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Ergodicity in a Harmonic Oscillator

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Ergodicity in a Harmonic Oscillator

By Mircea Andrecut

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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.