Langevin equation explained

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 dependent variables in a Langevin equation 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, which models the fluctuating motion of a small particle in a fluid.

Brownian motion as a prototype

The original Langevin equation^{[1] [2]} describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,

$$m\backslash frac=-\backslash lambda\; \backslash mathbf+\backslash boldsymbol\backslash left(t\backslash right).$$

Here,

v is the velocity of the particle, λ is its damping coefficient, and m is its 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{ η }\left( t\right) representing the effect of the collisions with the molecules of the fluid. The force \boldsymbol{ η }\left( t\right) has a Gaussian probability distribution with correlation function

$$\backslash left\backslash langle\; \backslash eta\_\backslash left(t\backslash right)\backslash eta\_\backslash left(t\text{'}\backslash right)\; \backslash right\backslash rangle\; =2\backslash lambda\; k\_\backslash textT\backslash delta\; \_\; \backslash delta\; \backslash left(t-t\text{'}\backslash right),$$where

k _{B} is Boltzmann's constant, T is the temperature and η _{i\left(} t\right) is the i-th component of the vector \boldsymbol{ η }\left( t\right) . The \delta -function form of the time correlation means that the force at a time t is 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 \delta -correlation and the Langevin equation becomes virtually exact.

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

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

Mathematical aspects

A strictly

\delta -correlated fluctuating force \boldsymbol{ η }\left( t\right) is not a function in the usual mathematical sense and even the derivative d v / d t is not defined in this limit. This problem disappears when the Langevin equation is written in integral form $$m\backslash mathbf\; =\; \backslash int^t\; \backslash left(-\backslash lambda\; \backslash mathbf\; +\; \backslash boldsymbol\backslash left(t\backslash right)\backslash right)\backslash mathrmt.$$

Therefore, the differential form is only an abbreviation for its time integral. The general mathematical term for equations of this type is "stochastic differential equation".

Another mathematical ambiguity occurs for Langevin equations with multiplicative noise, which refers to noise terms that are multiplied by a non-constant function of the dependent variables, e.g.,

\left|\boldsymbol{v}(t)\right| \boldsymbol{ η }(t) . 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. If the noise is external to the system, the appropriate interpretation is the Stratonovich one.^{[3] [4]}

Generic Langevin equation

There is a formal derivation of a generic Langevin equation from classical mechanics.^{[5] [6]} This generic equation plays a central role in the theory of critical dynamics,^[7] and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.

An essential step in the derivation is the division of 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. Thus, densities of conserved quantities, and in particular their long wavelength components, are slow variable candidates. This division can be expressed formally with the Zwanzig projection operator.^[8] Nevertheless, the derivation is not completely rigorous from a mathematical physics perspective because it relies on assumptions that lack rigorous proof, and instead are justified only as plausible approximations of physical systems.

Let

A = \{A _{i\}} denote the slow variables. The generic Langevin equation then reads$$\backslash frac=k\_\backslash textT\backslash sum\backslash limits\_-\backslash sum\backslash limits\_\backslash sum\backslash limits\_+\backslash eta\; \_\backslash left(t\backslash right).$$

The fluctuating force

η _{i\left(} t\right) obeys a Gaussian probability distribution with correlation function$$\backslash left\backslash langle\; \backslash right\backslash rangle\; =2\backslash lambda\; \_\backslash left(A\backslash right)\; \backslash delta\; \backslash left(t-t\text{'}\backslash right).$$λ _{i,j} =λ _{j,i} for the damping coefficients λ . The dependence d λ _{i,j} / d A _{j} of λ on A is negligible in most cases. The symbol l{H}=-ln\left(p _{0\right)} denotes the Hamiltonian of the system, where p _{0\left(A\right)} is the equilibrium probability distribution of the variables A . Finally, [A _{i,A} _{j]} is the projection of the Poisson bracket of the slow variables A _{i} and A _{j} onto the space of slow variables.

