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Advanced Molecular Dynamics

Advanced Molecular Dynamics. Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat. Naïve approach. Velocity scaling. Do we sample the canonical ensemble?. Partition function. Maxwell-Boltzmann velocity distribution. Fluctuations in the momentum:.

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Advanced Molecular Dynamics

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  1. Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian & Lagrangian Appendix A Nose-Hoover thermostat

  2. Naïve approach Velocity scaling Do we sample the canonical ensemble?

  3. Partition function Maxwell-Boltzmann velocity distribution

  4. Fluctuations in the momentum: Fluctuations in the temperature

  5. Andersen thermostat Every particle has a fixed probability to collide with the Andersen demon After collision the particle is give a new velocity The probabilities to collide are uncorrelated (Poisson distribution)

  6. Velocity Verlet:

  7. Andersen thermostat: static properties

  8. Andersen thermostat: dynamic properties

  9. x t1 t2 t Hamiltonian & Lagrangian The equations of motion give the path that starts at t1 at position x(t1) and end at t2at position x(t2) for which the action (S) is the minimum S<S S<S

  10. Example: free particle Consider a particle in vacuum: v(t)=vav Always > 0!! η(t)=0 for all t

  11. At the boundaries: η(t1)=0 and η(t2)=0 η(t) is small Calculus of variation True path for which S is minimum η(t) should be such the δS is minimal

  12. Newton This term should be zero for all η(t) so […] η(t) Integration by parts If this term 0, S has a minimum Zero because of the boundaries η(t1)=0 and η(t2)=0 A description which is independent of the coordinates

  13. Cartesian coordinates (Newton) → Generalized coordinates (?) Lagrangian Lagrangian Action The true path plus deviation

  14. Desired format […] η(t) Partial integration Should be 0 for all paths Equations of motion Conjugate momentum Lagrangian equations of motion

  15. Newton? Valid in any coordinate system: Cartesian Conjugate momentum

  16. Pendulum Equations of motion in terms of l and θ Conjugate momentum

  17. Lagrangian dynamics We have: 2nd order differential equation Two 1st order differential equations With these variables we can do statistical thermodynamics Change dependence:

  18. Legrendre transformation Example: thermodynamics We have a function that depends on and we would like We prefer to control T: S→T Legendre transformation Helmholtz free energy

  19. Hamiltonian Hamilton’s equations of motion

  20. Newton? Conjugate momentum Hamiltonian

  21. Lagrangian Nosé thermostat Hamiltonian Extended system 3N+1 variables Associated mass Conjugate momentum

  22. Nosé and thermodynamics Delta functions Recall MD MC Gaussian integral Constant plays no role in thermodynamics

  23. Lagrangian Equations of Motion Hamiltonian Conjugate momenta Equations of motion:

  24. Nosé Hoover

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