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

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Naïve approach Velocity scaling Do we sample the canonical ensemble?

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Maxwell-Boltzmann velocity distribution Partition function

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Fluctuations in the momentum: Fluctuations in the temperature

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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)

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Velocity Verlet:

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Andersen thermostat: static properties

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Andersen thermostat: dynamic properties

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Hamiltonian & Lagrangian The equations of motion give the path that starts at t 1 at position x(t 1 ) and end at t 2 at position x(t 2 ) for which the action (S) is the minimum t x t2t2 t1t1 S~~
~~

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

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Calculus of variation True path for which S is minimum η(t) should be such the δS is minimal At the boundaries: η(t 1 )=0 and η(t 2 )=0 η(t) is small

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A description which is independent of the coordinates This term should be zero for all η(t) so […] η(t) If this term 0, S has a minimum Newton Integration by parts Zero because of the boundaries η(t 1 )=0 and η(t 2 )=0

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This term should be zero for all η(t) so […] η(t) If this term 0, S has a minimum Newton A description which is independent of the coordinates

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Integration by parts Zero because of the boundaries η(t 1 )=0 and η(t 2 )=0

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

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

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Desired format […] η(t) Conjugate momentum Equations of motion Should be 0 for all paths

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Partial integration

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Newton? Conjugate momentum Valid in any coordinate system: Cartesian

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

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

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

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Example: thermodynamics We prefer to control T: S→T Legendre transformation Helmholtz free energy

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Hamiltonian Hamilton’s equations of motion

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Newton? Conjugate momentum Hamiltonian

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

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

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Nosé and thermodynamics Gaussian integral Constant plays no role in thermodynamics

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Recall MD MC

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Delta functions

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Equations of Motion Lagrangian Hamiltonian Conjugate momenta Equations of motion:

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Nosé Hoover

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