Subscribe free to our newsletters via your
. Space Travel News .

An explanation for the Fermi-Pasta-Ulam system problem
by Staff Writers
Troy NY (SPX) Mar 26, 2015

This is a graphical representation of the resonance manifold in the Fermi-Pasta-Ulam problem, plotted in Mathematica. Image courtesy Yuri Lvov.

A team of researchers, led by Rensselaer Polytechnic Institute professor Yuri Lvov, has found an elegant explanation for the long-standing Fermi-Pasta-Ulam (FPU) problem, first proposed in 1953, investigated with one of the world's first digital computers, and now considered the foundation of experimental mathematics.

The research, published in the Proceedings of the National Academy of Science, offers a mathematical explanation for how a level of energy sufficient to produce one complete wave in an idealized chain of masses connected by springs is gradually distributed to thermal equilibrium. In this system, 32 masses (or particles) can move only left or right, and the energy in the system cannot dissipate through friction or heat.

This system, famous among mathematicians and physicists, was introduced by Los Alamos National Laboratory researchers Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou as a means to study how heat is conducted in solids and metals.

At the time the researchers proposed the problem, they expected their numerical calculations would reveal that the system reaches a state of thermal equilibrium - similar to thermal equilibrium exhibited by gasses - in which the energy within the system is evenly divided among each possible movement (called "modes").

When the problem was first simulated with the Mathematical Numerical Integrator and Computer (MANIAC), one of the first digital computers, the researchers were puzzled to discover that after many iterations, energy within the system periodically dispersed and then concentrated to 97 percent of energy within a single mode.

This phenomena became known as "FPU recurrence" and spawned a wealth of questions and research, in addition to thousands of research papers, becoming the foundation of experimental mathematics.

Decades later, using far more powerful computers to run the simulation for greater lengths of time, researchers discovered that energy within the system did eventually reach equilibrium, but the question remained: how precisely was this happening? In the paper "Route to thermalization in the a-Fermi-Pasta-Ulam system," the researchers led by Lvov offer a simple and elegant explanation.

According to their calculations, the key lies in a gradual transfer of energy during coincidences of six modes in the system. When precisely six modes interact, the energy is transferred in a nonreversible way. Over many iterations, enough six-wave interactions occur, and enough energy is transferred, to reach complete thermal equilibrium. This conclusion is supported by extensive numerical simulations.

"The key approach of our research is to consider the FPU system as a collection of resonantly interacting waves, in other words - waves interact in groups," said Lvov, a professor of mathematical sciences in the School of Science at Rensselaer Polytechnic Institute. "My collaborators and I have shown that interactions of triads, quartets, and quintets are reversible; in other words, they do not bring the FPU system closer to thermal equilibrium.

However, the interaction of waves in sixtets does lead to irreversible transfer of energy. It takes the cooperation of six different waves to produce an interaction that is irreversible and, because of that, the process is extremely weak and very slow. That is why it takes so long to approach thermal equilibrium for the FPU system."

Lvov researches in the area of mathematical physics and nonlinear phenomena, in particular the ocean internal and surface water waves. These systems can be viewed as complex systems composed of interacting particles or waves, which can be described under the general theoretical framework of "wave turbulence theory." Lvov seeks to further develop this theory and to use these developments to study such complex systems.

Lvov's co-authors on the research at Miguel Onorato and Lara Vozella of the Universita di Torino in Italy; and Davide Proment of the University of East Anglia, in the United Kingdom.

Thanks for being here;
We need your help. The SpaceDaily news network continues to grow but revenues have never been harder to maintain.

With the rise of Ad Blockers, and Facebook - our traditional revenue sources via quality network advertising continues to decline. And unlike so many other news sites, we don't have a paywall - with those annoying usernames and passwords.

Our news coverage takes time and effort to publish 365 days a year.

If you find our news sites informative and useful then please consider becoming a regular supporter or for now make a one off contribution.

