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In statistical physics, a Langevin equation (Paul Langevin, 1908) 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.
Contents
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 degree of freedom of interest here is the position
where
Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient
Mathematical aspects
A strictly
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 because it relies on (plausible) assumptions akin to assumptions required elsewhere in basic statistical mechanics.
Let
The fluctuating force
This implies the Onsager reciprocity relation
In the Brownian motion case one would have
Harmonic oscillator in a fluid
A non-ideal harmonic oscillator is affected by some form of damping, from which it follows via the fluctuation-dissipation theorem that there must be some fluctuations in the system. The diagram at right shows a phase portrait of the time evolution of the momentum,
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
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
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.
Recovering Boltzmann statistics
Langevin equations must reproduce the Boltzmann distribution. 1-dimensional overdamped Brownian motion is an instructive example. The overdamped case is realized when the inertia of the particle is negligible in comparison to the damping force. The trajectory
where the noise is characterized by
If
where we make use of the probability density function
where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions
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
The equilibrium distribution
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
where
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).