## Thermostats by stochastic collision ### Andersen thermostat - System is coupled to heat bath - The coupling is represented by stochastic force. Impulsive force that act occasionally on random selected particles. - Coupling strength described by collision frequency. And the collision follows Poisson distribution (time uncorrelated collisions) $P(t, \nu)=\nu e^{(-\nu t)}$ $\nu$ is frequency of collision. >[!Note] > The redistribution of velocity follows Maxwell Boltzmann distribution, so it is also not suitable for calculating kinetic related term (e.g. diffusion coefficient), except no collision happened. Pros of Anderson thermostat: - Generates canonical ensemble (NVT) - Stochastic - Local thermostat Cons: - Destroy momentum transport. No true molecular kinetics gets presented. Cannot be used to calculate transport properties. ### Langevin thermostat - The system is sought to interact with the solvent (of small frictional particles), giving damping force. - Pipping coefficient and violence of random force connected to fluctuation, dispersion relation, generate Canonical ensemble. Pros: - Recover NVT - Local thermostat - Ergodic - Large time step possible Cons: - No transport properties >[!Note] > More suitable for simulating the reaction/process in solution. ### Nosé–Hoover thermostat In general, this method introduces an additional term to control the behavior of thermal baths (effective mass & coupling), and made a conserved quantity, i.e., extended Hamiltonian. #### Nosé–Hoover (2nd order on K): - Proper sampling - Deterministic (can be non-ergotic) - Take another on K, can be oscillating >[!Notice] > The deterministic here is a very good property, meaning it can be not stochastic. #### Nosé–Hoover chain (higher order on K): - Canonical - Ergodic - Additional term for chaotic behavior