Thermal Forces and Brownian Motion

Transcription

1 Theral Forces and Brownian Moion Ju Li GEM4 Suer School 006 Cell and Molecular Mechanics in BioMedicine Augus 7 18, 006, MIT, Cabridge, MA, USA

2 Ouline Meaning of he Cenral Lii Theore Diffusion vs Langevin equaion descripions (average vs individual) Diffusion coefficien and flucuaion-dissipaion heore

3 Cenral Lii Theore Y = X 1 + X + + X N X 1, X,, X N are rando variables E[Y] = E[X 1 ] + E[X ] + + E[X N ] If X 1, X,, X N are independen rando variables: var[y] = var[x 1 ] + var[x ] + + var[x N ] Noe: var[x] = σ X E[ (X-E[X]) ] 3

4 If X 1, X,, X N are independen rando variables sapled fro he sae disribuion: E[Y] = NE[X] var[y] = N var[x 1 ] = Nσ X Average of he su: y Y/N E[y] = E[X], var[y] = var[y]/n = σ X / N Law of large nubers: as N ges large, he average of he su becoes ore and ore deerinisic, wih variance σ X / N. 4

5 X 5 X 1, X,, X N ay be sapled fro Probabiliy densiy Probabiliy densiy -1 X Probabiliy densiy X

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

7 Why Gaussian? ρ 1 ( Y NE[ X]) large N ( Y ) exp πnσ Nσ X 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 aheaical operaion of convoluion. 7

8 Diffusion Equaion in 1D = ( D ) = D x x x ρ ρ ρ Rando walker view of diffusion: iagine (a) We release he walker a x=0 a =0, (b) Walker akes a ove of ±a, wih equal J probabiliy, every =1/ν fro hen on. Maheaically, we say ρ(x,=0)=δ(x). N= =ν independen rando seps Then, ( )... x = x+ x + + x 1 / 8

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

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

11 Fa droples suspended in ilk (fro Dave Walker). The droples range in size fro abou 0.5 o 3 µ. 11 Brownian Moion

12 In addiion o dissipaive force, here us be anoher, siulaive force. 1 viscous oil v Sokes' law: F=-6πrηv=-v v = F = v, v( = 0) = v v () = ve Einsein's Explanaion of Brownian Moion 0 0 Also, equi-pariion heore: v kbt =

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

14 vv () () = = 1 1 fluc ( ) ( ) fluc d F ( ) e d F ( ) e ( ) ( ) de de F ( ) F ( ) fluc ( ) ( ) fluc 1 ( ) ( ) = d e d e Bδ ( ) = 1 d e = B e H ( ) e B H( x) is Heaviside sep funcion: 1 if x > 0 H( x) = 0 if x 0 14

15 In paricular: vv ( ) ( ) = B However, fro equilibriu saisical echanics: equi-pariion heore: vv () () = kt B B = kt B The raio beween square of siulaive force and dissipaive force is fixed, T 15

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

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

18 Acually, he onse of acroscopic diffusion = D ( ρ ρ x ) is only valid only when inrinsic iescale of g( ) (Sae as cenral lii heore in rando walk) So he correc forula is D = 0 d v( ) v(0) The above is one of he flucuaion-dissipaion heores. 18

19 1 Theral conduciviy: κ = J ( ) (0) q Jq d Ω kt 0 B 1 Elecrical conduciviy: σ = J ( J ) (0) d Ω kt 0 B Ω Shear viscosiy: η = τxy( ) τxy(0) d kt 0 Flucuaion-dissipaion heore (Green-Kubo forula) is one of he os elegan and significan resuls of saisical echanics. I relaes ranspor properies (syse behavior if linearly perurbed fro equilibriu) o he ie-correlaion of equilibriu flucuaions. B 19

20 Coing back o diffusion (ass ranspor): kt B vv () () = e kt B So D = d v( ) v(0) =. 0 1 is acually he obiliy of he paricle, when driven by exernal (non-heral) force. D 1/ = kt B is called he Einsein relaion, firs derived in

21 References Kubo, Toda & Hashiue, Saisical Physics II: Nonequilibriu Saisical Mechanics (Springer-Verlag, New York, 199). Zwanzig, Nonequilibriu Saisical Mechanics (Oxford Universiy Press, Oxford, 001). van Kapen, Sochasic processes in physics and cheisry, rev. and enl. ed. (Norh-Holland, Aserda, 199). Reichl, A odern course in saisical physics (Wiley, New York, 1998). 1