Lecture 21: Physical Brownian Motion II

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1 Lecture 21: Physical Brownian Motion II Scribe: Ken Kamrin Department of Mathematics, MIT May 3, 25 Resources An instructie applet illustrating physical Brownian motion can be found at: For more information on this topic, see: H. Risken, The Fokker-Planck Equation (Springer, 2nd ed., 1989). Einstein s Theory Einstein s theory of Brownian motion (i.e. Mathematical Brownian motion) treats the process as a random walk with iid steps. Specifically, the motion considers both diffusion and drift such that = drift elocity = µf where µ = D kt. This description of Brownian motion was considered to a greater extent in past lectures. 1

2 Langein s Theory After Einstein s theory was deeloped, Langein published another model of Brownian motion (i.e. Physical Brownian motion). Langein s model emphasizes that a particle moing due to random collisions with, say, gas molecules, does not actually experience independent steps since its inertia tends to keep it moing in roughly the same direction as its preious step. To account for inertia, the model is based on an expression of Newton s second law: m = F tot = F ext α + mγ(t). (1) The α term accounts for drag forces and the function Γ(t) is a stochastic noise term which accounts for random collisions with gas molecules. The function Γ(t) is truly a white noise in that Γ(t) =, Γ(t)Γ(t ) = q δ(t t ). The aboe is like a continuous ersion of iid steps and the ariable q is equialent to 2D. In equation 1, the noise term is not multiplied by, so we can reduce the equation to the Weiner-Îto SDE when F ext = : d = γ dt + D dz In the aboe, γ = α m τ 1 where τ can be thought of as the time needed for drag to kill acceleration. The Fokker-Planck equation for the resultant PDF is therefore ρ t γ (ρ) = D 2 ρ 2 2

3 At this point, it is worthwhile to discuss the behaior for a general stochastic process of the form ẋ = a(x, t) + b(x, t)γ(t) with Γ(t) a white noise. There are actually two different treatments of the problem each inoking different methods of numerical solution. In soling iteratiely for x to arious orders of accuracy, it becomes necessary to define how we ealuate integrals of the form τ Φ(w(t), t) d(w(t)) for a non-stochastic function Φ where w(τ) t+τ t Γ(t ) dt for some fixed t. 1. Îto : Under Îto s definition, the integral is soled using ti+1 t i Φ(w(t), t) d(w(t)) Φ(w(t i ), t i )[w(t i+1 ) w(t i )] where i is the step index. Numerical solution ia the Îto definition resembles the forward Euler method for soling PDE s in one time deriatie. The resulting Fokker- Planck equation has drift coefficient D 1 = a(x, t). 2. Stratonoich: Under this definition, we sole the SDE in a manner analagous to the Crank-Nicholson scheme, using ti+1 ( w(ti ) + w(t i+1 ) Φ(w(t), t) d(w(t)) Φ, t ) i + t i+1 [w(t i+1 ) w(t i )]. 2 2 t i The resulting Fokker-Planck equation has drift coefficient D 1 = a(x, t)+b x (x, t)b(x, t). The additional term is called the noise induced drift. The diffusion coefficient D 2 is, howeer, the same under both definitions. For clarity, if the Îto definition is intended, the SDE will be displayed in terms of increments (e.g. dx = a dt+b dz), and if the Stratonoich definition is intended, the SDE will be written in terms of stochastic deriaties (e.g. ẋ = a + b Γ(t)). 3

4 We now return to the physical Brownian motion SDE, d = γ dt + D dz. Either of the preious definitions suffices here since b(x, t) = constant. Our SDE is actually the Ornstein-Uhlenbeck process for elocity. Its solution is ρ(, t) = ( exp 2πσ (t) 2 ) ( (t))2 2σ (t) 2 where (t) = exp( γt) and σ (t) 2 = D (1 exp( 2γt))/γ. σ 2 D / γ t=τ time t=τ time The figures aboe illustrates the ballistic to diffusie transition which occurs near t = γ 1 = τ. In a steady state, we hae =, σ 2 = D /γ. This yields: ( ) exp γ2 2D ρ(, ) = 2πD /γ exp( E/kT) = exp( m2 /2kT) Maxwell s Distribution. Thus we may deduce: γ 2 2D = m2 2kT = D = γkt/m = γ 2 thermal At this point we may now draw an interesting connection to mathematical Brownian motion. In steady state, the force balance becomes = F tot = F ext α + mγ(t). 4

5 When F ext = the Green-Kubo expression for D x yields D x = ()(t) dt = γ 2 Γ()Γ(t) dt = 2D γ 2 Recall that in Einstein s theory, δ(t) dt = γ 2 D = kt γm. µ = drift F ext = D x kt. The force expression aboe indicates that in steady state, drift F ext hae deduced (γm) 1 = D x kt = D x = kt γm. This was our precise result from analyzing the Langein Equation. = α 1 = (γm) 1. Thus we Kramer s Theory Kramer s solution to physical Brownian motion deals with phase space coordinates (x, ). This turns the problem into soling two coupled SDE s: (Îto) dx = dt, d = ( γ + F ext m ) dt + D dz or (Stratonoich) ẋ =, = γ + F ext m + D Γ(t). We proceed by writing the multi-dimensional Fokker-Planck equation for ρ(x,, t): ρ t + ( D 1 ρ) = ( ) : (D 2 ρ) where D 1 is the drift ector (first moment ector) and D 2 is the diffusion matrix (second moment matrix). They are defined naturally by D 1 = γ x for γ = D 2 =. D 1 γ, and 5

6 Applying the Fokker-Planck equation in phase space yields Kramer s equation for the time eolution of the PDF: ρ t + x (ρ) γ (ρ) = 1 m (F(x)ρ) + D 2 ρ 2. This is typically soled using characteristics. When F ext = k s x, the Kramer s equation can be soled exactly using principles from ector Ornstein-Uhlenbeck processes. Letting k s yields ρ(x,, t x,, t = ) = exp ( 1 z A z) 2 2π A where A = ariance matrix, z = x x, x(t) = x + (1 exp( γt)), γ (t) The components of the ariance matrix are = exp( γt). A xx = kt mγ2(2γt exp( γt) exp( 2γt)) A = kt (1 exp( 2γt)) m A x = A x = kt m (1 exp( γt))2. We can construct a PDF for position, c(x, t x, t = ), based on this extended PDF ρ(x,, t x,, t = ). We remoe dependence on the initial elocity by aeraging it out oer a Maxwell distribution of possible starting elocities. We then remoe dependence on the final elocity by integrating this result oer all possible final elocities. Thus, c(x, t) = = = exp ( x2 2πσx (t) 2 = 2σ x(t) 2 ) ρ(x,, t x,, t = ) exp( m2 2kT ) 2πkT/m }{{} Maxwell Distribution d d 6

7 where σ x (t) 2 = 22 thermal(γt 1 + exp( γt)). Note that: γ 2 σ x (t) 2 thermal 2 t2 for γt 1 2D x t for γt 1. This indicates the ballisitic diffusie transition akin to the result from Telegrapher s Equation in soling for persistent random walks. 7

1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1].

1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1]. 1 Introduction The content of these notes is also covered by chapter 3 section B of [1]. Diffusion equation and central limit theorem Consider a sequence {ξ i } i=1 i.i.d. ξ i = d ξ with ξ : Ω { Dx, 0,