Langevin dynamics

In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems using the Langevin equation. 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. Langevin dynamics simulations are a kind of Monte Carlo simulation.^[1]

Overview

A real world molecular system 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 as 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 ${\displaystyle N}$ particles with masses ${\displaystyle M}$, with coordinates ${\displaystyle X=X(t)}$ that constitute a time-dependent random variable, the resulting Langevin equation is^[2][3] $${\displaystyle M\,{\ddot {\mathbf {X} }}=-\mathbf {\nabla } U(\mathbf {X} )-\gamma \,M\,{\dot {\mathbf {X} }}+{\sqrt {2\,M\,\gamma \,k_{\rm {B}}T}}\,\mathbf {R} (t)\,,}$$ where ${\displaystyle U(\mathbf {X} )}$ is the particle interaction potential; ${\displaystyle \nabla }$ is the gradient operator such that ${\displaystyle -\mathbf {\nabla } U(\mathbf {X} )}$ is the force calculated from the particle interaction potentials; the dot is a time derivative such that ${\displaystyle {\dot {\mathbf {X} }}}$ is the velocity and ${\displaystyle {\ddot {\mathbf {X} }}}$ is the acceleration; ${\displaystyle \gamma }$ is the damping constant (units of reciprocal time), also known as the collision frequency; ${\displaystyle T}$ is the temperature, ${\displaystyle k_{\rm {B}}}$ is the Boltzmann constant; and ${\displaystyle \mathbf {R} (t)}$ is a delta-correlated stationary Gaussian process with zero-mean, satisfying $${\displaystyle \left\langle \mathbf {R} (t)\right\rangle =0}$$ $${\displaystyle \left\langle \mathbf {R} (t)\cdot \mathbf {R} (t')\right\rangle =\delta (t-t')}$$

Here, ${\displaystyle \delta }$ is the Dirac delta.

If the main objective is to control temperature, care should be exercised to use a small damping constant ${\displaystyle \gamma }$. As ${\displaystyle \gamma }$ 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.^[4]

The translational Langevin equation can be solved using various numerical methods ^[5][6] with difference in: sophistication of analytical solutions, allowed time-steps, time-reversibility (symplectic methods) in the limit of zero friction, etc.

The Langevin equation can be generalized to rotational dynamics of molecules, Brownian particles, etc. A standard (according to NIST^[7]) way to do it is to leverage a quaternion-based description of the stochastic rotational motion.^[8][9]

See also

• Hamiltonian mechanics
• Statistical mechanics
• Implicit solvation
• Stochastic differential equations
• Langevin equation
• Klein–Kramers equation