Langevin's equation

A Langevin equation (after Paul Langevin ) is a stochastic differential equation which describes the dynamics of a subset of the degrees of freedom of a physical system. These are typically “slow” (macroscopic) degrees of freedom, the “fast” (microscopic) degrees of freedom are responsible for the stochastic nature of the differential equation.

Brownian movement as a prototype

The original Langevin equation describes the movement of a particle suspended in a liquid as a result of collisions with liquid molecules,

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

The equation was established by Paul Langevin on the basis of heuristic considerations . The slow variable here is the speed of the particle, is the particle mass, the constant is the coefficient of friction . The size is the so-called fluctuating force , in this case a Gaussian white noise with a correlation function ${\displaystyle \mathbf {v} (t)}$ ${\displaystyle m}$ ${\displaystyle \lambda }$ ${\displaystyle {\boldsymbol {\eta }}(t)}$

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

Here is the Boltzmann constant , the temperature and the i-th component of the vector . The function for the correlation in time means that the force at the time is completely uncorrelated with the force at another time. This is of course an approximation. The actual fluctuating force is correlated at least over a time interval that corresponds to the duration of the impact. However, the Langevin equation is used to describe the motion of a “macroscopic” particle over much larger time scales, and in this limiting case the correlation and the Langevin equation give the correct stochastic process. ${\displaystyle k_{B}}$ ${\displaystyle T}$ ${\displaystyle \eta _{i}\left(t\right)}$ ${\displaystyle {\boldsymbol {\eta }}\left(t\right)}$ ${\displaystyle \delta }$ ${\displaystyle t}$ ${\displaystyle \delta }$

Another prototypical property of the Langevin equation is the appearance of the damping coefficient in the correlation function of the fluctuating force. The technical term for this is the Einstein-Smoluchowski relationship . ${\displaystyle \lambda }$

Mathematical Aspects

A fluctuating force precisely correlated in time is not a mathematical function in the usual sense and the derivative is also not defined in this case. This problem disappears if the Langevingequation is written in integral form , and a Langevingequation should always be interpreted in the simplest case as an abbreviation for its time integral. The mathematical name for equations of this type is stochastic differential equation . ${\displaystyle \delta }$ ${\displaystyle {\boldsymbol {\eta }}\left(t\right)}$ ${\displaystyle d\mathbf {v} /dt}$ ${\displaystyle m\mathbf {v} =\int ^{t}\left(-\lambda \mathbf {v} +{\boldsymbol {\eta }}\left(t\right)\right)dt}$

Another mathematical problem arises for (rather special) Langevin equations with a multiplicative fluctuating force, i.e. H. Terms as on the right. Such equations can be interpreted according to Ito or Stratonovich scheme (according to Itō Kiyoshi, Ruslan Stratonovich), and if the derivation of the Langevin equation does not provide any information on this, the derivation is questionable anyway. ${\displaystyle |{\boldsymbol {v}}(t)|{\boldsymbol {\eta }}(t)}$

Generic length equation

A generic length equation can be derived from classical mechanics with projection operator methods. This generic Langevin equation plays a central role in critical dynamics theory and other areas of non-equilibrium thermodynamics. The equation for Brownian motion above is a special case.

An essential prerequisite for the derivation is a criterion for dividing the degrees of freedom into the categories slow and fast. For example, local thermal equilibrium is achieved in a liquid within a few peak times. In contrast, it takes much longer for densities of conserved quantities such as mass or energy to relax to the equilibrium value. Densities of conserved quantities and in particular their proportions with long wavelengths are therefore candidates for slow variables. Technically, the subdivision is realized with the twenty projection operator, the essential tool of derivation. The derivation uses few (plausible) assumptions which are required in a similar way elsewhere in statistical physics and is therefore not strictly mathematical.

