and random matrix theory

Transfer matrices and circuit representation for the semiclassical traces of the baker map

Universality in Statistical Measures of Trajectories in Classical Billiard Systems

Contact Info

Product

Resources

About