Langevin equation

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.

Brownian motion as a prototype

The original Langevin equation[1] describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,

$m\frac{d^{2}\mathbf{x}}{dt^{2}}=-\lambda \frac{d\mathbf{x}}{dt}+\boldsymbol{\eta}\left( t\right).$

The degree of freedom of interest here is the position x of the particle, m 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 η(t) (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 η(t) has a Gaussian probability distribution with correlation function

$\left\langle \eta_{i}\left( t\right)\eta_{j}\left( t^{\prime}\right) \right\rangle =2\lambda k_{B}T\delta _{i,j}\delta \left(t-t^{\prime }\right) ,$

where kB is Boltzmann's constant and T is the temperature. The δ-function form of the correlations in time means that the force at a time t is assumed to be completely uncorrelated with it 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 δ-correlation and the Langevin equation become exact.

Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient λ in the correlation function of the random force, a fact also known as Einstein relation.

Generic Langevin equation

There is a formal derivation of a generic Langevin equation from classical mechanics.[2] This generic equation plays a central role in the theory of critical dynamics,[3] 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,[4] 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 A={Ai} denote the slow variables. The generic Langevin equation then reads

$\frac{dA_{i}}{dt}=k_{B}T\sum\limits_{j}{\left[ {A_{i},A_{j}}\right] \frac{{d}\mathcal{H}}{{dA_{j}}}}-\sum\limits_{j}{\lambda _{i,j}\left( A\right) \frac{d\mathcal{H}}{{dA_{j}}}+}\sum\limits_{j}{\frac{d{\lambda _{i,j}\left(A\right) }}{{dA_{j}}}}+\eta _{i}\left( t\right).$

The fluctuating force ηi(t) obeys a Gaussian probability distribution with correlation function

$\left\langle {\eta _{i}\left( t\right) \eta _{j}\left( t^{\prime }\right) }\right\rangle =2\lambda _{i,j}\left( A\right) \delta \left( t-t^{\prime}\right).$

This implies the Onsager reciprocity relation λi,jj,i for the damping coefficients λ. The dependence i,j/dAj of λ on A is negligible in most cases. The symbol $\mathcal{H}$=-ln(p0) denotes the Hamiltonian of the system, where p0(A) is the equilibrium probability distribution of the variables A. Finally, [Ai, Aj] is the projection of the Poisson bracket of the slow variables Ai and Aj onto the space of slow variables.

In the Brownian motion case one would have $\mathcal{H}$= p2/(2mkBT), A={p} or A={x, p} and [xi, pj]=δi,j. The equation of motion dx/dt=p/m for x is exact, there is no fluctuating force ηx and no damping coefficient λx,p.

Examples

Phase portrait of a harmonic oscillator showing spreading due to the Langevin Equation.

Harmonic oscillator in a fluid

The diagram at right shows a phase portrait of the time evolution of the momentum, p=mv, vs. position, r of a harmonic oscillator. Deterministic motion would follow along the ellipsoidal trajectories which cannot cross each other without changing energy. The presence of a molecular fluid environment (represented by diffusion and damping terms) continually adds and removes kinetic energy from the system, causing an initial ensemble of stochastic oscillators (dotted circles) to spread out, eventually reaching thermal equilibrium.

An electric circuit consisting of a resistor and a capacitor.

Thermal noise in an electrical resistor

Another application is 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 $\mathcal{H}$= E/kBT=CU2/(2kBT), and the Langevin equation becomes

$\frac{dU}{dt} =-\frac{U}{RC}+\eta \left( t\right),\;\; \left\langle \eta \left( t\right) \eta \left( t^{\prime }\right)\right\rangle = \frac{2k_{B}T}{RC^{2}}\delta \left(t-t^{\prime }\right).$

This equation may be used to determine the correlation function

$\left\langle U\left(t\right) U\left(t^{\prime }\right) \right\rangle =\left( k_{B}T/C\right) \exp \left( -\left\vert t-t^{\prime }\right\vert /RC\right) \approx 2Rk_{B}T\delta \left( t-t^{\prime }\right),$

which becomes a white noise (Johnson noise) when the capacitance C becomes negligibly small.

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 P(A,t) of stochastic variables A. The Fokker–Planck equation corresponding to the generic Langevin equation above may be derived with standard techniques (see for instance ref.[5]),

$\frac{\partial P\left(A,t\right)}{\partial t}=\sum_{i,j}\frac{\partial}{\partial A_{i}}\left(-k_{B}T\left[A_{i},A_{j}\right]\frac{\partial\mathcal{H}}{\partial A_{j}}+\lambda_{i,j}\frac{\partial\mathcal{H}}{\partial A_{j}}+\lambda_{i,j}\frac{\partial}{\partial A_{j}}\right)P\left(A,t\right).$

The equilibrium distribution P(A,t) = p0(A) = const×exp(-$\mathcal{H}$) 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 P(η)(η)dη of the fluctuating force η to a probability distribution of the slow variables, schematically P(A)dA = P(η)(η(A))det(dη/dA)dA. The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where A(t+Δt)-A(t) depends on A(t) but not on A(t+Δt). It turns out to be convenient to introduce auxiliary response variables $\tilde A$. The path integral equivalent to the generic Langevin equation then reads [6]

$\int P\left(A,\tilde{A}\right)dAd\tilde{A} = N\int exp\left(L\left(A,\tilde{A}\right)\right)dAd\tilde{A},$
$L = \int dt\sum_{i,j}\left\{ \tilde{A}_{i}\lambda_{i,j}\tilde{A}_{j}-\widetilde{A}_{i}\left[\delta_{i,j}\frac{dA_{j}}{dt}-k_{B}T\left[A_{i},A_{j}\right]\frac{d\mathcal{H}}{dA_{j}}+\lambda_{i,j}\frac{d\mathcal{H}}{dA_{j}}-\frac{d\lambda_{i,j}}{dA_{j}}\right]\right\},$

where N is a normalization factor. 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).

References

Notes

1. ^ Langevin, P. (1908). "Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]". C. R. Acad. Sci. (Paris) 146: 530–533. ; reviewed by D. S. Lemons & A. Gythiel: Paul Langevin’s 1908 paper "On the Theory of Brownian Motion" [...], Am. J. Phys. 65, 1079 (1997), DOI:10.1119/1.18725
2. ^ Grabert, H. (1982). Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Springer Tracts in Modern Physics 95. Berlin: Springer-Verlag. ISBN 3-540-11635-4.
3. ^ Hohenberg, P. C.; Halperin, B. I. (1977). "Theory of dynamic critical phenomena". Reviews of Modern Physics 49 (3): 435–479. Bibcode:1977RvMP...49..435H. doi:10.1103/RevModPhys.49.435.
4. ^ Zwanzig, R. (1961). "Memory effects in irreversible thermodynamics". Phys. Rev. 124 (4): 983–992. Bibcode:1961PhRv..124..983Z. doi:10.1103/PhysRev.124.983.
5. ^ Ichimaru, S. (1973), Basic Principles of Plasma Physics (1st. ed.), USA: Benjamin, p. 231, ISBN 0805387536
6. ^ Janssen, H. K. (1976). "Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties". Z. Phys. B 23: 377. Bibcode:1976ZPhyB..23..377J. doi:10.1007/BF01316547.