Semiclassical Theory of Chaotic Quantum Transport

We present a refined semiclassical approach to the Landauer conductance and Kubo conductivity of clean chaotic mesoscopic systems. We demonstrate for systems with uniformly hyperbolic dynamics that including off-diagonal contributions to double sums over classical paths gives a weak-localization correction in quantitative agreement with results from random matrix theory. We further discuss the magnetic field dependence. This semiclassical treatment accounts for current conservation.Comment: 4 pages, 1 figur

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

Single chain dimers of MASH-1 bind DNA with enhanced affinity

By designing recombinant genes containing tandem copies of the coding region of the BHLH domain of MASH-1 (MASH-BHLH) with intervening DNA sequences encoding linker sequences of 8 or 17 amino acids, the two subunits of the MASH dimer have been connected to form the single chain dimers MM8 and MM17. Despite the long and flexible linkers which connect the C-terminus of the first BHLH subunit to the N-terminus of the second, a distance of ∼55 Å, the single chain dimers could be produced in Escherichia coli at high levels. MM8 and MM17 were monomeric and no ‘cross-folding' of the subunits was observed. CD spectroscopy revealed that, like wild-type MASH-BHLH, MM8 and MM17 adopt only partly folded structures in the absence of DNA, but undergo a folding transition to a mainly α-helical conformation on DNA binding. Titrations by electrophoretic mobility shift assays revealed that the affinity of the single chain dimers for E box-containing DNA sequences was increased ∼10-fold when compared with wild-type MASH-BHLH. On the other hand, the affinity for heterologous DNA sequences was increased only 5-fold. Therefore, the introduction of the peptide linker led to a 4-fold increase in DNA binding specificity from −0.14 to −0.57 kcal/mo

Semiclassical universality of parametric spectral correlations

We consider quantum systems with a chaotic classical limit that depend on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor $K(\tau,x)$ which depends on a scaled parameter difference $x$. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small $\tau$ expansion of the form factor agrees with Random Matrix Theory for systems with and without time reversal symmetry.Comment: 18 pages, no figure

Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential equations and hence provides a better justification for using these delay-type models. The Mori-Zwanzig technique gives a formal rewriting of the system using a projection onto a set of resolved variables, where the rewritten system contains a memory term. The computation of this memory term requires solving the orthogonal dynamics equation, which represents the unresolved dynamics. For nonlinear systems, it is often not possible to obtain an analytical solution to the orthogonal dynamics and an approximate solution needs to be found. Here, we demonstrate the Mori-Zwanzig technique for a two-strip model of the El Nino Southern Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The resulting nonlinear delay model contains an additional term compared to previously proposed ad-hoc conceptual models. This new term leads to a larger ENSO period, which is closer to that seen in observations.Comment: Submitted to Proceedings of the Royal Society A, 25 pages, 10 figure

A generic C1C^1 map has no absolutely continuous invariant probability measure

Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense $G_\delta) set in the$C^1$ topology. In the course of the proof, we need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result may be of independent interest.Comment: 12 page

Classical orbit bifurcation and quantum interference in mesoscopic magnetoconductance

We study the magnetoconductance of electrons through a mesoscopic channel with antidots. Through quantum interference effects, the conductance maxima as functions of the magnetic field strength and the antidot radius (regulated by the applied gate voltage) exhibit characteristic dislocations that have been observed experimentally. Using the semiclassical periodic orbit theory, we relate these dislocations directly to bifurcations of the leading classes of periodic orbits.Comment: 4 pages, including 5 figures. Revised version with clarified discussion and minor editorial change

Periodic-Orbit Bifurcations and Superdeformed Shell Structure

We have derived a semiclassical trace formula for the level density of the three-dimensional spheroidal cavity. To overcome the divergences occurring at bifurcations and in the spherical limit, the trace integrals over the action-angle variables were performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell-correction energies are in good agreement with quantum-mechanical results. We find that the bifurcations of some dominant short periodic orbits lead to an enhancement of the shell structure for "superdeformed" shapes related to those known from atomic nuclei.Comment: 4 pages including 3 figure

On the stability of periodic orbits in delay equations with large delay

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.Comment: postprint versio

Representations of J-central J-Potapov functions in both nondegenerate and degenerate cases

AbstractLet J be an m×m signature matrix (i.e. J∗=J and J2=Im) and let D:={z∈C:|z|<1}. Denote PJ(D) the class of all J-Potapov functions in D, i.e. the set of all meromorphic m×m matrix-valued functions f in D with J-contractive values at all points of D at which f is holomorphic. Further, denote PJ,0(D) the subclass of all f∈PJ(D) which are holomorphic at the origin. Let f∈PJ,0(D), and let f(w)=∑j=0∞Ajwj be the Taylor series representation of f in some neighborhood of 0. Then it was proved in [B. Fritzsche, B. Kirstein, U. Raabe, On the structure of J-Potapov sequences, Linear Algebra Appl., in press] that for each n∈N the matrix An can be described by its position in a matrix ball depending on the sequence (Aj)j=0n-1. The J-Potapov function f is called J-central if there exists some k∈N such that for each integer j⩾k the matrix Aj coincides with the center of the corresponding matrix ball.In this paper, we derive left and right quotient representations of matrix polynomials for J-central J-Potapov functions in D. Moreover, we obtain recurrent formulas for the matrix polynomials involved in these quotient representations