HOME

TheInfoList



OR:

In physics, a Langevin equation (named after
Paul Langevin Paul Langevin (; ; 23 January 1872 – 19 December 1946) was a French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the ''Comité de vigilance des intellectuels antifascistes'', an an ...
) is a
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
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 Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, which models the fluctuating motion of a small particle in a fluid.


Brownian motion as a prototype

The original Langevin equation describes
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, m\frac=-\lambda \mathbf+\boldsymbol\left( t\right). Here, \mathbf is the velocity of the particle, 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\left( t\right) representing the effect of the collisions with the molecules of the fluid. The force \boldsymbol\left( t\right) has a Gaussian probability distribution with correlation function \left\langle \eta_\left( t\right)\eta_\left( t'\right) \right\rangle =2\lambda k_\textT\delta _ \delta \left(t-t'\right) , where k_\text is
Boltzmann's constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant ...
, T is the temperature and \eta_i\left( t\right) is the i-th component of the vector \boldsymbol\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 \lambda 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\left( t\right) is not a function in the usual mathematical sense and even the derivative d\mathbf/dt is not defined in this limit. This problem disappears when the Langevin equation is written in integral form m\mathbf = \int^t\left( -\lambda \mathbf + \boldsymbol\left( t\right)\right) dt. 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 A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
". 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(t)\ \boldsymbol(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
Itō calculus Itō may refer to: *Itō (surname), a Japanese surname *Itō, Shizuoka, Shizuoka Prefecture, Japan *Ito District, Wakayama Prefecture, Japan See also * Itô's lemma, used in stochastic calculus *Itoh–Tsujii inversion algorithm, in field theory ...
). 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.


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 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. 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=\ denote the slow variables. The generic Langevin equation then reads \frac=k_\textT\sum\limits_-\sum\limits_\sum\limits_+\eta _\left( t\right). The fluctuating force \eta_i\left( t\right) obeys a Gaussian probability distribution with correlation function \left\langle \right\rangle =2\lambda _\left( A\right) \delta \left( t-t'\right). This implies the Onsager reciprocity relation \lambda_=\lambda_ for the damping coefficients \lambda. The dependence d\lambda_/dA_j of \lambda on A is negligible in most cases. The symbol \mathcal=-\ln\left(p_0\right) denotes the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
of the system, where p_0\left(A\right) is the equilibrium probability distribution of the variables A. Finally, _i,A_j/math> 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 \mathcal=\mathbf^2/\left(2mk_\textT\right), A=\ or A=\ and _i, p_j\delta_. The equation of motion d\mathbf/dt=\mathbf/m for \mathbf is exact: there is no fluctuating force \eta_x and no damping coefficient \lambda_.


Examples


Thermal noise in an electrical resistor

There is a close analogy between the paradigmatic Brownian particle discussed above and
Johnson noise Johnson is a surname of Anglo-Norman origin meaning "Son of John". It is the second most common in the United States and 154th most common in the world. As a common family name in Scotland, Johnson is occasionally a variation of ''Johnston'', a ...
, the electric voltage generated by thermal fluctuations in a resistor. The diagram at the right shows an electric circuit consisting of a resistance ''R'' and a
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized ar ...
''C''. The slow variable is the voltage ''U'' between the ends of the resistor. The Hamiltonian reads \mathcal = E / k_\textT = CU^2 / (2k_\textT), and the Langevin equation becomes \frac =-\frac + \eta \left( t\right),\;\;\left\langle \eta \left( t\right) \eta \left( t'\right)\right\rangle = \frac\delta \left(t-t'\right). This equation may be used to determine the correlation function \left\langle U\left(t\right) U\left(t'\right) \right\rangle = \frac \exp \left(-\frac \right) \approx 2Rk_\textT \delta \left( t - t'\right), which becomes white noise (Johnson noise) when the capacitance becomes negligibly small.


