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| 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.
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| == Brownian motion as a prototype ==
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| The original Langevin equation<ref>{{cite journal | title = Sur la théorie du mouvement brownien [On the Theory of Brownian Motion] | journal = C. R. Acad. Sci. (Paris) | year = 1908 | first = P. | last = Langevin | volume = 146 | pages = 530–533| id = | accessdate = 2010-08-08}}; reviewed by D. S. Lemons & A. Gythiel: ''Paul Langevin’s 1908 paper "On the Theory of Brownian Motion" [...]'', Am. J. Phys. 65, 1079 (1997), [[DOI:10.1119/1.18725]]</ref> describes [[Brownian motion]], the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,
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| :<math>m\frac{d^{2}\mathbf{x}}{dt^{2}}=-\lambda \frac{d\mathbf{x}}{dt}+\boldsymbol{\eta}\left( t\right).</math>
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| The degree of freedom of interest here is the position '''''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 [[Wiener process|''noise term'']] '''''η'''(t)'' (the name given in physical contexts to terms in stochastic differential equations which are [[stochastic process]]es) representing the effect of the collisions with the molecules of the fluid. The force '''''η'''(t)'' has a [[Gaussian distribution|Gaussian probability distribution]] with correlation function
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| :<math>\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) ,</math>
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| where ''k<sub>B</sub>'' is [[Boltzmann constant|Boltzmann's constant]] and ''T'' is the temperature. The [[Dirac delta|δ-function]] form of the correlations in time means that the force at a time ''t'' is assumed to be completely uncorrelated with it 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 δ-correlation and the Langevin equation become exact.
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| Another prototypical feature of the Langevin equation is the occurrence of the damping coefficient λ in the correlation function of the random force, a fact also known as [[Einstein_relation_(kinetic_theory)|Einstein relation]].
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| == Generic Langevin equation ==
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| There is a formal derivation of a generic Langevin equation from classical mechanics.<ref>{{cite book |title=Projection Operator Techniques in Nonequilibrium Statistical Mechanics |series=Springer Tracts in Modern Physics |year=1982 |first=H. |last=Grabert |volume=95 |location=Berlin |publisher=Springer-Verlag |isbn=3-540-11635-4 }}</ref> This generic equation plays a central role in the theory of critical dynamics,<ref>{{cite journal |title=Theory of dynamic critical phenomena |journal=[[Reviews of Modern Physics]] |year=1977 |first=P. C. |last=Hohenberg |first2=B. I. |last2=Halperin |volume=49 |issue=3 |pages=435–479 |doi=10.1103/RevModPhys.49.435 |bibcode = 1977RvMP...49..435H }}</ref> and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.
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| 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]],<ref>{{cite journal |title=Memory effects in irreversible thermodynamics |journal=[[Physical Review|Phys. Rev.]] |year=1961 |first=R. |last=Zwanzig |volume=124 |issue=4 |pages=983–992 |doi=10.1103/PhysRev.124.983 |bibcode = 1961PhRv..124..983Z }}</ref> 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.
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| Let A={A<sub>i</sub>} denote the slow variables. The generic Langevin equation then reads
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| :<math>\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).</math>
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| The fluctuating force '''''η'''<sub>i</sub>(t)'' obeys a [[Gaussian distribution|Gaussian probability distribution]] with correlation function
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| :<math>\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).</math>
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| This implies the [[Onsager reciprocal relations|Onsager reciprocity relation]] ''λ<sub>i,j</sub>=λ<sub>j,i</sub>'' for the damping coefficients ''λ''. The dependence ''dλ<sub>i,j</sub>/dA<sub>j</sub>'' of ''λ'' on ''A'' is negligible in most cases.
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| The symbol <math>\mathcal{H}</math>''=-ln(p<sub>0</sub>)'' denotes the Hamiltonian of the system, where ''p<sub>0</sub>(A)'' is the equilibrium probability distribution of the variables ''A''. Finally, ''[A<sub>i</sub>, A<sub>j</sub>]'' is the projection of the [[Poisson bracket]] of the slow variables ''A<sub>i</sub>'' and ''A<sub>j</sub>'' onto the space of slow variables.
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| In the Brownian motion case one would have <math>\mathcal{H}</math>''= '''p'''<sup>2</sup>/(2mk<sub>B</sub>T)'',
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| ''A={'''p'''}'' or ''A={'''x''', '''p'''}'' and ''[x<sub>i</sub>, p<sub>j</sub>]=δ<sub>i,j</sub>''. The equation of motion d'''x'''/dt='''p'''/m for '''x''' is exact, there is no fluctuating force ''η<sub>x</sub>'' and no damping coefficient ''λ<sub>x,p</sub>''.
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| == Examples ==
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| [[File:Oscillator phase portrait.svg|right|250px|thumb|Phase portrait of a [[harmonic oscillator]] showing spreading due to the Langevin Equation.]]
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| === Harmonic oscillator in a fluid ===
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| 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 [[Canonical ensemble|thermal equilibrium]].
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| [[File:ResistorCapacitance.png|right|250px|thumb|An electric circuit consisting of a resistor and a capacitor.]]
