Langevin equation

Summary

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

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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,  

Here,   is the velocity of the particle,   is its damping coefficient, and   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   representing the effect of the collisions with the molecules of the fluid. The force   has a Gaussian probability distribution with correlation function   where   is the Boltzmann constant,   is the temperature and   is the i-th component of the vector  . The  -function form of the time correlation means that the force at a time   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  -correlation and the Langevin equation becomes virtually exact.

Another common feature of the Langevin equation is the occurrence of the damping coefficient   in the correlation function of the random force, which in an equilibrium system is an expression of the Einstein relation.

Mathematical aspects

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A strictly  -correlated fluctuating force   is not a function in the usual mathematical sense and even the derivative   is not defined in this limit. This problem disappears when the Langevin equation is written in integral form  

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.,  . 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

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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   denote the slow variables. The generic Langevin equation then reads  

The fluctuating force   obeys a Gaussian probability distribution with correlation function  

This implies the Onsager reciprocity relation   for the damping coefficients  . The dependence   of   on   is negligible in most cases. The symbol   denotes the Hamiltonian of the system, where   is the equilibrium probability distribution of the variables  . Finally,   is the projection of the Poisson bracket of the slow variables   and   onto the space of slow variables.

In the Brownian motion case one would have  ,   or   and  . The equation of motion   for   is exact: there is no fluctuating force   and no damping coefficient  .

Examples

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Thermal noise in an electrical resistor

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An electric circuit consisting of a resistor and a capacitor.

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  , and the Langevin equation becomes  

This equation may be used to determine the correlation function   which becomes white noise (Johnson noise) when the capacitance C becomes negligibly small.

Critical dynamics

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The dynamics of the order parameter   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]

 
Figure 1: Phase portrait of a harmonic oscillator showing spreading due to the Langevin Equation.
 
Figure 2: Equilibrium probability for Langevin dynamics in Harmonic Potential

Harmonic oscillator in a fluid

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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 ( ) is plotted with the Boltzmann probabilities for velocity (green) and position (red). In particular, the late time behavior depicts thermal equilibrium.

 
Simulated squared displacements of free Brownian particles (semi-transparent wiggly lines) as a function of time, for three selected choices of initial squared velocity which are 0, 3kBT/m, and 6kBT/m respectively, with 3kBT/m being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared displacements for the corresponding parameter choices.

Trajectories of free Brownian particles

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Consider a free particle of mass   with equation of motion described by   where   is the particle velocity,   is the particle mobility, and   is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale   of particle collisions, i.e.  . The general solution to the equation of motion is   where   is the correlation time of the noise term. It can also be shown that the autocorrelation function of the particle velocity   is given by[10]   where we have used the property that the variables   and   become uncorrelated for time separations  . Besides, the value of   is set to be equal to   such that it obeys the equipartition theorem. If the system is initially at thermal equilibrium already with  , then   for all  , meaning that the system remains at equilibrium at all times.

The velocity   of the Brownian particle can be integrated to yield its trajectory  . If it is initially located at the origin with probability 1, then the result is  

Hence, the average displacement   asymptotes to   as the system relaxes. The mean squared displacement can be determined similarly:  

This expression implies that  , indicating that the motion of Brownian particles at timescales much shorter than the relaxation time   of the system is (approximately) time-reversal invariant. On the other hand,  , which indicates an irreversible, dissipative process.

 
This plot corresponds to solutions of the complete Langevin equation for a lightly damped harmonic oscillator, obtained using the Euler–Maruyama method. The left panel shows the time evolution of the phase portrait at different temperatures. The right panel captures the corresponding equilibrium probability distributions. At zero temperature, the velocity slowly decays from its initial value (the red dot) to zero, over the course of a handful of oscillations, due to damping. For nonzero temperatures, the velocity can be kicked to values higher than the initial value due to thermal fluctuations. At long times, the velocity remains nonzero, and the position and velocity distributions correspond to that of thermal equilibrium.

Recovering Boltzmann statistics

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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   is described by the overdamped Langevin equation   where   is the damping constant. The term   is white noise, characterized by   (formally, the Wiener process). One way to solve this equation is to introduce a test function   and calculate its average. The average of   should be time-independent for finite  , leading to  

Itô's lemma for the Itô drift-diffusion process   says that the differential of a twice-differentiable function f(t, x) is given by  

Applying this to the calculation of   gives  

This average can be written using the probability density function  ;   where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions  , it follows that   thus recovering the Boltzmann distribution  

Equivalent techniques

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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

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A Fokker–Planck equation is a deterministic equation for the time dependent probability density   of stochastic variables  . The Fokker–Planck equation corresponding to the generic Langevin equation described in this article is the following:[11]   The equilibrium distribution   is a stationary solution.