SpaceDaily Contributor
$5 Billed Once

credit card or paypal
SpaceDaily Monthly Supporter
$5 Billed Monthly

paypal only


Related Links
Rensselaer Polytechnic Institute
Space Technology News - Applications and Research

Comment on this article via your Facebook, Yahoo, AOL, Hotmail login.

Share this article via these popular social media networks DiggDigg RedditReddit GoogleGoogle

Memory Foam Mattress Review
Newsletters :: SpaceDaily :: SpaceWar :: TerraDaily :: Energy Daily
XML Feeds :: Space News :: Earth News :: War News :: Solar Energy News

New optical materials break digital connectivity barriers
Tel Aviv, Israel (SPX) Mar 23, 2015
From computers, tablets, and smartphones to cars, homes, and public transportation, our world is more digitally connected every day. The technology required to support the exchange of massive quantities of data is critical. That's why scientists and engineers are intent on developing faster computing units capable of supporting much larger amounts of data transfer and data processing. A ne ... read more

Arianespace selected by Airbus to launch EDRS-C Satellite

US to Scrap Delta IV Launch Vehicle in Favor of Russian-Made Rocket

Proton launches Express AM-7 satellite for Russian Government

DoD Works to Build Competition Into Space Launches

Ancient Martian lake system records 2 water-related events

Curiosity Rover Finds Biologically Useful Nitrogen on Mars

NASA's Opportunity Mars Rover Passes Marathon Distance

NASA Reformats Memory of Longest-Running Mars Rover

Extent of Moon's giant volcanic eruption is revealed

Yutu Changes Everything We Thought We Knew About Our Moon

Extent of moon's giant volcanic eruption is revealed

NASA's LRO Spacecraft Finds March 17, 2013 Impact Crater and More

Name the features on Pluto and its moon Charon

Science Shorts: Why Pluto?

Pluto Science, on the Surface

Science Shorts: How Big Is Pluto's Atmosphere?

Our Solar System May Have Once Harbored Super-Earths

SOFIA Finds Missing Link Between Supernovae and Planet Formation

ESA's CHEOPS Satellite: The Pharaoh of Exoplanet Hunting

Some habitable exoplanets could experience wildly unpredictable climates

Aerojet Rocketdyne Hot-Fire Tests Additive Components for the AR1 Engine

Sierra Nevada Corporation Unveils New Dream Chaser Cargo System

NASA's Space Launch System Booster Passes Major Ground Test

Replacing Russian Rocket Engine to Take 7 Years

China completes second test on new carrier rocket's power system

China's Yutu rover reveals Moon's "complex" geological history

China's Space Laboratory Still Cloaked

China has ability but no plan for manned lunar mission: expert

Next Steps on Journey to Mars: Progress on Asteroid Initiative

NASA plans to bring boulder into moon orbit

Comet 67P's Speed of Rotation Shows Signs of Slowing Down

Rosetta makes first detection of molecular nitrogen at a comet

The content herein, unless otherwise known to be public domain, are Copyright 1995-2014 - Space Media Network. All websites are published in Australia and are solely subject to Australian law and governed by Fair Use principals for news reporting and research purposes. AFP, UPI and IANS news wire stories are copyright Agence France-Presse, United Press International and Indo-Asia News Service. ESA news reports are copyright European Space Agency. All NASA sourced material is public domain. Additional copyrights may apply in whole or part to other bona fide parties. Advertising does not imply endorsement, agreement or approval of any opinions, statements or information provided by Space Media Network on any Web page published or hosted by Space Media Network. Privacy Statement All images and articles appearing on Space Media Network have been edited or digitally altered in some way. Any requests to remove copyright material will be acted upon in a timely and appropriate manner. Any attempt to extort money from Space Media Network will be ignored and reported to Australian Law Enforcement Agencies as a potential case of financial fraud involving the use of a telephonic carriage device or postal service.