GEM4 Summer School OpenCourseWare Given August 11,

GEM4 Summer School w.gem4.org/Lecture: “Thermal Forces and Brownian Motion” by Ju Li.Given August 11, 2006 during the GEM4 session at MIT in Cambridge, MA.Please use the following citation format:Li, Ju. “Thermal Forces and Brownian Motion.” Lecture, GEM4 session at MIT, Cambridge, MA, August 11, 2006.http://gem4.educommons.net/ (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-ShareAlike.Note: Please use the actual date you accessed this material in your citation.

Thermal Forces and Brownian MotionJu LiGEM4 Summer School 2006Cell and Molecular Mechanics in BioMedicineAugust 7–18, 2006, MIT, Cambridge, MA, USA

Outline Meaning of the Central Limit Theorem Diffusion vs Langevin equation descriptions(average vs individual) Diffusion coefficient andfluctuation-dissipation theorem2

Central Limit TheoremY X1 X2 XNX1, X2, , XN are random variablesE[Y] E[X1] E[X2] E[XN]If X1, X2, , XN are independent randomvariables:var[Y] var[X1] var[X2] var[XN]Note: var[X] σ2X E[ (X-E[X])2 ]3

If X1, X2, , XN are independent randomvariables sampled from the samedistribution:E[Y] NE[X]var[Y] N var[X1] Nσ2XAverage of the sum: y Y/NE[y] E[X], var[y] var[Y]/N2 σ2X / NLaw of large numbers: as N gets large, theaverage of the sum becomes more andmore deterministic, with variance σ2X / N.4

X1, X2, , XN may be sampled ydensity-12XXX5

We know the probability distribution of Yis shifting (NE[X]), as well as getting fat(Nσ2X). But how about its shape ?The central limit theorem says thatirrespective of the shape of X,ProbabilitydensityNσ2XNE[X])Y6

Why Gaussian ?large Nρ (Y ) (Y NE[ X ]) exp 222N σ X2π N σ X 12Gaussian is special(Maxwellian velocity distribution, etc).While proof is involved,here we note that Gaussian is an invariantshape (attractor in shape space) in themathematical operation of convolution.7

Diffusion Equation in 1DJ 2 t ρ x ( D x ρ ) D x ρRandom walker view of diffusion: imagine(a) We release the walker at x 0 at t 0,(b) Walker makes a move of a, with equalprobability, every t 1/ν from then on.Mathematically, we say ρ(x,t 0) δ(x).N t ν t independent random steps Then, x (t ) x1 x2 . xt / t t8

When N νt 1,the central limit theorem applies:E[x(t)] 0, var[x(t)] νt var[ x] νta2So we can directly write down ρ ( x (t )) as2 x 1ρ G ( x, t ) exp 2 22πν a t 2ν a t It is the probability of finding the walker at xat time t, knowing he was at 0 at time 0.9

By plugging in, we can directly verifyρ G ( x, t ) satisfies2 t ρ D x ρ , ρ ( x,0) δ ( x).2vawith macroscopic D identified as.2ρ G ( x, t ) x 1exp 2π (2 Dt) 2(2 Dt) 2is called Green's function solutionto diffusion equation.10

Brownian MotionCourtesy of Microscopy-UK. Used with permission.Fat droplets suspended in milk (from Dave Walker).The droplets range in size from about 0.5 to 3 µm.11

viscous oilStokes' law:F -6πrηv -λvvmv F λ v, v (t 0) v0 v (t ) v0eλ tmEinstein's Explanation of Brownian MotionAlso, equi-partition theorem:mv22k BT 2In addition to dissipative force, there must beanother, stimulative force.12

mv Fdissipative Fstimulative/fluctuation λ v Ffluc (t )Ffluc (t ) 0Ffluc (t ) Ffluc (t ′) b(t t ′)If b(t t ′) Bδ ( t t ′) : white noiseExact Green's function solution of v (t ):λ (t t ′ )1 tm′′v (t ) dtFte()fluc m13

v (t )v (t )1 2m1 2m tdt ′Ffluc (t ′)e t 1 2mdt ′e t 1 2mλ (t t ′ )m t dt ′e t λλ (t t ′ )m (t t ′ )m dt ′e t dt ′eλ t (t t ′ )mdt ′Ffluc (t ′)eλ (t t ′ )mdt ′eFfluc (t ′) Ffluc (t ′)λ (t t ′ )mH (t t ′)eBe 2mλ λmλ (t t ′ )mBδ ( t ′ t ′)λ (t t ′ )mBH ( x ) is Heaviside step function:t t 1H ( x) 0if x 0if x 014

In particular:Bv (t )v (t ) 2mλHowever, from equilibrium statistical mechanics:equi-partition theorem:m v (t )v (t ) k BTB k BT2λThe ratio between square of stimulative forceand dissipative force is fixed, T15

kTBv (t )v (t ) em λmt t Previously, from the Gaussian solution to2 t ρ D x ρ , ρ ( x,0) δ ( x) :2 x1ρ G ( x, t ) exp 2π (2 Dt) 2(2 Dt) we know if the particle is released at x 0 at t 0 :x (t ) x (t ) 2Dttx (t ) 0 dt ′v (t ′),0x (t ) v (t )16

dx (t ) x (t ) 2 x (t ) x (t ) 2 x (t )v (t )dtd (2Dt) 2DdtD x (t )v (t ) ( dt′v(t′)) v(t)t0t dt ′ v (t ′)v (t )0t dt ′ v (t ′)v (0)0Velocity auto-correlation function: g (t ) v (t )v (0)17

Actually, the onset of macroscopic diffusion( t ρ D ρ ) is only valid only when2xt intrinsic timescale of g (t ) mλ(Same as central limit theorem in random walk)So the correct formula is D dt ′ v (t ′)v (0)0The above is one of thefluctuation-dissipation theorems.18

1Thermal conductivity: κ J q (t ) J q (0) dt2 0Ωk BT 1Electrical conductivity: σ J (t ) J (0) dt Ωk BT 0Ω Shear viscosity: η τ xy (t )τ xy (0) dt k BT 0Fluctuation-dissipation theorem (Green-Kuboformula) is one of the most elegant andsignificant results of statistical mechanics. Itrelates transport properties (system behavior iflinearly perturbed from equilibrium) to thetime-correlation of equilibrium fluctuations.19

Coming back to diffusion (mass transport):λ t tkTBv (t )v (t ) e mm k BTSo D dt ′ v (t ′)v (0) .01λλis actually the mobility of the particle, whendriven by external (non-thermal) force.D k BT is called the Einstein relation,1/λfirst derived in 1905.20

ReferencesKubo, Toda & Hashitume,Statistical Physics II: Nonequilibrium Statistical Mechanics(Springer-Verlag, New York, 1992).Zwanzig, Nonequilibrium Statistical Mechanics(Oxford University Press, Oxford, 2001).van Kampen, Stochastic processes in physics and chemistry,rev. and enl. ed. (North-Holland, Amsterdam, 1992).Reichl, A modern course in statistical physics(Wiley, New York, 1998).21

Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag, New York, 1992). Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, Oxford, 2001). van Kampen, Stochastic processes in physics and chemistry, rev. and enl. ed. (North-Holland, Amsterdam, 1992). Reichl, A modern course in statistical physics