Structure of Quantum Chaotic Wavefunctions: Ergodicity, Localization, and Transport

We discuss recent developments in the study of quantum wavefunctions and transport in classically ergodic systems. Surprisingly, short-time classical dynamics leaves permanent imprints on long-time and stationary quantum behavior, which are absent from the long-time classical motion. These imprints can lead to quantum behavior on single-wavelength or single-channel scales which are very different from random matrix theory expectations. Robust and quantitative predictions are obtained using semiclassical methods. Applications to wavefunction intensity statistics and to resonances in open systems are discussed.Comment: 8 pages, including 2 figures; talk given at `Dynamics of Complex Systems' workshop in Dresden, 1999 and submitted for conference proceedings to appear in Physica

Semiclassical transmission across transition states

It is shown that the probability of quantum-mechanical transmission across a phase space bottleneck can be compactly approximated using an operator derived from a complex Poincar\'e return map. This result uniformly incorporates tunnelling effects with classically-allowed transmission and generalises a result previously derived for a classically small region of phase space.Comment: To appear in Nonlinearit

Chaos and Quantum Thermalization

We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey {\it Berry's conjecture}. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell--Boltzmann, Bose--Einstein, or Fermi--Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon {\it eigenstate thermalization}. We show that a generic initial state will approach thermal equilibrium at least as fast as $O(\hbar/\Delta)t^{-1}$, where $\Delta$ is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a new foundation for quantum statistical mechanics.Comment: 28 pages in Plain TeX plus 2 uuencoded PS figures (included); minor corrections only, this version will be published in Phys. Rev. E; UCSB-TH-94-1

Amplitude distribution of eigenfunctions in mixed systems

We study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space. For an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and good agreement with numerical computations for the family of limacon billiards is found. The natural extension of our result to more general systems, e.g. with a potential, is also discussed.Comment: 13 pages, 3 figures. Some of the pictures are included in low resolution only. For a version with pictures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab

Scar functions in the Bunimovich Stadium billiard

In the context of the semiclassical theory of short periodic orbits, scar functions play a crucial role. These wavefunctions live in the neighbourhood of the trajectories, resembling the hyperbolic structure of the phase space in their immediate vicinity. This property makes them extremely suitable for investigating chaotic eigenfunctions. On the other hand, for all practical purposes reductions to Poincare sections become essential. Here we give a detailed explanation of resonances and scar functions construction in the Bunimovich stadium billiard and the corresponding reduction to the boundary. Moreover, we develop a method that takes into account the departure of the unstable and stable manifolds from the linear regime. This new feature extends the validity of the expressions.Comment: 21 pages, 10 figure

Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear relations amongst the gradients of the integrals of motion, coincide with points where there are k (doubly) degenerate eigenvalues of representatives L and Lbar of the two inequivalent classes of Lax matrices (corresponding to degenerate periodic or antiperiodic solutions of the associated second-order difference equation). The singularities are shown to be nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold. Sigma_k is shown to be of elliptic type, and the frequencies of transverse oscillations under Hamiltonians which fix Sigma_k are computed in terms of spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is given by the product of the holonomies (equal to +/- 1) of the even- (or odd-) indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio

Semiclassical theory for many-body Fermionic systems

We present a treatment of many-body Fermionic systems that facilitates an expression of the well-known quantities in a series expansion of the Planck's constant. The ensuing semiclassical result contains to a leading order of the response function the classical time correlation function of the observable followed by the Weyl-Wigner series, on top of these terms are the periodic-orbit correction terms. The treatment given here starts from linear response assumption of the many-body theory and in its connection with semiclassical theory, it makes no assumption of the integrability of classical dynamics underlying the one-body quantal system. Applications of the framework are also discussed.Comment: 18 pages, Te

Toxicology and Epidemiology: Improving the Science with a Framework for Combining Toxicological and Epidemiological Evidence to Establish Causal Inference

Historically, toxicology has played a significant role in verifying conclusions drawn on the basis of epidemiological findings. Agents that were suggested to have a role in human diseases have been tested in animals to firmly establish a causative link. Bacterial pathogens are perhaps the oldest examples, and tobacco smoke and lung cancer and asbestos and mesothelioma provide two more recent examples. With the advent of toxicity testing guidelines and protocols, toxicology took on a role that was intended to anticipate or predict potential adverse effects in humans, and epidemiology, in many cases, served a role in verifying or negating these toxicological predictions. The coupled role of epidemiology and toxicology in discerning human health effects by environmental agents is obvious, but there is currently no systematic and transparent way to bring the data and analysis of the two disciplines together in a way that provides a unified view on an adverse causal relationship between an agent and a disease. In working to advance the interaction between the fields of toxicology and epidemiology, we propose here a five-step "Epid-Tox” process that would focus on: (1) collection of all relevant studies, (2) assessment of their quality, (3) evaluation of the weight of evidence, (4) assignment of a scalable conclusion, and (5) placement on a causal relationship grid. The causal relationship grid provides a clear view of how epidemiological and toxicological data intersect, permits straightforward conclusions with regard to a causal relationship between agent and effect, and can show how additional data can influence conclusions of causalit

Chaotic eigenfunctions in momentum space

We study eigenstates of chaotic billiards in the momentum representation and propose the radially integrated momentum distribution as useful measure to detect localization effects. For the momentum distribution, the radially integrated momentum distribution, and the angular integrated momentum distribution explicit formulae in terms of the normal derivative along the billiard boundary are derived. We present a detailed numerical study for the stadium and the cardioid billiard, which shows in several cases that the radially integrated momentum distribution is a good indicator of localized eigenstates, such as scars, or bouncing ball modes. We also find examples, where the localization is more strongly pronounced in position space than in momentum space, which we discuss in detail. Finally applications and generalizations are discussed.Comment: 30 pages. The figures are included in low resolution only. For a version with figures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp99-2.htm