Thermostat: Langevin MD

Thermostat: Langevin MD

The number of particles N, the volume V, and the temperature T are fixed in a canonical ensemble. However, since the temperature is defined by the ensemble average of kinetic energies of all particles as Eq. 1, it is impossible to fix T exactly at a set point.

..........(1)

Therefore, various types of thermostats, such as Berendsen, Langevin and Nosé-Hoover thermostats, have been proposed to control the motions. A Berendsen thermostat is a proportional type of thermostat, and corrects deviations of T from the set point T₀ by multiplying the velocities by a factor to control the value of T.[1] Nosé-Hoover thermostat is an integral type of thermostat, and it introduces additional degrees of freedom (momentum) into the Hamiltonian of a system.[2-4] The Langevin thermostat was used in the studies discussed below. The Langevin thermostats follow the Langevin equation of motion instead of Newton's equation of motion.[5] In the Langevin equation of motion, a frictional force added to the conservative force is proportional to the velocity, and it adjusts the kinetic energy of the particle so that the temperature matches the set temperature.

..........(2)

where m is mass of a particle, a is acceleration, f(r) is conservative force acting on the particle, v is the velocity of the particle, ξ is a frictional constant, and f ′ is a random force. The frictional force - ξv decreases the temperature because ξ is a fixed positive value. The random force is randomly determined from a Gaussian distribution to add kinetic energy to the particle, and its variance is the function of set temperature and time step. Therefore, the random force is balanced with the frictional force and maintains the system temperature at the set value.

References
[1] Berendsen, H.J.C., et al., Molecular-Dynamics with Coupling to an External Bath. Journal of Chemical Physics, 1984. 81(8): p. 3684-3690.
[2] Nosé, S., A Molecular-Dynamics Method for Simulations in the Canonical Ensemble. Molecular Physics, 1984. 52(2): p. 255-268.
[3] Hoover, W.G., Canonical Dynamics - Equilibrium Phase-Space Distributions. Physical Review A, 1985. 31(3): p. 1695-1697.
[4] Nosé, S., A Unified Formulation of the Constant Temperature Molecular-Dynamics Methods. Journal of Chemical Physics, 1984. 81(1): p. 511-519.
[5] Adelman, S.A. and J.D. Doll, Generalized Langevin Equation Approach for Atom-Solid-Surface Scattering - General Formulation for Classical Scattering Off Harmonic Solids. Journal of Chemical Physics, 1976. 64(6): p. 2375-2388.