In the Brownian motion case one would have

l{H}= p ^{2 /\left( 2 mk} _{B T\right)} , A = \{ p \} or A = \{ x , p \} and [x _{i,} p _{j]=\delta} _{i,j} . The equation of motion d x / d t = p /m for x is exact: there is no fluctuating force η _{x} and no damping coefficient λ _{x,p} .

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 a resistor.^[9] 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

l{H} = E / k _{BT} = CU ^{2} / ( 2 k _{BT)} , and the Langevin equation becomes$$\backslash frac\; =-\backslash frac\; +\; \backslash eta\; \backslash left(t\backslash right),\backslash ;\backslash ;\backslash left\backslash langle\; \backslash eta\; \backslash left(t\backslash right)\; \backslash eta\; \backslash left(t\text{'}\backslash right)\backslash right\backslash rangle\; =\; \backslash frac\backslash delta\; \backslash left(t-t\text{'}\backslash right).$$

This equation may be used to determine the correlation function$$\backslash left\backslash langle\; U\backslash left(t\backslash right)\; U\backslash left(t\text{'}\backslash right)\; \backslash right\backslash rangle\; =\; \backslash frac\; \backslash exp\; \backslash left(-\backslash frac\; \backslash right)\; \backslash approx\; 2Rk\_\backslash textT\; \backslash delta\; \backslash left(t\; -\; t\text{'}\backslash right),$$which becomes white noise (Johnson noise) when the capacitance becomes negligibly small.

Critical dynamics

\varphi of a second order phase transition slows down near the critical point and can be described with a Langevin equation.^[7] 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.^[7]

Harmonic oscillator in a fluid

See also: Phase space. $$m\backslash frac\; =\; -\backslash lambda\; v\; +\; \backslash eta\; (t)-k\; x$$

A particle in a fluid is described by a Langevin equation with a potential energy function, a damping force, and thermal fluctuations given by the fluctuation dissipation theorem. If the potential is quadratic then the constant energy curves are ellipses, as shown in the figure. If there is dissipation but no thermal noise, a particle continually loses energy to the environment, and its time-dependent phase portrait (velocity vs position) corresponds to an inward spiral toward 0 velocity. By contrast, thermal fluctuations continually add energy to the particle and prevent it from reaching exactly 0 velocity. Rather, the initial ensemble of stochastic oscillators approaches a steady state in which the velocity and position are distributed according to the Maxwell–Boltzmann distribution. In the plot below (figure 2), the long time velocity distribution (blue) and position distributions (orange) in a harmonic potential ($U\; =\; \backslash frac\; k\; x^2$) is plotted with the Boltzmann probabilities for velocity (green) and position (red). In particular, the late time behavior depicts thermal equilibrium.

Trajectories of free Brownian particles

Consider a free particle of mass

m with equation of motion described by$$m\; \backslash frac\; =\; -\backslash frac\; +\; \backslash boldsymbol(t),$$where v = d r /dt is the particle velocity, \mu is the particle mobility, and \boldsymbol{ η }(t) = m a (t) is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale t _{c} of particle collisions, i.e. \overline{\boldsymbol{ η }(t)} = 0 . The general solution to the equation of motion is$$\backslash mathbf(t)\; =\; \backslash mathbf(0)\; e^\; +\; \backslash int\_0^t\; \backslash mathbf(t\text{'})\; e^\; dt\text{'},$$where \tau = m\mu is the correlation time of the noise term. It can also be shown that the autocorrelation function of the particle velocity v is given by^[10]

where we have used the property that the variables

a (t _{1')} and a (t _{2')} become uncorrelated for time separations t _{2'-t} _{1'} \gg t _{c} . Besides, the value of $\backslash lim\_\; \backslash langle\; v^2\; (t)\; \backslash rangle\; =\; \backslash lim\_\; R\_(t,t)$ is set to be equal to 3 k _{BT/m} such that it obeys the equipartition theorem. If the system is initially at thermal equilibrium already with v ^{2(0)} = 3 k _{B} T/m , then \langle v ^{2(t)} \rangle = 3 k _{B} T/m for all t , meaning that the system remains at equilibrium at all times.

