Langevin equation

In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces.

The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation.

However, the Langevin equation is used to describe the motion of a "macroscopic" particle at a much longer time scale, and in this limit the

Another common feature of the Langevin equation is the occurrence of the damping coefficient

in the correlation function of the random force, which in an equilibrium system is an expression of the Einstein relation.

This problem disappears when the Langevin equation is written in integral form

If a multiplicative noise is intrinsic to the system, its definition is ambiguous, as it is equally valid to interpret it according to Stratonovich- or Ito- scheme (see Itô calculus).

Nevertheless, physical observables are independent of the interpretation, provided the latter is applied consistently when manipulating the equation.

This is necessary because the symbolic rules of calculus differ depending on the interpretation scheme.

[4][5] There is a formal derivation of a generic Langevin equation from classical mechanics.

[6][7] This generic equation plays a central role in the theory of critical dynamics,[8] and other areas of nonequilibrium statistical mechanics.

An essential step in the derivation is the division of the degrees of freedom into the categories slow and fast.

Thus, densities of conserved quantities, and in particular their long wavelength components, are slow variable candidates.

There is a close analogy between the paradigmatic Brownian particle discussed above and Johnson noise, the electric voltage generated by thermal fluctuations in a resistor.

[10] The diagram at the right shows an electric circuit consisting of a resistance R and a capacitance C. The slow variable is the voltage U between the ends of the resistor.

of a second order phase transition slows down near the critical point and can be described with a Langevin equation.

[8] The simplest case is the universality class "model A" with a non-conserved scalar order parameter, realized for instance in axial ferromagnets,

If the potential is quadratic then the constant energy curves are ellipses, as shown in the figure.

By contrast, thermal fluctuations continually add energy to the particle and prevent it from reaching exactly 0 velocity.

) is plotted with the Boltzmann probabilities for velocity (green) and position (red).

is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale

, indicating that the motion of Brownian particles at timescales much shorter than the relaxation time

If the external potential is conservative and the noise term derives from a reservoir in thermal equilibrium, then the long-time solution to the Langevin equation must reduce to the Boltzmann distribution, which is the probability distribution function for particles in thermal equilibrium.

In the special case of overdamped dynamics, the inertia of the particle is negligible in comparison to the damping force, and the trajectory

In some situations, one is primarily interested in the noise-averaged behavior of the Langevin equation, as opposed to the solution for particular realizations of the noise.

This section describes techniques for obtaining this averaged behavior that are distinct from—but also equivalent to—the stochastic calculus inherent in the Langevin equation.

The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where

The path integral equivalent to the generic Langevin equation then reads[16]

The path integral formulation allows for the use of tools from quantum field theory, such as perturbation and renormalization group methods.

The mathematical formalism for this representation can be developed on abstract Wiener space.

An electric circuit consisting of a resistor and a capacitor.
Figure 1: Phase portrait of a harmonic oscillator showing spreading due to the Langevin Equation.
Figure 2: Equilibrium probability for Langevin dynamics in Harmonic Potential
Simulated squared displacements of free Brownian particles (semi-transparent wiggly lines) as a function of time, for three selected choices of initial squared velocity which are 0, 3 k B T / m , and 6 k B T / m respectively, with 3 k B T / m being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared displacements for the corresponding parameter choices.
This plot corresponds to solutions of the complete Langevin equation for a lightly damped harmonic oscillator, obtained using the Euler–Maruyama method . The left panel shows the time evolution of the phase portrait at different temperatures. The right panel captures the corresponding equilibrium probability distributions. At zero temperature, the velocity slowly decays from its initial value (the red dot) to zero, over the course of a handful of oscillations, due to damping. For nonzero temperatures, the velocity can be kicked to values higher than the initial value due to thermal fluctuations. At long times, the velocity remains nonzero, and the position and velocity distributions correspond to that of thermal equilibrium.