Langevin equation

In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, calculating the statistics of the random motion of a small particle in a fluid due to collisions with the surrounding molecules in thermal motion.
Brownian motion as a prototype.
The original Langevin equation describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,
The degrees of freedom of interest here is the velocity formula_2 of the particle, formula_3 denotes the particle's mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity (Stokes' law), and a "noise term" formula_4 (the name given in physical contexts to terms in stochastic differential equations which are stochastic processes) representing the effect of the collisions with the molecules of the fluid. The force formula_4 has a Gaussian probability distribution with correlation function
where formula_7 is Boltzmann's constant, formula_8 is the temperature and formula_9 is the i-th component of the vector formula_4. The formula_11-function form of the correlations in time means that the force at a time formula_12 is assumed to be completely uncorrelated with the force at any other time. This is an approximation; the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. 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 formula_11-correlation and the Langevin equation becomes virtually exact.
Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient formula_14 in the correlation function of the random force, a fact also known as Einstein relation.
Mathematical aspects.
A strictly formula_11-correlated fluctuating force formula_4 is not a function in the usual mathematical sense and even the derivative formula_17 is not defined in this limit. This problem disappears when the Langevin equation is written in integral form 
formula_18 and a Langevin equation always should be interpreted as an abbreviation for its time integral. The general mathematical term for equations of this type is "stochastic differential equation".
Another mathematical ambiguity occurs for (rather special) Langevin equations with a multiplicative noise, that is terms like formula_19 on the r.h.s.. 
Such equations can be interpreted according to Stratonovich- or Ito- scheme, and if the derivation of the Langevin equation does not tell which one to use it is questionable anyhow. See Itō calculus.
Generic Langevin equation.
There is a formal derivation of a generic Langevin equation from classical mechanics. This generic equation plays a central role in the theory of critical dynamics, and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.
An essential condition of the derivation is a criterion dividing the degrees of freedom into the categories slow and fast. For example, local thermodynamic equilibrium in a liquid is reached within a few collision times. But it takes much longer for densities of conserved quantities like mass and energy to relax to equilibrium. Densities of conserved quantities, and in particular their long wavelength components, thus are slow variable candidates. Technically this division is realized with the Zwanzig projection operator, the essential tool in the derivation. The derivation is not completely rigorous from a mathematical physics perspective because it relies on certain plausible assumptions (akin to assumptions required elsewhere in basic statistical mechanics), but is otherwise acceptable from a theoretical physics perspective.
Let formula_20 denote the slow variables. The generic Langevin equation then reads
The fluctuating force formula_9 obeys a Gaussian probability distribution with correlation function
This implies the Onsager reciprocity relation formula_24 for the damping coefficients formula_14. The dependence formula_26 of formula_14 on formula_28 is negligible in most cases.
The symbol formula_29 denotes the Hamiltonian of the system, where formula_30 is the equilibrium probability distribution of the variables formula_28. Finally, formula_32 is the projection of the Poisson bracket of the slow variables formula_33 and formula_34 onto the space of slow variables.
In the Brownian motion case one would have formula_35, formula_36 or formula_37 and formula_38. The equation of motion formula_39 for formula_40 is exact, there is no fluctuating force formula_41 and no damping coefficient formula_42.
Examples.
Thermal noise in an electrical resistor.
There is a close analogy between the paradigmatic Brownian particle discussed above and Johnson noise, the electric voltage generated by thermal fluctuations in every resistor. 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. The Hamiltonian reads formula_43, and the Langevin equation becomes
This equation may be used to determine the correlation function
which becomes a white noise (Johnson noise) when the capacitance C becomes negligibly small.
Critical dynamics.
The dynamics of the order parameter formula_46 of a second order phase transition slows down near the critical point and can be described with a Langevin equation. The simplest case is the universality class "model A" with a non-conserved scalar order parameter, realized for instance in axial ferromagnets,
Other universality classes (the nomenclature is "model A"..., "model J") contain a diffusing order parameter, order parameters with several components, other critical variables and/or contributions from Poisson brackets.
Harmonic oscillator in a fluid.