Klein–Kramers equation

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The Fokker–Planck equation for an underdamped Brownian particle is called the Klein–Kramers equation.[12][13] If the Langevin equations are written as   where   is the momentum, then the corresponding Fokker–Planck equation is   Here   and   are the gradient operator with respect to r and p, and   is the Laplacian with respect to p.

In  -dimensional free space, corresponding to   on  , this equation can be solved using Fourier transforms. If the particle is initialized at   with position   and momentum  , corresponding to initial condition  , then the solution is[13][14]   where   In three spatial dimensions, the mean squared displacement is  

Path integral

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A path integral equivalent to a Langevin equation can be obtained from the corresponding Fokker–Planck equation or by transforming the Gaussian probability distribution   of the fluctuating force   to a probability distribution of the slow variables, schematically  . The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where   depends on   but not on  . It turns out to be convenient to introduce auxiliary response variables  . The path integral equivalent to the generic Langevin equation then reads[15]   where   is a normalization factor and   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 [16] or the Janssen-De Dominicis [15][17] formalism after its developers. The mathematical formalism for this representation can be developed on abstract Wiener space.

See also

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References

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  1. ^ Langevin, P. (1908). "Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]". C. R. Acad. Sci. Paris. 146: 530–533.
  2. ^ Lemons, Don S.; Gythiel, Anthony (1997). "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. 65 (11). American Association of Physics Teachers (AAPT): 1079–1081. Bibcode:1997AmJPh..65.1079L. doi:10.1119/1.18725. ISSN 0002-9505.
  3. ^ van Kampen, N. G. (1981). "Itô versus Stratonovich". Journal of Statistical Physics. 24 (1). Springer Science and Business Media LLC: 175–187. Bibcode:1981JSP....24..175V. doi:10.1007/bf01007642. ISSN 0022-4715. S2CID 122277474.
  4. ^ van Kampen, N. G. (2007). Stochastic Processes in Physics and Chemistry. Elsevier. doi:10.1016/b978-0-444-52965-7.x5000-4. ISBN 978-0-444-52965-7.
  5. ^ Kawasaki, K. (1973). "Simple derivations of generalized linear and nonlinear Langevin equations". J. Phys. A: Math. Nucl. Gen. 6 (9): 1289–1295. Bibcode:1973JPhA....6.1289K. doi:10.1088/0305-4470/6/9/004.
  6. ^ Dengler, R. (2015). "Another derivation of generalized Langevin equations". arXiv:1506.02650v2 [physics.class-ph].
  7. ^ a b c 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. S2CID 122636335.
  8. ^ Zwanzig, R. (1961). "Memory effects in irreversible thermodynamics". Phys. Rev. 124 (4): 983–992. Bibcode:1961PhRv..124..983Z. doi:10.1103/PhysRev.124.983.
  9. ^ Johnson, J. (1928). "Thermal Agitation of Electricity in Conductors". Phys. Rev. 32 (1): 97. Bibcode:1928PhRv...32...97J. doi:10.1103/PhysRev.32.97.
  10. ^ Pathria RK (1972). Statistical Mechanics. Oxford: Pergamon Press. pp. 443, 474–477. ISBN 0-08-018994-6.
  11. ^ Ichimaru, S. (1973), Basic Principles of Plasma Physics (1st. ed.), USA: Benjamin, p. 231, ISBN 0805387536
  12. ^ Kramers, H.A. (1940). "Brownian motion in a field of force and the diffusion model of chemical reactions". Physica. 7 (4). Elsevier BV: 284–304. Bibcode:1940Phy.....7..284K. doi:10.1016/s0031-8914(40)90098-2. ISSN 0031-8914. S2CID 33337019.
  13. ^ a b Risken, H. (1989). The Fokker–Planck Equation: Method of Solution and Applications. New York: Springer-Verlag. ISBN 978-0387504988.
  14. ^ Chandrasekhar, S. (1943). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics. 15 (1): 1–89. Bibcode:1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1. ISSN 0034-6861.
  15. ^ a b Janssen, H. K. (1976). "Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties". Z. Phys. B. 23 (4): 377–380. Bibcode:1976ZPhyB..23..377J. doi:10.1007/BF01316547. S2CID 121216943.
  16. ^ Martin, P. C. and Siggia, E. D. and Rose, H. A. (1973). "Statistical Dynamics of Classical Systems". Phys. Rev. A. 8 (1): 423–437. doi:10.1103/PhysRevA.8.423.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  17. ^ De Dominicis, C. (1976). "Technqiues de Renormalisation de la Théorie des Champs et Dynamique des Phénomènes Critiques". J. Phys. Colloques. 37 (C1): 247–253. doi:10.1051/jphyscol:1976138.

Further reading

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  • 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.