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04-10-2013
08:27 AM

04-10-2013
08:27 AM

Ergodicity in a Harmonic Oscillator

By Mircea Andrecut

**A Harmonic Oscillator Coupled with a Nose-Hoover Thermostat**

- Uses numerical simulations to show how a harmonic oscillator coupled with a Nose-Hoover thermostat has ergodic behavior that can only be achieved by extending Nose-Hoover dynamics to the Bulgax-Kusnetov dynamics
- Uses molecular dynamics, harmonic oscillators, and phase space
- Performs time-centered finite difference algorithms (leap frogs), time reversible centered-difference equations

Numerical simulations that are run in PTC Mathcad software show that in a harmonic osciallator coupled with a Nose-Hoover thermostat, the ergodic behavior can be achieved only by extending the Nose-Hoover dynamics to the Bulgac-Kusnetov dynamics. The Nosé-Hoover thermostat is a constant temperature MD simulation technique that can produce the canonical distribution in a classical system.

This worksheet models a classical particle described by a Hamiltonian and is an example of how one can conduct experiments involving mathematics and physics in practical, realistic scenarios. In solving, the equations of motion from Nose-Hoover is used. After the application of the forces for a harmonic oscillator, you can step through the time iterations. Results show that the behavior is not ergodic. By modifying the Nose-Hoover dynamics, simulating the Bulgac-Kusnezoc oscillator shows results consistent with our initial assumptions. It proves through numerical simulations that** ergodic behavior **can be achieved only by extending the Nosé-Hoover dynamics in the way proposed by **Bulgac and Kusnetov.**

This worksheet includes step by step notation, formulas, equations, figures and graphs, and solutions for the experimental question.

* Download and explore this worksheet yourself! You can download a free lifetime copy of PTC Mathcad Express and get 30 days of full functionality.*

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