A particle in a fluid is also described by the Langevin equation with a potential, a damping force and thermal fluctuations given by the fluctuation dissipation theorem. If the potential is a harmonic oscillator potential then the constant energy curves are ellipses as shown in Figure 1 below. However, in the presence of a dissipation force a particle keeps losing energy to the environment. On the other hand, the thermal fluctuation randomly adds energy to the particle. In the absence of the thermal fluctuations the particle continuously loses kinetic energy and the phase portrait of the time evolution of the velocity vs. position looks like an ellipse that is spiraling in until it reaches zero velocity. Conversely, the thermal fluctuations provide kicks to the particles that do not allow the particle to lose all its energy. So, at long times, the initial ensemble of stochastic oscillators to spread out, eventually reaching thermal equilibrium, for whom the distribution of velocity and position is given by the Maxwell–Boltzmann distribution. In the plot below (Figure 2), the long time velocity distribution (orange) and position distributions (blue) in a harmonic potential ( formula_49) is plotted with the Boltzmann probabilities for velocity (red) and position (green). We see that the late time behavior depicts thermal equilibrium. 
Trajectories of free Brownian particles.
Consider a free particle of mass formula_3 with equation of motion described by
where formula_52 is the particle velocity, formula_53 is the particle mobility, and formula_54 is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale formula_55 of particle collisions, i.e. formula_56. The general solution to the equation of motion is
where formula_58 is the relaxation time of the Brownian motion. As expected from the random nature of Brownian motion, the average drift velocity formula_59 quickly decays to zero at formula_60. It can also be shown that the autocorrelation function of the particle velocity formula_2 is given by
where we have used the property that the variables formula_63 and formula_64 become uncorrelated for time separations formula_65. Besides, the value of formula_66 is set to be equal to formula_67 such that it obeys the equipartition theorem. Note that if the system is initially at thermal equilibrium already with formula_68, then formula_69 for all formula_12, meaning that the system remains at equilibrium at all times.
The velocity formula_71 of the Brownian particle can be integrated to yield its trajectory (assuming it is initially at the origin)
Hence, the resultant average displacement formula_73 asymptotes to formula_74 as the system relaxes and randomness takes over. In addition, the mean squared displacement can be determined similarly to the preceding calculation to be
It can be seen that formula_76, indicating that the motion of Brownian particles at timescales much shorter than the relaxation time formula_77 of the system is (approximately) time-reversal invariant. On the other hand, formula_78, which suggests that the long-term random motion of Brownian particles is an irreversible dissipative process. Here we have made use of the Einstein–Smoluchowski relation formula_79, where formula_80 is the diffusion coefficient of the fluid.
Recovering Boltzmann statistics.
The Langevin equations for a system in which each particle is affected by an external force formula_81 and also random collisions with other particles must reproduce the Boltzmann distribution, which is the probability distribution function for particles in thermal equilibrium. One-dimensional overdamped This is Brownian motion in the presence of an external forcing. The overdamped case is realized when the inertia of the particle is negligible in comparison to the damping force. The trajectory formula_82 of the particle in a potential formula_83 is described by the Langevin equation
where the noise is characterized by the white noise formula_85, or equivalently in terms of the Wiener process formula_86, and formula_14 is the damping constant. We would like to compute the distribution formula_88 of the particle's position in the course of time. A direct way to determine this distribution is to introduce a test function formula_89, and to look at the average of this function over all realizations (ensemble average) which should vanish,
because the average of formula_91 should be time-independent for finite formula_82. Now, according to Itô's lemma for Itô drift-diffusion process satisfying the stochastic differential equation formula_93, the differential of a twice-differentiable function is given by 
Applying this to the above calculation, and considering that formula_95 is not an explicit function of time, i.e., formula_96, one gets 
This average can be written using the probability density function formula_88;
where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions formula_89, we must have:
thus recovering the Boltzmann distribution
Equivalent techniques.
A solution of a Langevin equation for a particular realization of the fluctuating force is of no interest by itself; what is of interest are correlation functions of the slow variables after averaging over the fluctuating force. Such correlation functions also may be determined with other (equivalent) techniques.
Fokker–Planck equation.
A Fokker–Planck equation is a deterministic equation for the time dependent probability density formula_103 of stochastic variables formula_28. The Fokker–Planck equation corresponding to the generic Langevin equation above may be derived with standard techniques (see for instance ref.),
The equilibrium distribution formula_106 is a stationary solution.
Path integral.
A path integral equivalent to a Langevin equation may be obtained from the corresponding Fokker–Planck equation or by transforming the Gaussian probability distribution formula_107 of the fluctuating force formula_108 to a probability distribution of the slow variables, schematically formula_109.
The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where formula_110 depends on formula_111 but not on formula_112. It turns out to be convenient to introduce auxiliary "response variables" formula_113. The path integral equivalent to the generic Langevin equation then reads
where formula_115 is a normalization factor and
The path integral formulation doesn't add anything new, but it does allow for the use of tools from quantum field theory; for example perturbation and renormalization group methods (if these make sense).