（From  http://en.wikipedia.org/wiki/Langevin_dynamics)

In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems, originally developed by the French physicist Paul Langevin. The approach is characterized by the use of simplified models while accounting for omitted degrees of freedom by the use of stochastic differential equations.

A molecular system in the real world is unlikely to be present in vacuum. Jostling of solvent or air molecules causes friction, and the occasional high velocity collision will perturb the system. Langevin dynamics attempts to extend molecular dynamics to allow for these effects. Also, Langevin dynamics allows controlling the temperature like a thermostat, thus approximating the canonical ensemble.

Langevin dynamics mimics the viscous aspect of a solvent. It does not fully model an implicit solvent; specifically, the model does not account for the electrostatic screening and also not for thehydrophobic effect. It should also be noted that for denser solvents, hydrodynamic interactions are not captured via Langevin dynamics.

For a system of particles with masses , with coordinates that constitute a time-dependent random variable, the resulting Langevin equation is ^[1]

where is the particle interaction potential; is the gradient operator such that is the force calculated from the particle interaction potentials; the dot is a time derivative such that is the velocity and is the acceleration; T is the temperature, k_B is Boltzmann's constant; and is a delta-correlated stationary Gaussian process with zero-mean, satisfying

Here, is the Dirac delta.

If the main objective is to control temperature, care should be exercised to use a small damping constant . As grows, it spans the inertial all the way to the diffusive (Brownian) regime. The Langevin dynamics limit of non-inertia is commonly described as Brownian dynamics.

The Langevin equation can be reformulated as a Fokker–Planck equation that governs the probability distribution of the random variable X.

FOR VASP:

(from http://cms.mpi.univie.ac.at/vasp/vasp/Langevin_thermostat.html#sec_langevin)

Langevin thermostat In Langevin dynamics [63], the temperature is maintained by modifying the Newton's equations of motion:

(6.35)

where is the force acting on atom due to the interaction potential, is a friction coefficient, and is a random force with dispersion related to the friction coefficient via:

(6.36)

with being the time-step used in MD to integrate equations of motion. Obviously, Langevin dynamics is identical to classical Hamiltonian in the limit of vanishing .

(FROM http://cms.mpi.univie.ac.at/vasp/vasp/Langevin_dynamics_in_NVT_ensemble.html )

Langevin dynamics in NVT ensemble See Sec. 6.62.5 for brief description of Langevin thermostat.

1. Set the standard MD-related flags: IBRION=0, TEBEG, POTIM, NSW

2. Set ISIF=2

3. Set MDALGO to 3 to invoke Langevin dynamics 6.62.3.

4. Set friction coefficients for all species defined in POSCAR using LANGEVIN_GAMMA (see 6.62.3)

Note that geometric constraints and metadynamics are not available for Langevin dynamics in the current version of VASP.

(From http://cms.mpi.univie.ac.at/vasp/vasp/Parrinello_Rahman_NpT_dynamics_with_Langevin_thermostat.html)

Parrinello-Rahman (NpT) dynamics with Langevin thermostat The Parrinello-Rahman dynamics is currently available only in connection with Langevin thermostat 6.62.5. The geometric constraints and metadynamics are not supported in the current version of VASP. See Sec. 6.62.6 and 6.62.5 for brief description of the Parrinello-Rahman dynamics and Langevin thermostat, respectively.

1. Use the same setup as for Langevin dynamics in NVT ensemble (see Sec. 6.62.2) but set ISIF=3 to allow for the cell volume and cell shape variations. At the moment, dynamics with fixed volume + variable shape (ISIF=4) and fixed shape + variable volume (ISIF=7) are not available.

2. Use LANGEVIN_GAMMA_L to set friction coefficient for lattice degrees of freedom (see 6.62.3)

3. Set mass for the lattice degrees of freedom using the parameter using the parameter PMASS 6.62.3

4. Optionally, external pressure (in kB) can be defined using the parameter PSTRESS

Note that the temperatures listed in the file OSZICAR are computed using the kinetic energy including contribution from both atomic and lattice degrees of freedom. The external pressure for a simulation can be computed as one third of trace of stress-tensor corrected for kinetic contribution listed in OUTCAR. This can be achieved e.g. by using the following command: grep "Total+kin" OUTCAR| awk 'BEGIN {p=0.} {p+=(\$2+\$3+\$4)/3.} END {print "pressure (kB):",p}'

IMPORTANT: In Parrinello-Rahman dynamics, components of stress tensor are used to define forces acting on lattice degrees of freedom (see Ref. [79,80] for details). In order to achieve a reasonable quality of sampling (or even to avoid numerical problems), it is essential to eliminate Pulay stress. Unfortunately, this usually requires rather large value of ENCUT. The setting with PREC=low (add the corresponding link) frequently used in NVT MD is not recommended for molecular dynamics with variable cell volume. For more details on the Pulay stress see 7.6.