It denotes the slow variables. The generic Langevinequation is then ${\displaystyle A=\{A_{i}\}}$

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

The fluctuating force obeys a normal distribution with a correlation function ${\displaystyle \eta _{i}\left(t\right)}$

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

This implies Onsager's reciprocity relations for the damping coefficients . The dependence of on is negligible in most cases. The symbol denotes the Hamiltonian of the system, is the equilibrium probability distribution of the variables . Finally, the bracket is the Poisson bracket projection of the slow variable and into the space of the slow variable. ${\displaystyle \lambda _{i,j}=\lambda _{j,i}}$ ${\displaystyle \lambda }$ ${\displaystyle d\lambda _{i,j}/dA_{j}}$ ${\displaystyle \lambda }$ ${\displaystyle A}$ ${\displaystyle {\mathcal {H}}=-\ln \left(p_{0}\right)}$ ${\displaystyle p_{0}\left(A\right)}$ ${\displaystyle A}$ ${\displaystyle [A_{i},A_{j}]}$ ${\displaystyle A_{i}}$ ${\displaystyle A_{j}}$

In the case of Brownian motion one would have , or and . The equation of motion for is exact, there is no fluctuating force and no damping coefficient . ${\displaystyle {\mathcal {H}}=\mathbf {p} ^{2}/\left(2mk_{B}T\right)}$ ${\displaystyle A=\{\mathbf {p} \}}$ ${\displaystyle A=\{\mathbf {x} ,\mathbf {p} \}}$ ${\displaystyle [x_{i},p_{j}]=\delta _{i,j}}$ ${\displaystyle d\mathbf {x} /dt=\mathbf {p} /m}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle \eta _{x}}$ ${\displaystyle \lambda _{x,p}}$

Examples

An electrical circuit with a resistor and a capacitor.

Resistance Noise

The figure on the right shows an electrical circuit consisting of a resistor with electrical resistance and a capacitor with capacitance . The slow variable is the voltage across the capacitor or resistor. The Hamiltonian is , and the Langevin equation becomes ${\displaystyle R}$ ${\displaystyle C}$ ${\displaystyle U}$ ${\displaystyle {\mathcal {H}}=E/k_{B}T=CU^{2}/(2k_{B}T)}$

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

The resulting correlation function

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

becomes white noise ( Nyquist noise, Johnson noise ) when the capacitance becomes small.

Overdamped Brownian movement

A borderline case of the classical Langevin equation is the case of strong friction (or small mass ). In this case the inertia term is negligible and one speaks of overdamping. The Langevinequation gets the form ${\displaystyle m}$ ${\displaystyle md\mathbf {v} /dt}$

${\displaystyle \lambda {\frac {d\mathbf {x} }{dt}}={\boldsymbol {\eta }}\left(t\right).}$

Integration and averaging of the square over the fluctuating force yields where the spatial dimension is. With the help of this equation, the Boltzmann constant can be determined from experimentally measured diffusion if the coefficient of friction is known . Incidentally, the not overdamped Langevinequation delivers the same law of diffusion. ${\displaystyle \left\langle \mathbf {x} (t)^{2}\right\rangle =2k_{B}TDt/\lambda ,}$ ${\displaystyle D}$ ${\displaystyle k_{B}}$

Micro rotary balance (harmonic oscillator)

This is a mirror hanging on a quartz thread. A light beam reflected on the mirror enables precise measurements. The slow variables are angular momentum and deflection angle with Hamiltonian ${\displaystyle j}$ ${\displaystyle \phi }$

${\displaystyle {\mathcal {H}}={\frac {1}{k_{B}T}}\left({\frac {j^{2}}{2N}}+{\frac {w\phi ^{2}}{2}}\right).}$

The damping constant can be adjusted via the air pressure. With the help of the corresponding Langevin equation and experiments, an exact value for the Boltzmann constant can be obtained. Conversely, the Langevin equation describes how thermal fluctuations limit the sensitivity of pointer instruments.

Critical Dynamics

The order parameter of a continuous phase transition changes only slowly in the vicinity of the critical point and its dynamics are with an i. A. nonlinear length equation describable. The simplest case is the universality class “Model A” with a scalar order parameter that has not been preserved . B. in an axial ferromagnet. The Langevinequation is ${\displaystyle \varphi }$

${\displaystyle {\begin{aligned}{\frac {\partial \varphi \left(\mathbf {x} ,t\right)}{\partial t}}&=-\lambda {\frac {\delta {\mathcal {H}}}{\delta \varphi }}+\eta \left(\mathbf {x} ,t\right),\\{\mathcal {H}}&=\int d^{d}x\left\{{\frac {1}{2}}\varphi \left[r_{0}-\nabla ^{2}\right]\varphi +u\varphi ^{4}\right\},\\\left\langle \eta \left(\mathbf {x} ,t\right)\eta \left(\mathbf {x} ',t'\right)\right\rangle &=2\lambda \delta \left(\mathbf {x} -\mathbf {x} '\right)\delta \left(t-t'\right).\end{aligned}}}$

Other universality classes (the nomenclature is "Model A", ..., "Model J") have a diffusing order parameter, order parameters with several components, additional other slow variables and mostly also contributions from Poisson brackets.