Critical dynamics

The dynamics of the
order parameter In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
\varphi of a second order phase transition slows down near the critical point and can be described with a Langevin equation. The simplest case is the
universality class In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...
"model A" with a non-conserved scalar order parameter, realized for instance in axial ferromagnets, \begin \frac&=-\lambda\frac+\eta\left(\mathbf,t\right),\\ \mathcal&=\int d^x \left \frac r_0 \varphi^2 + u \varphi^4 + \frac (\nabla\varphi)^2 \right\\ pt\left\langle \eta\left(\mathbf,t\right)\eta\left(\mathbf',t'\right)\right\rangle &= 2\lambda\delta\left(\mathbf-\mathbf'\right)\delta\left(t-t'\right). \end 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.


Harmonic oscillator in a fluid

m\frac = -\lambda v + \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 The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the the ...
. 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 physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
. In the plot below (figure 2), the long time velocity distribution (orange) and position distributions (blue) in a harmonic potential ( U = \frac k x^2 ) is plotted with the Boltzmann probabilities for velocity (red) and position (green). 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 \frac = -\frac + \boldsymbol(t), where \mathbf = d\mathbf/dt is the particle velocity, \mu is the particle mobility, and \boldsymbol(t) = m \mathbf(t) is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale t_c of particle collisions, i.e. \overline = 0. The general solution to the equation of motion is \mathbf(t) = \mathbf(0) e^ + \int_0^t \mathbf(t') e^ dt', where \tau = m\mu is the correlation time of the noise term. It can also be shown that the
autocorrelation function Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variabl ...
of the particle velocity \mathbf is given by \begin R_(t_1,t_2) & \equiv \langle \mathbf(t_1) \cdot \mathbf(t_2) \rangle \\ & = v^2(0) e^ + \int_0^ \int_0^ R_(t_1',t_2') e^ dt_1' dt_2' \\ & \simeq v^2(0) e^ + \left frac - v^2(0)\right\Big ^ - e^\Big \end where we have used the property that the variables \mathbf(t_1') and \mathbf(t_2') become uncorrelated for time separations t_2'-t_1' \gg t_c. Besides, the value of \lim_ \langle v^2 (t) \rangle = \lim_ R_(t,t) is set to be equal to 3k_\textT/m such that it obeys the
equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
. If the system is initially at thermal equilibrium already with v^2(0) = 3 k_\text T/m, then \langle v^2(t) \rangle = 3 k_\text T/m for all t, meaning that the system remains at equilibrium at all times. The velocity \mathbf(t) of the Brownian particle can be integrated to yield its trajectory \mathbf(t). If it is initially located at the origin with probability 1, then the result is \mathbf(t) = \mathbf(0) \tau \left(1-e^\right) + \tau \int_0^t \mathbf(t') \left - e^\rightdt'. Hence, the average displacement \langle \mathbf(t) \rangle = \mathbf(0) \tau \left(1-e^\right) asymptotes to \mathbf(0) \tau as the system relaxes. The
mean squared displacement In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference positi ...
can be determined similarly: \langle r^2(t) \rangle = v^2(0) \tau^2 \left(1 - e^\right)^2 - \frac \tau^2 \left(1 - e^\right) \left(3 - e^\right) + \frac \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_\text T \tau t/m = 6 \mu k_\text T t = 6Dt, 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 In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability th ...
, which is the probability distribution function for particles in thermal equilibrium. In the special case of
overdamped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
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 \lambda \frac = - \frac + \eta(t)\equiv - \frac+\sqrt \frac, where \lambda is the damping constant. The term \eta(t) is white noise, characterized by \left\langle\eta(t) \eta(t')\right\rangle = 2 k_\text T \lambda \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 \frac \left\langle f(x(t))\right\rangle = 0, Itô's lemma for the Itô drift-diffusion process dX_t = \mu_t \, dt + \sigma_t \, dB_t says that the differential of a twice-differentiable function is given by df = \left(\frac + \mu_t\frac + \frac\frac\right)dt + \sigma_t\frac\,dB_t. Applying this to the calculation of \langle f(x(t)) \rangle gives \left\langle -f'(x)\frac + k_\text T f''(x) \right\rangle=0. This average can be written using the probability density function p(x); \int \left( -f'(x)\fracp(x) + f''(x)p(x) \right) dx = \int \left( -f'(x)\frac p(x) - f'(x)p'(x) \right) dx = 0, where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions f, it follows that \frac p(x) + p'(x) = 0, thus recovering the Boltzmann distribution p(x) \propto \exp \left( \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 In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag force ...
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: \frac = \sum_ \frac \left(-k_\textT\left _i,A_j\right\frac + \lambda_ \frac + \lambda_\frac\right) P\left(A,t\right). The equilibrium distribution P(A) = p_0(A) = \text\times\exp(-\mathcal) is a stationary solution.