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| === Thermal noise in an electrical resistor ===
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| 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 [[Electrical resistance|resistance]] ''R'' and a [[capacitance]] ''C''. The slow variable is the voltage ''U'' between the ends of the resistor. The Hamiltonian reads <math>\mathcal{H}</math>''= E/k<sub>B</sub>T=CU<sup>2</sup>/(2k<sub>B</sub>T)'', and the Langevin equation becomes | |
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| :<math>\frac{dU}{dt} =-\frac{U}{RC}+\eta \left( t\right),\;\;
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| \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).</math>
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| This equation may be used to determine the correlation function
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| :<math>\left\langle U\left(t\right) U\left(t^{\prime }\right) \right\rangle
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| =\left( k_{B}T/C\right) \exp \left( -\left\vert t-t^{\prime }\right\vert
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| /RC\right) \approx 2Rk_{B}T\delta \left( t-t^{\prime }\right),</math>
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| which becomes a white noise (Johnson noise) when the capacitance C becomes negligibly small.
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| == Equivalent techniques ==
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| 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. Such correlation functions also may be determined with other (equivalent) techniques.
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| === Fokker Planck equation ===
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| A [[Fokker–Planck equation]] is a deterministic equation for the time dependent probability density ''P(A,t)'' of stochastic variables ''A''. The Fokker–Planck equation corresponding to the generic Langevin equation above may be derived with standard techniques (see for instance ref.<ref name=Ichimaru1973>{{citation|last=Ichimaru|first=S.|title=Basic Principles of Plasma Physics|publisher = Benjamin|year=1973|location=USA|edition=1st.|pages=231|isbn=0-805-38753-0}}</ref>),
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| :<math>
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| \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).</math>
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| The equilibrium distribution ''P(A,t)'' = ''p<sub>0</sub>(A)'' = ''const×exp(-<math>\mathcal{H}</math>)'' is a stationary solution.
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| === Path integral ===
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| A [[Path integral formulation|path integral]] equivalent to a Langevin equation may be obtained from the corresponding [[Fokker–Planck equation]] or by transforming the Gaussian probability distribution ''P<sup>(η)</sup>(η)dη'' of the fluctuating force ''η'' to a probability distribution of the slow variables, schematically ''P(A)dA'' = ''P<sup>(η)</sup>(η(A))det(dη/dA)dA''.
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| The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where ''A(t+Δt)-A(t)'' depends on ''A(t)'' but not on ''A(t+Δt)''. It turns out to be convenient to introduce auxiliary ''response variables'' <math>\tilde A</math>. The path integral equivalent to the generic Langevin equation then reads
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| <ref name="Janssen1976">{{cite journal | title = Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties | journal = Z. Phys. B | year = 1976 | first = H. K. | last = Janssen | volume = 23 | pages = 377|bibcode = 1976ZPhyB..23..377J |doi = 10.1007/BF01316547 }}</ref>
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| :<math>\int P\left(A,\tilde{A}\right)dAd\tilde{A} = N\int exp\left(L\left(A,\tilde{A}\right)\right)dAd\tilde{A},</math>
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| :<math>
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| L = \int dt\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\},</math>
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| where ''N'' is a normalization factor. The path integral formulation doesn't add anything new, but it does allow for the use of tools from [[quantum field theory]]; for example perturbation and renormalization group methods (if these make sense).
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| ==See also==
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| *[[Langevin dynamics]]
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| ==References==
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| '''Notes'''
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| {{Reflist}}
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| '''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.
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| *Reif, F. ''Fundamentals of Statistical and Thermal Physics'', McGraw Hill New York, 1965. See section 15.5 Langevin Equation
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| *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)
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| *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.
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| {{DEFAULTSORT:Langevin Equation}}
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| [[Category:Statistical mechanics]]
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| [[Category:Equations]]
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Perfect life is 1 that is clear of any bodily anguish. A patient enduring pain caused by hemorrhoids may yearn for several treatment for hemorrhoids to ease his pain. However, nothing is more uplifting than to realize that we needn't undergo any type of medicine or surgery. In reality, there are a number of signifies to avoid the appearance of hemorrhoids and these preventive procedures are awfully easy.
Before you finally find out the number one hemorrhoid surgery, it happens to be important to recognize initially the cause of the hemorrhoid. Actually there are lots of causes of hemorrhoid including irregularity, prolong sitting or standing, pregnancy and a lot more. The initially thing that we should do is to cure the cause and for sure getting rid of the hemorrhoid usually follow.
For example, inside the case of constipation we would ensure a diet involved more fiber and water plus less processed food. This is not the entire picture and is a bit too simplistic, nevertheless we receive the general idea. Another benefit to using natural techniques whenever dealing with the irregularity problem is the fact that not just would this cure the constipation, nevertheless there are definite total health advantages to eating properly.
Constipation is the well-known cause of hemorrhoid. In purchase to get over with this irregularity, you must change the diet. Before should you employ to eat those instant foods or processed food, then it is very the most perfect time for you to stop eating these foods. You have to change your die with those fibrous foods inside order to cure your irregularity. Another superior thing to do is to strengthen your fluid consumption. This will moreover enable you to soften a stool plus avoid from straining during bowel movement.
A sitz shower, taken 2-3 occasions for about 15 minutes a day will sooth painful piles plus usually keep this delicate region well cleansed. Add several Epson salt to the water.
Or, try to apply phenylephrine or Preparation H to the region where you have hemorrhoid. According to some experts, employ of the ointment will be also very powerful. It can actually constrict the blood vessels and lessen the redness and all.
It is extremely challenging now to suffer from hemorrhoid. We deal with severe redness, irritation, swelling, and pain. We are usually uncomfortable. However have faith that we will feel better with a hemorrhoid treatment. You might do away with the hemorrhoid in no time maybe by utilizing petroleum jelly, utilizing ointment phenylephrine or Preparation H, or using soft cotton underwear. We only have to provide a try any of them.