The velocity

v (t) of the Brownian particle can be integrated to yield its trajectory r (t) . If it is initially located at the origin with probability 1, then the result is$$\backslash mathbf(t)\; =\; \backslash mathbf(0)\; \backslash tau\; \backslash left(1-e^\backslash right)\; +\; \backslash tau\; \backslash int\_0^t\; \backslash mathbf(t\text{'})\; \backslash left[1\; -\; e^\{-(t-t\text{'})\; /\; \backslash tau\}\backslash right]\; dt\text{'}.$$

Hence, the average displacement $\backslash langle\; \backslash mathbf(t)\; \backslash rangle\; =\; \backslash mathbf(0)\; \backslash tau\; \backslash left(1-e^\backslash right)$ asymptotes to

v ( 0 ) \tau as the system relaxes. The mean squared displacement can be determined similarly:

$$\backslash langle\; r^2(t)\; \backslash rangle\; =\; v^2(0)\; \backslash tau^2\; \backslash left(1\; -\; e^\backslash right)^2\; -\; \backslash frac\; \backslash tau^2\; \backslash left(1\; -\; e^\backslash right)\; \backslash left(3\; -\; e^\backslash right)\; +\; \backslash frac\; \backslash tau\; t.$$

This expression implies that

\langle r ^{2(t} \ll \tau) \rangle \simeq v ^{2(0)} t ^{2} , indicating that the motion of Brownian particles at timescales much shorter than the relaxation time \tau of the system is (approximately) time-reversal invariant. On the other hand, \langle r ^{2(t} \gg \tau) \rangle \simeq 6 k _{B} T \tau t/m = 6 \mu k _{B} T t = 6 Dt , which indicates an irreversible, dissipative process.

Recovering Boltzmann statistics

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

x(t) is described by the overdamped Langevin equation

$$\backslash lambda\; \backslash frac\; =\; -\; \backslash frac\; +\; \backslash eta(t)\backslash equiv\; -\; \backslash frac+\backslash sqrt\; \backslash frac,$$

where

λ is the damping constant. The term η (t) is white noise, characterized by \left\langle η (t) η (t')\right\rangle = 2 k _{B} T λ \delta(t - t') (formally, the Wiener process). One way to solve this equation is to introduce a test function f and calculate its average. The average of f(x(t)) should be time-independent for finite x(t) , leading to

$$\backslash frac\; \backslash left\backslash langle\; f(x(t))\backslash right\backslash rangle\; =\; 0,$$

dX _{t} = \mu _{t} dt + \sigma _{t} dB _{t} says that the differential of a twice-differentiable function is given by $$df\; =\; \backslash left(\backslash frac\; +\; \backslash mu\_t\backslash frac\; +\; \backslash frac\backslash frac\backslash right)dt\; +\; \backslash sigma\_t\backslash frac\backslash ,dB\_t.$$

Applying this to the calculation of

\langle f(x(t)) \rangle gives $$\backslash left\backslash langle\; -f\text{'}(x)\backslash frac\; +\; k\_\backslash text\; T\; f$$(x) \right\rangle=0.

This average can be written using the probability density function

p(x) ;(x)p(x) \right) dx \\= &\int \left(-f'(x)\frac p(x) - f'(x)p'(x) \right) dx \\ = & \; 0\endwhere the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions f , it follows that$$\backslash frac\; p(x)\; +\; p\text{'}(x)\; =\; 0,$$thus recovering the Boltzmann distribution$$p(x)\; \backslash propto\; \backslash exp\; \backslash left(\backslash right).$$

Equivalent techniques

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.

