Advanced Molecular Dynamics

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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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Partition function
Maxwell-Boltzmann velocity distribution
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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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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 t1 at position x(t1) and end at t2 at
position x(t2) for which the action (S) is the
minimum
SltS
SltS
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Example free particle
Consider a particle in vacuum
v(t)vav
Always gt 0!!
?(t)0 for all t
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Lagrangian
Lagrangian
Action
The true path plus deviation
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Should be 0 for all paths
Equations of motion
Lagrangian equations of motion
Conjugate momentum
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Newton?
Valid in any coordinate system Cartesian
Conjugate momentum
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Lagrangian dynamics
We have
2nd order differential equation
Two 1st order differential equations
With these variables we can do statistical
thermodynamics
Change dependence
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Hamiltonian
Hamiltons equations of motion
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Newton?
Conjugate momentum
Hamiltonian
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Nosé thermostat
Lagrangian
Extended system 3N1 variables
Hamiltonian
Associated mass
Conjugate momentum
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Nosé and thermodynamics
Recall
MD
MC
Gaussian integral
Constant plays no role in thermodynamics
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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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