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

Naïve approach Velocity scaling Do we sample the canonical ensemble?

Maxwell-Boltzmann velocity distribution Partition function

Fluctuations in the momentum: Fluctuations in the temperature

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)

Velocity Verlet:

Andersen thermostat: static properties

Andersen thermostat: dynamic properties

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<S

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

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

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

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

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

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

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

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

Partial integration

Newton? Conjugate momentum Valid in any coordinate system: Cartesian

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

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

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

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

Hamiltonian Hamilton’s equations of motion

Newton? Conjugate momentum Hamiltonian

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

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

Nosé and thermodynamics Gaussian integral Constant plays no role in thermodynamics

Recall MD MC

Delta functions

Equations of Motion Lagrangian Hamiltonian Conjugate momenta Equations of motion:

Nosé Hoover