Quantum mechanical analysis of the equilateral triangle billiard: periodic orbit theory and wave packet revivals

Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by $T_{rev} = 9 \mu a^2/4\hbar \pi$ where $a$ and $\mu$ are the length of one side and the mass of the point particle respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral ($60^{\circ}-60^{\circ}-60^{\circ}$) triangle, such as the half equilateral ($30^{\circ}-60^{\circ}-90^{\circ}$) triangle and other `foldings', which have related energy spectra and revival structures.Comment: 34 pages, 9 embedded .eps figure

Non-Euclidean geometry in nature

I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random graphs. For these systems I discuss the following interlinked questions: (i) the optimal embedding of plants leaves in the three-dimensional space, (ii) the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to chaotic Hamiltonian systems is adde

The influence of cam geometry and operating conditions on chaotic mixing of viscous fluids in a twin cam mixer

Smooth particle hydrodynamica (SPH) simulations were used to better understand the mixing performance of a class of two-dimensional Twin Cam mixers. The chaotic manifolds of the flow are used to describe the mixing and to identify isolated regions. For an equilateral triangle cam geometry, a figure-eight manifold structure traps a layer of fluid against the cam boundaries. Changes in the differential rotation and phase offsets between the cams results in modest improvements in the mixing rate across the manifold barrier. Reducing the apex angle of the triangle changes the manifold structure and allows the trapped layer of fluid to mix more effectively with the rest of the domain. This article shows that examining the chaotic manifolds within a typical industrial mixer can provide valuable insight into both the transient and long-term mixing processes, leading to a more focused exploration of possible mixer configurations and to practical improvements in mixing efficienc

Chaos in music: historical developments and applications to music theory and composition

The Doctoral Dissertation submitted by Jonathan R. Salter, in partial fulfillment of the requirements for the degree Doctor of Musical Arts at the University of North Carolina at Greensboro comprises the following: 1. Doctoral Recital I, March 24, 2007: Chausson, Andante et Allegro; Tomasi, Concerto for Clarinet; Bartok, Contrasts; Fitkin, Gate. 2. Doctoral Recital II, December 2, 2007: Benjamin, Le Tombeau de Ravel ; Mandat, Folk Songs; Bolcom, Concerto for Clarinet; Kovacs, Sholem-alekhem, rov Fiedman! 3. Doctoral Recital III, May 3, 2009: Kalliwoda, Morceau du Salon; Shostakovich, Sonata, op. 94 (transcription by Kennan); Tailleferre, Arabesque; Schoen eld, Trio for Clarinet, Violin, and Piano. 4. Dissertation Document: Chaos in Music: Historical Developments and Applications to Music Theory and Composition. Chaos theory, the study of nonlinear dynamical systems, has proven useful in a wide-range of applications to scienti c study. Here, I analyze the application of these systems in the analysis and creation of music, and take a historical view of the musical developments of the 20th century and how they relate to similar developments in science. I analyze several 20th century works through the lens of chaos theory, and discuss how acoustical issues and our interpretation of music relate to the theory. The application of nonlinear functions to aspects of music including organization, acoustics and harmonics, and the role of chance procedures is also examined toward suggesting future possibilities in incorporating chaos theory in the act of composition. Original compositions are included, in both sheet music and recorded form

Hearing shapes of drums - mathematical and physical aspects of isospectrality

In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.Comment: 42 pages, 60 figure