Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

First Law: Particles in Two-dimensional Equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic equilibrium involves objects in motion without acceleration; but it is important to remember that these conditions are relative. For instance, an object may be at rest when viewed from one frame of reference, but that same object would appear to be in motion when viewed by someone moving at a constant velocity.
Newton's first law tells us about the...

Conservation of Linear Momentum for a System of Particles

Conservation of Linear Momentum for a System of Particles

In the dynamic realm of billiards, a fascinating interplay of forces governs the motion of cue balls and stationary balls. When the cue ball collides with a stationary ball, linear momentum is exchanged. The cue ball imparts a fraction of its linear momentum to the stationary ball, causing the cue ball to decelerate while initiating the motion of the stationary ball.
The impulsive force at play during this interaction is of extremely short duration, rendering its impulse negligible. When...

First Law: Particles in One-dimensional Equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton's first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a net external force. It also states that there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This cause is a net external force. For example, consider what happens to an object sliding along a rough horizontal surface. The object quickly grinds to a halt, due to the net force of friction. If we...

The de Broglie Wavelength

The de Broglie Wavelength

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...

Woodward–Hoffmann Selection Rules and Microscopic Reversibility

Woodward–Hoffmann Selection Rules and Microscopic Reversibility

Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...

The Quantum-Mechanical Model of an Atom

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

One-dimensional invariant measure in periodicity hubs: A tribute to Professor Jason A. C. Gallas.

Chaos (Woodbury, N.Y.)·2025

Same author

Mixing property of symmetrical polygonal billiards.

Physical review. E·2024

Same author

Time-reversal-invariant hexagonal billiards with a point symmetry.

Physical review. E·2022

Same author

Experimental Microwave Scattering in Polygonal Billiards.

Scientific reports·2019

Same author

Influence of different anoxic time exposures on active biomass, protozoa and filamentous bacteria in activated sludge.

Water science and technology : a journal of the International Association on Water Pollution Research·2016

Same author

Classical billiards and quantum fluids.

Physical review. E, Statistical, nonlinear, and soft matter physics·2015

Search research articles

Related Experiment Video

Updated: May 9, 2026

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

Published on: September 5, 2019

Ergodicity and quantum correlations in irrational triangular billiards.

T Araújo Lima¹, S Rodríguez-Pérez, F M de Aguiar

¹Departamento de Física, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 16, 2013

Summary

This summary is machine-generated.

This study reveals that irrational triangular billiards exhibit varying ergodic dynamics, impacting their spectral statistics. These findings bridge classical and quantum mechanics for nonchaotic systems.

More Related Videos

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Published on: August 2, 2019

Related Experiment Videos

Last Updated: May 9, 2026

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

A Photonic System for Generating Unconditional Polarization-Entangled Photons Based on Multiple Quantum Interference

Published on: September 5, 2019

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Scalable Quantum Integrated Circuits on Superconducting Two-Dimensional Electron Gas Platform

Published on: August 2, 2019

Area of Science:

Mathematical Physics
Dynamical Systems Theory
Quantum Chaos

Background:

Irrational triangular billiards are nonchaotic systems whose ergodic properties are not fully understood.
Previous research suggested unique dynamics, but a comprehensive analysis across a parameter family was lacking.

Purpose of the Study:

To systematically investigate pseudochaotic properties in a one-parameter family of irrational triangular billiards.
To explore the relationship between classical dynamics (position correlation decay) and quantum spectral statistics.
To establish a classical-quantum correspondence for nonchaotic systems exhibiting strong mixing properties.

Main Methods:

Numerical calculation of 150,000 energy eigenvalues for each billiard.
Analysis of the position correlation function C(x)(t) decay rate (α).
Computation and analysis of spectral statistical properties, comparing them to the Gaussian orthogonal ensemble (GOE).

Main Results:

Observed position correlation function decay rates ranging from slow (0<α<1) to fast (α≈1).
Demonstrated that dynamics vary smoothly, not uniquely, between strong mixing (α=1) and regular (α=0) behaviors.
Identified GOE spectral fluctuations for fast decay (α≈1) and intermediate statistics otherwise.
Confirmed zero Kolmogorov-Sinai entropy and infinite genus for these irrational billiards.

Conclusions:

Irrational triangular billiards do not possess a unique ergodic dynamic; it varies smoothly with parameter changes.
A clear classical-quantum correspondence exists, linking correlation decay rates to spectral statistics.
These nonchaotic systems can exhibit strong mixing, providing insights into the ergodic hierarchy.