Klein–Kramers equation

The Fokker–Planck equation for an underdamped Brownian particle is called the
Klein–Kramers equation In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation is a partial differential equation that describes the probability density function of a Brownian particle in phase space . In one ...
. If the Langevin equations are written as \begin \dot &= \frac \\ \dot &= -\xi \, \mathbf - \nabla V(\mathbf) + \sqrt \boldsymbol(t), \qquad \langle \boldsymbol^(t) \boldsymbol(t') \rangle = \mathbf \delta(t-t') \end where \mathbf is the momentum, then the corresponding Fokker–Planck equation is \frac + \frac \mathbf \cdot \nabla_ f = \xi \nabla_ \cdot \left( \mathbf \, f \right) + \nabla_ \cdot \left( \nabla V(\mathbf) \, f \right) + m \xi k_ T \, \nabla_^2 f Here \nabla_ and \nabla_ are the
gradient operator Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
with respect to and , and \nabla_^2 is the Laplacian with respect to . In d-dimensional free space, corresponding to V(\mathbf) = \text on \mathbb^, this equation can be solved using Fourier transforms. If the particle is initialized at t = 0 with position \mathbf' and momentum \mathbf', corresponding to initial condition f(\mathbf, \mathbf, 0) = \delta(\mathbf-\mathbf')\delta(\mathbf-\mathbf'), then the solution is \begin f(\mathbf, \mathbf, t) = \frac \exp\left \frac \left( \frac + \frac - \frac \right) \right\end where \begin &\sigma^2_X = \frac \left + 2 \xi t - \left(2 - e^\right)^2 \right \qquad \sigma^2_P = m k_ T \left(1 - e^ \right) \\ &\beta = \frac \left(1 - e^\right)^2 \\ &\boldsymbol_X = \mathbf' + (m \xi)^ \left(1 - e^ \right) \mathbf' ; \qquad \boldsymbol_P = \mathbf' e^. \end In three spatial dimensions, the mean squared displacement is \langle \mathbf(t)^2 \rangle = \int f(\mathbf, \mathbf, t) \mathbf^2 \, d\mathbfd\mathbf = \boldsymbol_X^2 + 3 \sigma_X^2


Path integral

A path integral equivalent to a Langevin equation can be obtained from the corresponding
Fokker–Planck equation In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag force ...
or by transforming the Gaussian probability distribution P^(\eta)d\eta of the fluctuating force \eta to a probability distribution of the slow variables, schematically P(A)dA = P^(\eta(A))\det(d\eta/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+\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 \int P(A,\tilde)\,dA\,d\tilde = N\int \exp \left( L(A,\tilde)\right) dA\,d\tilde, where N is a normalization factor and L(A,\tilde) = \int \sum_ \left\ dt. The path integral formulation allows for the use of tools from quantum field theory, such as perturbation and renormalization group methods.


See also

*
Langevin dynamics In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems. It was originally developed by French physicist Paul Langevin. The approach is characterized by the use of simplified models while acco ...
* Stochastic thermodynamics


References


Further reading

*W. T. Coffey ( Trinity College, Dublin, Ireland) and Yu P. Kalmykov (
Université de Perpignan The University of Perpignan (french: Université de Perpignan; ca, Universitat de Perpinyà Via Domitia) is a French university, located in Perpignan. History The first university of Perpignan was established in 1349 by King Peter IV of Ara ...
,
France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan area ...
, ''The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering'' (Third edition),
World Scientific Series in Contemporary Chemical Physics In its most general sense, the term "world" refers to the totality of entities, to the whole of reality or to everything that is. The nature of the world has been conceptualized differently in different fields. Some conceptions see the worl ...
- 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. {{DEFAULTSORT:Langevin Equation Statistical mechanics Stochastic differential equations