Brownian motion 18.S995 - L03 & 04

Transcription

1 Brownian motion 18.S995 - L03 & 04

2 Typical length scales

3 Brownian motion

4 Brownian motion Jan Ingen-Housz ( ) 1784/1785:

5 Robert Brown ( ) Linnean society, London 1827: irreguläre Eigenbewegung von Pollen in Flüssigkeit irregular motion of pollen in fluid

6

7 !"#$%&'()*+,-#./0102/3//4 5"#67,8&(7,#./0932/3114 D! 2 RT RT 1 32 mc D! D! 6"#aC N 6" k P 243 "$ R A'7*"#:+B"#!C#90/#./3D14 5,,"#A'E8"#"#C#1F3#./3D14 5,,"#A'E8"#$"C#91G#./3DG4

8 Jean Baptiste Perrin ( , Nobelpreis prize 1926) colloidal particles of radius 0.53µm Mouvement brownien et réalité moléculaire, Annales de chimie et de physique VIII 18, (1909) successive positions every 30 seconds joined by straight line segments mesh size is 3.2µm Les Atomes, Paris, Alcan (1913) Experimenteller Nachweis der atomistischen Struktur der Materie experimental evidence for atomistic structure of matter

9 Norbert Wiener ( ) MIT

10 Relevance in biology intracellular transport intercellular transport microorganisms must beat BM to achieve directed locomotion tracer diffusion = important experimental tool generalized BMs (polymers, membranes, etc.) dunkel@math.mit.edu

11 Polymers & filaments (D=1) Dogic Lab, Brandeis Physical parameters (e.g. bending rigidity) from fluctuation Drosophila oocyte analysis Goldstein lab, PNAS 2012

12 Brownian tracer particles in a bacterial suspension Bacillus subtilis Tracer colloids PRL 2013

13

14 Basic idea Split dynamics into deterministic part (drift) random part (diffusion)

15 Typical problems Determine noise structure transport coefficients first passage (escape) times D

16 Probability space A B [0, 1] P[;] =0

17 Expectation values of discrete random variables

18 Expectation values of continuous random variables

19 Random walk model X N = x 0 + ` NX S i

20 1.1 Random walks Unbiased random walk (RW) Consider the one-dimensional unbiased RW (fixed initial position X 0 length `) = x 0, N steps of X N = x 0 + ` NX S i (1.1) where S i 2 {±1} are iid. random variables (RVs) with P[S i = ±1] = 1/2. Noting that 1 E[S i ] = =0, (1.2) 2 apple E[S i S j ] = ij E[Si 2 ]= ij ( 1) (1)2 1 2 = ij, (1.3) we find for the first moment of the RW E[X N ] = x 0 + ` NX E[S i ]=x 0 (1.4) R

21 1.1 Random walks Unbiased random walk (RW) Consider the one-dimensional unbiased RW (fixed initial position X 0 length `) = x 0, N steps of X N = x 0 + ` NX S i (1.1) where S i 2 {±1} are iid. random variables (RVs) with P[S i = ±1] = 1/2. Noting that 1 E[S i ] = =0, (1.2) 2 apple E[S i S j ] = ij E[Si 2 ]= ij ( 1) (1)2 1 2 = ij, (1.3) we find for the first moment of the RW E[X N ] = x 0 + ` NX E[S i ]=x 0 (1.4) R

22 1.1 Random walks Unbiased random walk (RW) Consider the one-dimensional unbiased RW (fixed initial position X 0 length `) = x 0, N steps of X N = x 0 + ` NX S i (1.1) where S i 2 {±1} are iid. random variables (RVs) with P[S i = ±1] = 1/2. Noting that 1 E[S i ] = =0, (1.2) 2 apple E[S i S j ] = ij E[Si 2 ]= ij ( 1) (1)2 1 2 = ij, (1.3) we find for the first moment of the RW E[X N ] = x 0 + ` NX E[S i ]=x 0 (1.4) R