Equivalent techniques

As a rule, a solution to a length equation for a specific realization of the fluctuating force is of no interest. Instead, one is interested in the correlation functions of the slow variables after averaging over the fluctuating force. Such mean values ​​can also be obtained in other ways.

Fokker-Planck equation

A Fokker-Planck equation is a deterministic equation for the time-dependent probability distribution of the slow variables . The Fokker-Planck equation corresponding to the generic Langevin equation above can be obtained using standard techniques (e.g. Ref.), ${\displaystyle P\left(A,t\right)}$ ${\displaystyle A}$

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

The equilibrium distribution is a stationary solution. ${\displaystyle P(A,t)=p_{0}(A)={\text{const}}\times \exp(-{\mathcal {H}})}$

Path integral

A path integral equivalent to a Langevin equation can be obtained from the corresponding Fokker-Planck equation or directly from the Langevin equation. The path integral corresponding to the generic Langevin equation is

${\displaystyle \int P(A,{\tilde {A}})\,dA\,d{\tilde {A}}=N\int \exp \left(L(A,{\tilde {A}})\right)dA\,d{\tilde {A}},}$

where is a normalization factor and ${\displaystyle N}$

${\displaystyle L(A,{\tilde {A}})=\int \sum _{i,j}\left\{{\tilde {A}}_{i}\lambda _{i,j}{\tilde {A}}_{j}-{\widetilde {A}}_{i}\left\{\delta _{i,j}{\frac {dA_{j}}{dt}}-k_{B}T\left[A_{i},A_{j}\right]{\frac {d{\mathcal {H}}}{dA_{j}}}+\lambda _{i,j}{\frac {d{\mathcal {H}}}{dA_{j}}}-{\frac {d\lambda _{i,j}}{dA_{j}}}\right\}\right\}dt.}$

The auxiliary variables are called response variables . The path integral representation simplifies the application of quantum field theory techniques , such as perturbation calculus and the renormalization group. ${\displaystyle {\tilde {A}}}$

literature

• Don S. Lemons, Anthony Gythiel: Paul Langevin's 1908 paper "On the Theory of Brownian Motion" ["Sur la théorie du mouvement brownien," CR Acad. Sci. (Paris) 146, 530-533 (1908)], Am. J. Phys. 65, 1079 (1997), DOI: 10.1119 / 1.18725
• NG Van Kampen: "Stochastic Processes in Physics and Chemistry.". 3. Edition. North Holland, 2007.
• Schwabl, Franz: Statistical Mechanics . Springer, ISBN 3-540-31095-9
• Huang, Kerson: Statistical Mechanics . Wiley, ISBN 978-81-265-1849-4
• Huang, Kerson: Introduction to Statistical Physics . CRC Press, ISBN 0-7484-0942-4

Individual evidence

1. ↑ Stochastic Processes in Physics and Chemistry . Elsevier, 2007, ISBN 978-0-444-52965-7 , doi : 10.1016 / b978-0-444-52965-7.x5000-4 .
2. ↑ K. Kawasaki: Simple derivations of generalized linear and nonlinear Langevin equations . In: J. Phys. A: Math. Nucl. Gene. . 6, 1973, p. 1289. bibcode : 1973JPhA .... 6.1289K . doi : 10.1088 / 0305-4470 / 6/9/004 .
3. ↑ arxiv : 1506.02650v2
4. ^ ^{A b c} P. C. Hohenberg, BI Halperin: Theory of dynamic critical phenomena . In: Reviews of Modern Physics . 49, No. 3, 1977, pp. 435-479. bibcode : 1977RvMP ... 49..435H . doi : 10.1103 / RevModPhys.49.435 .
5. ↑ R. Zwanzig: Memory effects in irreversible thermodynamics . In: Phys. Rev. . 124, No. 4, 1961, pp. 983-992. bibcode : 1961PhRv..124..983Z . doi : 10.1103 / PhysRev.124.983 .
6. ↑ S. Ichimaru: Basic Principles of Plasma Physics. Benjamin, 1973, ISBN 3-540-61236-X .
7. ^ HK Janssen: Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties . In: Z. Phys. B . 23, 1976, p. 377. bibcode : 1976ZPhyB..23..377J . doi : 10.1007 / BF01316547 .