1 Brownian motion and the Langevin equation

Transcription

1 Figure 1: The robust appearance of Robert Brown ( ) 1 Brownian otion and the Langevin equation In 1827, while exaining pollen grains and the spores of osses suspended in water under a icroscope, Brown observed inute particles within vacuoles in the pollen grains executing a jittery otion. He then observed the sae otion in particles of dust, enabling hi to rule out the hypothesis that the otion was due to pollen being alive. Although he did not hiself provide a theory to explain the otion, the phenoenon is now known as Brownian otion in his honour. 1.1 Derivation The Langevin equation is Newtons second law for a Brownian particle, where the forces include both the viscous drag due to the surrounding fluid and the fluctuations caused by the individual collisions with the fluid olecules. In order to describe the statistical nature of the force fluctuations we state the Central liit theore, which we have already proved in the special case of a rando walk: 1

2 δp i Figure 2: Fluid olecules hitting a suspended particle When a variable P is a su of independent increents,δp i, like a rando walk N P = δp i (1) where δp i = and δp 2 i = σ, then, when N is large, the rando variable P has a Gaussian distribution with P 2 = Nσ. Now, take δp i to be the x-coponent of the oentu transfer to the Brownian particle fro a olecular collision, and let N be the nuber of collisions during the tie t. Then P is the x-coponent of the oentu received by the Brownian particle during the tie t, and P 2 t. The corresponding force P / t will have a variance i=1 F 2 x = P 2 t 2 1 t. (2) In a coputer siulation of a Brownian particle one would have to integrate Newtons second law for the particle picking a rando force F (t) fro a Gaussian distribution at every tiestep t. Will this force be correlated 2

3 fro tie step to tie step? No, not as long as t is larger than the correlation tie of the forces fro the olecules which is of the order a few ean free ties of the olecules. When this requireent is fullfilled we ay hence write the correlation function as { F x (t) F a when t < t/2 x () = t (3) when t > t/2 where the constant a could in principle be deterined fro the variance of p i. In stead we will deterine it fro the equipartition principle. In the t liit the above equation becoes F x (t) F x () = aδ(t), (4) and likewise for the other spatial conponents of the force which will all be independent of each other. The regression hypothesis requires that on the average a particle velocity fluctuation will decay as a acroscopic velocity perturbation. Since F = the fluctuating force by itself will cause no average decay of the velocity. The acroscopic decay we need is the one caused by viscosity, i.e. v = αv where is the particle ass, v is the velocity relative to the surrounding fluid, and α is a constant coefficient, which for a sphere which is sall (but larger than a Brownian particle), has the exact for α = 6πηr, where η is the viscosity, r is the spheres radius. This friction law is known as Stokes law, and it is valid when the sphere oves at oderate speeds relative to a viscous fluid. A version of Newtons 2. law which is consistent with the regression hypothesis, and at the sae tie takes the fluctations into account is the Langevin equation dv dt = αv + F (5) where each coponent of F has a Gaussian distribution of agnitudes, and a correlation function given by Eq. (4). Both forces on the right hand side above coe fro the olecular fluid. The drag force αv represents the velocity dependence, and it is therefore reasonable to postulate that F is velocity-independent, and therfore where i and j are Cartesian indices. F i v j = (6) 3

4 1.2 Velocity auto-correlation function The first thing we copute fro Eq. (5) is the velocity autocorrelation function. For siplicity we will start out in one diension, since the generalization to higher diensions is straightforward. Multiplying Eq. (5) by e αt/ / we can then write it as d dt ( ) αt αt v(t)e = e which can be intergrated fro t = to give F (t) (7) v(t) = dt e α(t t ) F (t ) (8) where e αt plays the role of a response function. The velocity auto-correlation function now follows directly as v(t)v() = = = dt dt dt e α(t 2t ) dt e α(t t t ) dt e α(t t t ) a 2 = F (t ) F (t ) 2 (9) aδ(t t ) 2 (1) a 2α e αt (11) where we have used Eq. (4) and the fact that the average... coutes with the integration. Note that the first integration above is over t. Assuing t > this guarantees that the t = t at soe point during the integration and the a non-zero value of the δ-function is sapled. The above result is in agreeent with the regression hypothesis since a acroscopic perturbation v, which is governed by dv/dt = αv, decays as v(t) = e αt/ v(). We ay cobine Eq. (11) with the equipartition principle to fix a. Since we get 1 2 v2 = kt 2 (12) a = 2αkT. (13) In other words, the agnitude of the fluctuations increase both with teperature and friction. 4

5 1.3 The Langevin equation and diffusion We ay integrate the velocity over tie to get the displaceent x(t), and the diffusive behavior linked to the variance x 2 (t). Note that this is a refineent of the rando walker, since now the step length is governed by the integration of Newtons 2. law. Starting fro Eq. (8) and using Eq. (4) we get x 2 (t) = = 2αkT 2 dt dt t dt dt dt t dt e α(t t +t t ) 2αkT δ(t t ) (14) 2 dt Θ(t t )e α(t +t 2t ) (15) where the Heaviside function Θ(t), which is 1 for t > and zero otherwise, appeared because the integration over the δ-function is nonzero only if its arguent passes. The Θ-function will be, however, only if t < t, so we get x 2 (t) = 2αkT dt dt in(t,t ) dt e α(t +t 2t ) 2. (16) The three reaining integrals are straightforward and gives x 2 (t) = 2 kt α ( t α ( ) ) αt 1 e. (17) When t /α only the t-ter survives and x 2 (t) = 2Dt with D = kt α. (18) This relation is known as the Einstein relation. Like the fluctuation-disspation theore it relates quantities related to spontaneous fluctuations, D and kt, with a quantitt that describe acroscopic decay, α. We could also have derived the Einstein relation directly fro the Green-Kubo relation between D and the integral over the velocity autocorrelation function. When t /α we Taylor expand the exponential to second order to get x 2 (t) = kt t2, (19) that is x 2 (t) = v th t where vth 2 = kt / is exactly the theral velocity that follows fro the equipartition principle. Having a finite correlation tie /α 5

6 for the velocity the Langevin equation thus describes the crossover between the ballistic regie x 2 (t) t to the diffusive regie x 2 (t) t. The Langevin equation is based on the approxiation that the friction force has the instantaneous value αv and does not depend on the history of the otion. However, the aount of oentu given off by the Brownian particle does depend on the otion-history, and this oentu will change the velocity of the surrounding fluid, which in turn changes the force on the Brownian particle. A correct description then does take the history dependence of the force into account, and this changes the velocity correlation function: In the long tie liit the correlation function changes fro the e αt behavior to an 1/t d/2 behavior, where d is the diension. This was discovered in the late 196 s, first through coputer siulations, then analytically. Langevin equations show up also outside physics. Stock prises for instance are often described by a Langevin equation. 6