23 Second moment (uncentered) NX E[XN] 2 = E[(x 0 + ` S i ) 2 ] = E[x x 0` NX NX NX S i + `2 S i S j ] j=1 j=1 NX NX = x x 0 0+`2 E[S i S j ] NX NX = x x 0 0+`2 j=1 ij = x `2N. (1.5)

24 Second moment (uncentered) NX E[XN] 2 = E[(x 0 + ` S i ) 2 ] = E[x x 0` NX NX NX S i + `2 S i S j ] j=1 j=1 NX NX = x x 0 0+`2 E[S i S j ] NX NX = x x 0 0+`2 j=1 ij = x `2N. (1.5)

25 Second moment (uncentered) NX E[XN] 2 = E[(x 0 + ` S i ) 2 ] = E[x x 0` NX NX NX S i + `2 S i S j ] j=1 j=1 NX NX = x x 0 0+`2 E[S i S j ] NX NX = x x 0 0+`2 j=1 ij = x `2N. (1.5)

26 Second moment (uncentered) NX E[XN] 2 = E[(x 0 + ` S i ) 2 ] = E[x x 0` NX NX NX S i + `2 S i S j ] j=1 j=1 NX NX = x x 0 0+`2 E[S i S j ] NX NX = x x 0 0+`2 j=1 ij = x `2N. (1.5)

27 Second moment (uncentered) NX E[XN] 2 = E[(x 0 + ` S i ) 2 ] = E[x x 0` NX NX NX S i + `2 S i S j ] j=1 j=1 NX NX = x x 0 0+`2 E[S i S j ] NX NX = x x 0 0+`2 j=1 ij = x `2N. (1.5)

28 Continuum limit X N = x 0 + ` NX S i Let

29 Continuum limit P (N,K) = = N 1 N N K 2 N N! ((N + K)/2)! ((N K)/2)!. (1.8) The associated probability density function (PDF) can be found by defining p(t, x) := P (N,K) 2` and considering limit, `! 0suchthat = P (t/,x/`) 2` (1.9) D := `2 2 = const, (1.10)

30 Continuum limit yielding the Gaussian p(t, x) ' r 1 x 2 4 Dt exp 4Dt (1.11) (pset1) Eq. (1.11) is the fundamental solution to the di usion t p t = D@ xx p, (1.12) xx,... denote partial derivatives. The mean square displacement of the continuous process described by Eq. (1.11) is Z E[X(t) 2 ]= dx x 2 p(t, x) =2Dt, (1.13) in agreement with Eq. (1.7).

31 Z Remark One often classifies di usion processes by the (asymptotic) power-law growth of the mean square displacement, µ =0:Staticprocesswithnomovement. E[(X(t) X(0)) 2 ] t µ. (1.14) 0 <µ<1:sub-di usion, arises typically when waiting times between subsequent jumps can be long and/or in the presence of a su ciently large number of obstacles (e.g. slow di usion of molecules in crowded cells). µ = 1 : Normal di usion, corresponds to the regime governed by the standard Central Limit Theorem (CLT). 1 <µ<2: Super-di usion, occurs when step-lengths are drawn from distributions with infinite variance (Lévy walks; considered as models of bird or insect movements). µ =2:Ballisticpropagation(deterministicwave-likeprocess).

32 Brownian motion non-brownian Levy-flight

33 1.1.2 Biased random walk (BRW) Consider a one-dimensional hopping process on a discrete lattice (spacing `), defined such that during a time-step aparticleatpositionx(t) =`j 2 `Z can either (i) jump a fixed distance ` to the left with probability,or (ii) jump a fixed distance ` to the right with probability, or (iii) remain at its position x with probability (1 ). Assuming that the process is Markovian (does not depend on the past), the evolution of the associated probability vector P (t) =(P (t, x)) = (P j (t)), where x = `j, is governed by the master equation P (t +,x)=(1 ) P (t, x)+ P (t, x `)+ P (t, x + `). (1.15)

