Advanced Molecular Dynamics Velocity scaling Andersen Thermostat Hamiltonian
- Slides: 22
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 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 x S<S t 1 t 2 t
Example: free particle Consider a particle in vacuum: v(t)=vav η(t)=0 for all t Always > 0!!
Lagrangian Cartesian coordinates (Newton) → Generalized coordinates (? ) Lagrangian Action S[q+η]= S[q] The true path plus deviation
Should be 0 for all paths S[q+η]= S[q] Equations of motion Lagrangian equations of motion Conjugate momentum
Newton? Valid in any coordinate system: Cartesian Conjugate momentum
Lagrangian dynamics We have: 2 nd order differential equation Two 1 st order differential equations With these variables we can do statistical thermodynamics Change dependence:
Hamiltonian Hamilton’s equations of motion
Newton? Conjugate momentum Hamiltonian
Nosé thermostat Lagrangian Hamiltonian Extended system 3 N+1 variables Associated mass Conjugate momentum
Nosé and thermodynamics Recall MD MC Gaussian integral Constant plays no role in thermodynamics
Lagrangian Hamiltonian Conjugate momenta Equations of motion: Equations of Motion
Nosé Hoover
- Thermostat
- Andersen thermostat
- Lagrangian
- Darcy's law
- Final velocity initial velocity acceleration time
- Rotational motion and the law of gravity
- Hydraulic conductivity
- Speed unit
- Angular acceleration formula with radius
- Angular vs linear velocity
- Instantaneous velocity vs average velocity
- Initial velocity and final velocity formula
- Giant molecular structure vs simple molecular structure
- Zinc oxide + nitric acid → zinc nitrate + water
- Covalently bonded substances
- Molecular dynamics limitations
- Hamiltonian circuit
- Hamiltonian operator
- Hamilton's equations of motion
- Hamiltonian circuit
- If the lagrangian is cyclic in qj then
- Hamiltonian operator
- Particle on a ring angular momentum