Fokker–Planck equation

A Fokker - Planck equation is a deterministic equation for the time dependent probability density

P\left(A,t\right) of stochastic variables A . The Fokker - Planck equation corresponding to the generic Langevin equation described in this article is the following:$$\backslash frac\; =\; \backslash sum\_\; \backslash frac\; \backslash left(-k\_\backslash textT\backslash left[A\_i,A\_j\backslash right]\; \backslash frac\; +\; \backslash lambda\_\; \backslash frac\; +\; \backslash lambda\_\backslash frac\backslash right)\; P\backslash left(A,t\backslash right).$$The equilibrium distribution P(A) = p _{0(A)} = const x \exp( - l{H}) is a stationary solution.

Klein–Kramers equation

See main article: Klein–Kramers equation. The Fokker–Planck equation for an underdamped Brownian particle is called the Klein–Kramers equation.^{[11] [12]} If the Langevin equations are written aswhere

p is the momentum, then the corresponding Fokker–Planck equation is$$\backslash frac\; +\; \backslash frac\; \backslash mathbf\; \backslash cdot\; \backslash nabla\_\; f\; =\; \backslash xi\; \backslash nabla\_\; \backslash cdot\; \backslash left(\backslash mathbf\; \backslash ,\; f\; \backslash right)\; +\; \backslash nabla\_\; \backslash cdot\; \backslash left(\backslash nabla\; V(\backslash mathbf)\; \backslash ,\; f\; \backslash right)\; +\; m\; \backslash xi\; k\_\; T\; \backslash ,\; \backslash nabla\_^2\; f$$Here \nabla _{r} and \nabla _{p} are the gradient operator with respect to and, and \nabla ^{2} _{p} is the Laplacian with respect to .

In

d -dimensional free space, corresponding to V( r ) = constant on R ^{d} , this equation can be solved using Fourier transforms. If the particle is initialized at t = 0 with position r ' and momentum p ' , corresponding to initial condition f( r , p , 0 ) = \delta( r - r ')\delta( p - p ') , then the solution is^{[12] [13]} whereIn three spatial dimensions, the mean squared displacement is$$\backslash langle\; \backslash mathbf(t)^2\; \backslash rangle\; =\; \backslash int\; f(\backslash mathbf,\; \backslash mathbf,\; t)\; \backslash mathbf^2\; \backslash ,\; d\backslash mathbfd\backslash mathbf\; =\; \backslash boldsymbol\_X^2\; +\; 3\; \backslash sigma\_X^2$$

Path integral

A path integral equivalent to a Langevin equation can be obtained from the corresponding Fokker - Planck equation or by transforming the Gaussian probability distribution

P ^{(η)} ( η ) d η of the fluctuating force η to a probability distribution of the slow variables, schematically P(A) d A = P ^{(η)} ( η (A))\det( d η / d A) d A .The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where A(t + \Delta t) - A(t) depends on A(t) but not on A(t + \Delta t) . It turns out to be convenient to introduce auxiliary response variables \tilde A . The path integral equivalent to the generic Langevin equation then reads^[14] $$\backslash int\; P(A,\backslash tilde)\backslash ,\backslash mathrmA\backslash ,\backslash mathrm\backslash tilde\; =\; N\backslash int\; \backslash exp\; \backslash left(L(A,\backslash tilde)\backslash right)\; \backslash mathrmA\backslash ,\backslash mathrm\backslash tilde,$$where N is a normalization factor and$$L(A,\backslash tilde)\; =\; \backslash int\; \backslash sum\_\; \backslash left\backslash \; \backslash mathrmt.$$The path integral formulation allows for the use of tools from quantum field theory, such as perturbation and renormalization group methods. This formulation is typically referred to as either the Martin-Siggia-Rose formalism ^[15] or the Janssen-De Dominicis ^[16] formalism after its developers. The mathematical formalism for this representation can be developed on abstract Wiener space.

See also

• Grote–Hynes theory
• Langevin dynamics
• Stochastic thermodynamics

Further reading

• W. T. Coffey (Trinity College, Dublin, Ireland) and Yu P. Kalmykov (Université de Perpignan, France, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering (Third edition), World Scientific Series in Contemporary Chemical Physics - Vol 27.
• 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.