34 Master equations P (t +,x)=(1 ) P (t, x)+ P (t, x `)+ P (t, x + `). (1.15) Technically,, and (1 ) arethenon-zero-elementsofthecorrespondingtransition matrix W =(W ij )withw ij > 0thatgovernstheevolutionofthecolumnprobability vector P (t) =(P j (t)) = (P (t, y)) by P i (t + ) =W ij P j (t) (1.16a) or, more generally, for n steps P (t + n ) =W n P (t). (1.16b) The stationary solutions are the eigenvectors of W with eigenvalue 1. To preserve normalization, one requires P i W ij =1.

35 Continuum limit Define the density p(t,x) = P (t,x)/`. Assume, ` are small, so that we can Taylor-expand p(t +,x) ' p(t, t p(t, x) (1.17a) p(t, x ± `) ' p(t, x) ± `@ x p(t, x)+`2 xxp(t, x) (1.17b) Neglecting the higher-order terms, it follows from Eq. (1.15) that p(t, t p(t, x) ' (1 ) p(t, x)+ Dividing by, one obtains the advection-di usion equation with drift velocity u and di usion constant D given by 2 [p(t, x) `@ x p(t, x)+`2 xxp(t, x)] + [p(t, x)+`@ x p(t, x)+`2 xxp(t, x)]. t p = x p + xx p (1.19a) u := ( ) `, D := ( + ) `2 2. (1.19b)

36 Continuum limit Define the density p(t,x) = P (t,x)/`. Assume, ` are small, so that we can Taylor-expand p(t +,x) ' p(t, t p(t, x) (1.17a) p(t, x ± `) ' p(t, x) ± `@ x p(t, x)+`2 xxp(t, x) (1.17b) Neglecting the higher-order terms, it follows from Eq. (1.15) that p(t, t p(t, x) ' (1 ) p(t, x)+ Dividing by, one obtains the advection-di usion equation with drift velocity u and di usion constant D given by 2 [p(t, x) `@ x p(t, x)+`2 xxp(t, x)] + [p(t, x)+`@ x p(t, x)+`2 xxp(t, x)]. t p = x p + xx p (1.19a) u := ( ) `, D := ( + ) `2 2. (1.19b)

37 Continuum limit Define the density p(t,x) = P (t,x)/`. Assume, ` are small, so that we can Taylor-expand p(t +,x) ' p(t, t p(t, x) (1.17a) p(t, x ± `) ' p(t, x) ± `@ x p(t, x)+`2 xxp(t, x) (1.17b) Neglecting the higher-order terms, it follows from Eq. (1.15) that p(t, t p(t, x) ' (1 ) p(t, x)+ Dividing by, one obtains the advection-di usion equation with drift velocity u and di usion constant D given by 2 [p(t, x) `@ x p(t, x)+`2 xxp(t, x)] + [p(t, x)+`@ x p(t, x)+`2 xxp(t, x)]. t p = x p + xx p (1.19a) u := ( ) `, D := ( + ) `2 2. (1.19b)

38 Time-dependent solution Dividing by, one obtains the advection-di usion equation with drift velocity u and di usion constant D given by t p = x p + xx p (1.19a) u := ( ) `, D := ( + ) `2 2. We recover the classical di usion equation (1.12) from Eq. (1.19a) for = time-dependent fundamental solution of Eq. (1.19a) reads r 1 (x ut) 2 p(t, x) = 4 Dt exp 4Dt (1.19b) = 0.5. The (1.20)

39 r Remarks Note that Eqs. (1.12) and Eq. (1.19a) can both be written in the t p x j x =0 (1.21) with j x = up D@ x p, (1.22) reflecting conservation of probability. Another commonly-used representation t p = Lp, (1.23) where L is a linear di erential operator; in the above example (1.19b) L := x + xx. (1.24) Stationary solutions, if they exist, are eigenfunctions of L with eigenvalue 0. (useful later when discussing Brownian motors)