Langevin dynamics

In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. It was originally developed by 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.
Overview.
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 temperature to be controlled like with 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 the hydrophobic effect. For denser solvents, hydrodynamic interactions are not captured via Langevin dynamics.
For a system of formula_1 particles with masses formula_2, with coordinates formula_3 that constitute a time-dependent random variable, the resulting Langevin equation is 
where formula_5 is the particle interaction potential; formula_6 is the gradient operator such that formula_7 is the force calculated from the particle interaction potentials; the dot is a time derivative such that formula_8 is the velocity and formula_9 is the acceleration; formula_10 is the damping constant (units of reciprocal time); formula_11 is the temperature, formula_12 is Boltzmann's constant; and formula_13 is a delta-correlated stationary Gaussian process with zero-mean, satisfying
Here, formula_16 is the Dirac delta.
If the main objective is to control temperature, care should be exercised to use a small damping constant formula_10. As formula_10 grows, it spans from the inertial all the way to the diffusive (Brownian) regime. The Langevin dynamics limit of non-inertia is commonly described as Brownian dynamics. Brownian dynamics can be considered as overdamped Langevin dynamics, i.e. Langevin dynamics where no average acceleration takes place.
The Langevin equation can be
reformulated as a Fokker–Planck equation that governs the probability distribution of the random variable "X".