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: 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 t1 at position x(t1) and end at t2 at position x(t2) for which the action (S) is the minimum S<S t x t2 t1 S<S

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

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

A description which is independent of the coordinates 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 Newton A description which is independent of the coordinates

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

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

Newton? Valid in any coordinate system: Cartesian Conjugate momentum

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

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

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

Hamilton’s equations of motion Hamiltonian Hamilton’s equations of motion

Newton? Conjugate momentum Hamiltonian

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

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

Equations of Motion Lagrangian Hamiltonian Conjugate momenta

Nosé Hoover