GEM4 Summer School OpenCourseWare

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1 GEM4 Summer School OpenCourseWare hp://gem4.educommons.ne/ hp:// Lecure: Thermal Forces and Brownian Moion by Ju Li. Given Augus 11, 2006 during he GEM4 session a MIT in Cambridge, MA. Please use he following ciaion forma: Li, Ju. Thermal Forces and Brownian Moion. Lecure, GEM4 session a MIT, Cambridge, MA, Augus 11, hp://gem4.educommons.ne/ (accessed MM DD, YYYY). License: Creaive Commons Aribuion-Noncommercial-Share Alike. Noe: Please use he acual dae you accessed his maerial in your ciaion.

2 Thermal Forces and Brownian Moion Ju Li GEM4 Summer School 2006 Cell and Molecular Mechanics in BioMedicine Augus 7 18, 2006, MIT, Cambridge, MA, USA

3 2 Ouline Meaning of he Cenral Limi Theorem Diffusion vs Langevin equaion descripions (average vs individual) Diffusion coefficien and flucuaion-dissipaion heorem

4 Cenral Limi Theorem Y = X 1 + X X N X 1, X 2,, X N are random variables E[Y] = E[X 1 ] + E[X 2 ] + + E[X N ] If X 1, X 2,, X N are independen random variables: var[y] = var[x 1 ] + var[x 2 ] + + var[x N ] Noe: var[x] = σ 2 X E[ (X-E[X])2 ] 3

5 more deerminisic, wih variance σ 2 X / N. 4 If X 1, X 2,, X N are independen random variables sampled from he same disribuion: E[Y] = NE[X] var[y] = N var[x 1 ] = Nσ 2 X Average of he sum: y Y/N E[y] = E[X], var[y] = var[y]/n 2 = σ 2 X / N Law of large numbers: as N ges large, he average of he sum becomes more and

6 Probabiliy densiy X 1, X 2,, X N may be sampled from Probabiliy densiy -1 2 X Probabiliy densiy X X 5

7 We know he probabiliy disribuion of Y is shifing (NE[X]), as well as geing fa (Nσ 2 ). Bu how abou is shape? X The cenral limi heorem says ha irrespecive of he shape of X, Probabiliy densiy Nσ 2 X NE[X]) Y 6

8 mahemaical operaion of convoluion. 7 Why Gaussian? 1 (Y N E[X ]) 2 large N ρ( Y ) exp 2Nσ 2 X 2 πnσ 2 X Gaussian is special (Maxwellian velociy disribuion, ec). While proof is involved, here we noe ha Gaussian is an invarian shape (aracor in shape space) in he

9 Diffusion Equaion in 1D ρ = x ( D ρ ) = D 2 x J xρ Random walker view of diffusion: imagine (a) We release he walker a x=0 a =0, (b) Walker makes a move of ±a, wih equal probabiliy, every =1/ν from hen on. Mahemaically, we say ρ(x,=0)=δ(x). N = =ν independen random seps Then, x ( ) = x + x x 1 2 / 8

10 When N=ν >>1, he cenral limi heorem applies: E[x()] = 0, var[x()] = ν var[ x] = ν a 2 So we can direcly wrie down ρ( x( )) as 1 x 2 ρ ( x, ) = exp G 2 2πν a 2 2ν a I is he probabiliy of finding he walker a x a ime, knowing he was a 0 a ime 0. 9

11 By plugging in, we can direcly verify ρ ( x, ) saisfies G =, ρ x,0) ρ D 2 ρ ( = δ (x). x 2 wih macroscopic D idenified as va. 2 ρ ( x, ) = G 1 x 2 exp 2 π (2 D) 2(2 D) is called Green's funcion soluion o diffusion equaion. 10

12 Brownian Moion Couresy of Microscopy-UK. Used wih permission. Fa droples suspended in milk (from Dave Walker). The droples range in size from abou 0.5 o 3 µm. 11

13 In addiion o dissipaive force, here mus be anoher, simulaive force. 12 viscous oil v Sokes' law: F=-6πrηv=-λv mv = F = λv, v( = 0) = v 0 v () = ve 0 λ m Einsein's Explanaion of Brownian Moion Also, equi-pariion heorem: 2 mv kb T = 2 2

14 mv = F dissipaive + F simulaive/flucuaion = λv + F () fluc F () = 0 fluc F () F fluc fluc ( ) = b( ) If b ( ) = Bδ ( ) : whie noise Exac Green's funcion soluion of v( ): v () = 1 df e fluc m ( ) λ ( ) m 13

15 vv () () 1 m m = d F fluc ( ) e d F fluc ( ) e 2 m λ ( ) λ ( ) 1 λ ( ) λ ( ) d e m = 2 d e m m F fluc ( ) F ( ) fluc 1 λ ( ) λ ( ) m d e 2 = d e m = m Bδ ( ) 1 λ ( ) ( m m d 2 H ) e λ ) e ( B m H( x ) is Heaviside sep funcion: B λ 1 if x > 0 ( ) = = e m H x 0 if x 0 2mλ 14

16 B In paricular: vv ( ) ( ) = 2mλ However, from equilibrium saisical mechanics: equi-pariion heorem: mvv () () = kt B B = kb T 2λ The raio beween square of simulaive force and dissipaive force is fixed, T 15

17 vv () () kt λ B = e m m Previously, from he Gaussian soluion o ρ D 2 ρ, ρ(x,0) = δ = x ( x): 1 x 2 ρ G ( x, ) = exp 2 π (2 D) 2(2 D) we know if he paricle is released a x = 0 a = 0 : xx () () = 2D x () = 0 + dv ( ), x () = v () 0 16

18 17 d d xx () () = 2 xx () () = 2 xv () () d = (2D) = 2D d D = x() v() ) ( dv 0 = ( ) v() = d v( ) v( ) 0 = d v( ) v (0) 0 Velociy auo-correlaion funcion: g( ) vv ( ) (0)

19 Acually, he onse of macroscopic diffusion ( ρ = D 2 ρ) is only valid only when x inrinsic imescale of g( ) m λ (Same as cenral limi heorem in random walk) So he correc formula is D = d v( ) v (0) 0 The above is one of he flucuaion-dissipaion heorems. 18

20 Thermal conduciviy: κ = 1 Ωk T B J q ( J ) q (0) d Elecrical conduciviy: σ = J ( J ) (0) ΩkT 0 Shear viscosiy: η = Ω τ xy ( ) τ xy (0) d kt 0 B Flucuaion-dissipaion heorem (Green-Kubo formula) is one of he mos elegan and significan resuls of saisical mechanics. I relaes ranspor properies (sysem behavior if linearly perurbed from equilibrium) o he ime-correlaion of equilibrium flucuaions. 19 B d

21 Coming back o diffusion (mass ranspor): kt λ vv () () B = e m m kt So D = B d v( ) v (0) =. 0 λ 1 is acually he mobiliy of he paricle, when λ driven by exernal (non-hermal) force. D 1/λ = k T is calle d he Einsein relaion, B firs derived in

22 References Kubo, Toda & Hashiume, Saisical Physics II: Nonequilibrium Saisical Mechanics (Springer-Verlag, New York, 1992). Zwanzig, Nonequilibrium Saisical Mechanics (Oxford Universiy Press, Oxford, 2001). van Kampen, Sochasic processes in physics and chemisry, rev. and enl. ed. (Norh-Holland, Amserdam, 1992). Reichl, A modern course in saisical physics (Wiley, New York, 1998). 21