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 $\mathbf{x}$ 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 $\boldsymbol{\eta}(t)$ (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 $\boldsymbol{\eta}(t)$ 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 k_B is Boltzmann's constant and T is the temperature. The δ-function of the time difference is an approximation, the actual random force has a finite 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.

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={A_i} 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 η_i(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 λ_i,j = λ_j,i for the damping coefficients λ. The dependence dλ_i,j / dA_j of λ on A is negligible in most cases. The symbol $\mathcal{H}=-\ln p_{0}\left( A\right)$ denotes the Hamiltonian of the system, where p₀(A) is the equlibribium probability distribution of the variables A. Finally, [A_i, A_j] is the Poisson bracket of the slow variables A_i and A_j.

In the Brownian motion case one would have $\mathcal{H}=\mathbf{p}^{2}/(2mk_{B}T)$, A={p} or A={x, p} and [x_i, p_j]=δ_i,j. The equation of motion dx/dt=p/m for x is exact, there is no fluctuating force $\mathbf{\eta }_{x}$ and no damping coefficient λ_x,p.

Other examples and additional notes

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.

One method of solution makes use of the Fokker–Planck equation, which provides a deterministic equation satisfied by the time dependent probability density. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques, such as path integrals have also been used, drawing on the analogy between statistical physics and quantum mechanics (the Fokker-Planck equation is formally equivalent to the Schrödinger equation).

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 $\mathcal{H}=E/k_{B}T=CU^{2}/\left( 2k_{B}T\right)$, 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.

References

1. ^ Langevin, P. (1908). "On the Theory of Brownian Motion". C. R. Acad. Sci. (Paris) 146: 530–533. 
2. ^ Grabert, H. (1982). Projection Operator Techniques in Nonequilibrium Statistical Mechanics. Springer Tracts in Modern Physics. 95. Berlin: Springer-Verlag. ISBN 3540116354. 
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. 

See also

• Langevin dynamics

Further reading

• W. T. Coffey (Trinity College, Dublin, Ireland), Yu P. Kalmykov (Université de Perpignan, France) & J. T. Waldron (Trinity College, Dublin, Ireland), The Langevin Equation, With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering (Second Edition), World Scientific Series in Contemporary Chemical Physics - Vol 14. (The First Edition is Vol 10)
• Reif, F. Fundamentals of Statistical and Thermal Physics, McGraw Hill New York, 1965. See section 15.5 Langevin Equation
• R. Friedrich, J. Peinke and Ch. Renner. How to Quantify Deterministic and Random Influences on the Statistics of the Foreign Exchange Market, Phys. Rev. Lett. 84, 5224 - 5227 (2000)
• L.C.G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Cambridge Mathematical Library, Cambridge University Press, Cambridge, reprint of 2nd (1994) edition, 2000.