Notes and References

1. Sur la théorie du mouvement brownien [On the Theory of Brownian Motion] . C. R. Acad. Sci. Paris . 1908 . P. . Langevin . 146 . 530–533.
2. Lemons . Don S. . Gythiel . Anthony . Paul Langevin's 1908 paper "On the Theory of Brownian Motion" ["Sur la théorie du mouvement brownien," C. R. Acad. Sci. (Paris) 146, 530–533 (1908)] . American Journal of Physics . American Association of Physics Teachers (AAPT) . 65 . 11 . 1997 . 0002-9505 . 10.1119/1.18725 . 1079–1081. 1997AmJPh..65.1079L .
3. van Kampen . N. G. . Itô versus Stratonovich . Journal of Statistical Physics . Springer Science and Business Media LLC . 24 . 1 . 1981 . 0022-4715 . 10.1007/bf01007642 . 175–187. 1981JSP....24..175V . 122277474 .
4. Book: van Kampen, N. G. . Stochastic Processes in Physics and Chemistry . Elsevier . 2007 . 978-0-444-52965-7 . 10.1016/b978-0-444-52965-7.x5000-4 .
5. Kawasaki. K.. 1973. Simple derivations of generalized linear and nonlinear Langevin equations . J. Phys. A: Math. Nucl. Gen.. 6. 9. 1289–1295. 1973JPhA....6.1289K. 10.1088/0305-4470/6/9/004.
6. Dengler . R. . 1506.02650v2 . Another derivation of generalized Langevin equations . 2015 . physics.class-ph .
7. Theory of dynamic critical phenomena . . 1977 . P. C. . Hohenberg . B. I. . Halperin . 49 . 3 . 435–479 . 10.1103/RevModPhys.49.435 . 1977RvMP...49..435H . 122636335 .
8. Memory effects in irreversible thermodynamics . . 1961 . R. . Zwanzig . 124 . 4 . 983–992 . 10.1103/PhysRev.124.983 . 1961PhRv..124..983Z .
9. Johnson . J. . 1928 . Thermal Agitation of Electricity in Conductors . Phys. Rev. . 32 . 1. 97 . 1928PhRv...32...97J . 10.1103/PhysRev.32.97 .
10. Book: Pathria RK . Raj Pathria. 1972 . Statistical Mechanics . Pergamon Press . Oxford . 0-08-018994-6 . 443, 474 - 477.
11. Kramers . H.A. . Brownian motion in a field of force and the diffusion model of chemical reactions . Physica . Elsevier BV . 7 . 4 . 1940 . 0031-8914 . 10.1016/s0031-8914(40)90098-2 . 284–304. 1940Phy.....7..284K . 33337019 .
12. Book: Risken, H. . The Fokker–Planck Equation: Method of Solution and Applications . Springer-Verlag . New York . 1989 . 978-0387504988 .
13. Chandrasekhar. S.. Stochastic Problems in Physics and Astronomy. Reviews of Modern Physics. 15. 1. 1943. 1–89. 0034-6861. 10.1103/RevModPhys.15.1. 1943RvMP...15....1C .
14. Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties . Z. Phys. B . 1976 . H. K. . Janssen . 23 . 4 . 377–380. 1976ZPhyB..23..377J . 10.1007/BF01316547 . 121216943 .
15. Statistical Dynamics of Classical Systems . Martin, P. C. and Siggia, E. D. and Rose, H. A. . Phys. Rev. A . 8 . 1 . 423-437 . 1973 . 10.1103/PhysRevA.8.423.
16. Technqiues de Renormalisation de la Théorie des Champs et Dynamique des Phénomènes Critiques . De Dominicis, C. . J. Phys. Colloques . 1976 . 37 . C1 . 247-253 . 10.1051/jphyscol:1976138.