SNAPSHOTS OF BROWNIAN MOTION: A PRIMER

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SNAPSHOTS OF BROWNIAN MOTION:

A PRIMERFrank Munley

FEBRUARY 2011

The following discussion of Brownian motion is appropriate for students who have had algebra-basedcalculus-based courses in physics. A few questions are interspersed in the discussion; the answers ar

given at the end.

I. A FEW WORKHORSES OF PHYSICSF = ma

KE = mv2/2

KEthermal

= 3kBT/2, or :

mv themal2

/2 = 3kBT/2, so :

vthermal

" vm= 3k

BT/m,

where vm is the rms or average thermal speed of the center of mass of a body of mass m.

The body can be a molecule, macroscopic particle, or even a parcel of liquid.

II. A TYPICAL BROWNIAN PARTICLEIn the last equation, m can be the mass of anything--a small particle suspended in a thermal bath (e.gliquid or gas) or even a parcel of liquid for as long as the parcel has integrity before mixing with

surrounding liquid.

Consider a small particle, a Brownian Particle (BP):Particle density equal to waters = 1000 kg/m3;

Particle radiusR = 0.5x10-6

m, which is large enough to be visible under a visible-light microscope;Particle mass M= 5.24x10

-16kg (thats about 10

10atoms); and

Particle is suspended in water at 300 K:FIGURE 1:

Particles center of mass (cm)

obeys:

MvM2 /2= 3k

BT/2,

vM =3kBT

M=0.00487 m/s

For comparison: water

molecules thermal speed:vm= 643m/s

ABOUT10

-6m

cm

particlemass = M

water molecule,

vm = 3kBT/m

= 643 m/s


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GENESIS OF BROWNIAN MOTIONThe motion of Brownian particles suspended in a fluid (e.g., particles like the one just describ

that are visible under a visible-light microscope) is caused essentially by the fact that thermal motion

fluid molecules is random. The molecules, either individually or as a group, interact with a much lar

Brownian particle and exert random forces on it, thus causing the particle itself to behave randomly.The surprising fact of Brownian motion in ordinary fluids like air and water is that the particles execumotions in which they change direction only gradually. This is much unlike the motion of molecule

colliding with each other, in which case the molecules suffer abrupt and random changes in directionmotion. One purpose of this essay is to make a convincing case that Brownian particles do indeed

execute smooth, gradually changing motions.

In a low-density gas, the mechanism of Brownian motion is most simply understood: molecucollide with the Brownian particle randomly, and at any instant of time, there will be more collisions

one side than another. This fluctuation in impacts causes the particles direction of motion to changeslightly. Over time, the accumulated fluctuations dont completely cancel (see Section V below), and

their effect grows with the result that the particles direction gradually and constantly changes over tiA similar process occurs in a liquid too, though the more appropriate mechanism to describe theBrownian motion is fluctuation in pressure.

EXPERIMENTAL SCENARIOTake a snapshot of the Brownian particles position every t

sseconds (strobe the particle).

it for a total oft=N tSseconds so we observe the particleNtimes in time t:

The total displacement is:

r= ri

i = 1

N

" (A)

Squaring both sides, we get:

r1

r=

displacementafterNsteps in

time t

r3

r2

rN-1

r8

rN

FIGURE 2:

Browniansteps in a 2-

dimensional

walk.


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r2 = ri2+

i=1

N

" ri# rjj$i

N

"i=1

N

" . (B)

Now repeat the identical experiment (same snapshot time tS,Nsnapshots, total time t = NtS) many tim

and take averages of Eq. (B):

r2(t) = r

i

2=

i=1

N

" NLts

2= (t/ t

s)L

ts

2= (L

ts

2/ t

s)t, (C)

where LTS

2" r

i

2=average step size of step i. For details of the averaging process, see below for

answers to questions. Note that there is nothing special about step i, and the average value for all ste

in the experiment is the same: ri

2= L

tS

2 for all i.

LtSis a measure of how far the particle travels before its direction of motion significantly changes. L

about this big for the above diagram:

Example: for a typical Brownian particle in water, and an observation or strobe time of tS = 6

Lts=60 = 1.26x10

-5m,

about 10 times the diameter of the Brownian particle and easy to observe with a microscope, as was

done laboriously by Jean Perrin in 1911 for which he won the Nobel Prize).

III. A KEY EXPERIMENTAL RESULTRemember from Eq. (C):

r2(t) = Lt

S

2/ tS( )t. For given liquid, particle size, and temperature,

Lts

2/ ts in Eq. (C) is observed to be constant: if tS is quadrupled, Lt

s

only doubles, thus keeping the r

the same,etc. This constant ratio defines the diffusion constantD. For one

dimension:

x2(t) = (x

ts

2/ t

s)t = 2Dt, so:

D "

xts

2

2ts

. (D)

Particle moves about 1.26x10-5

m in 60 s.

Apparent average speed: 2.10x10-7

m/sMUCHsmaller than 0.00487 m/s!

QUESTION 1: What happened toi=1

N

" ri# rjj$i

N

" in going from (B) to (C)?

QUESTION 2: Explain why ri

2= L

tS

2 for all steps if in any one

experiment different steps have different lengths.

LtS


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(The factor of 2 is a pure convention.) For three dimensions: r2(t) = (Lts

2/ t

s)t = 6Dt, so:

D "

Lts

2

6ts

. (E)

IV. SELF-SIMILARITY OF AT SMALLER SCALESSuppose we had done the original experiment with tS one-fourth as large as the original. The

we would see that each step in Figure 2 above is really four steps:

So for a given total observation time tand snapshot time tS, we takeNsnapshots, while if the snapsho

time is shortened to tS/4, we take 4Nsnapshots in time t:

t= NtS = (4N)(tS /4) .

In the above diagram on the left, look at r1, the first step after the start, resulting from the firs

two snapshots. On the right, with four times as many snapshots, we see that the particle really didnttravel on a straight line along r1. It traveled a longer distance, so it is really traveling faster than we

thought! In general, the more frequently we look, the greater the distance we see the particle move in

the total time for the experiment. Mathematically: for a given tS, the average speed over a step lengt

vs= 6D / t

s .

So iftS is made smaller and smaller, vS gets bigger and bigger and will eventually be greater than thespeed of light, because the path length between two points grows bigger and bigger without limit as t

approaches zero.

Technically, the jagged Brownian path is afractalof dimension 2: it would cover the plane ifjaggedness persisted down to vanishingly small snapshot times!

CONCLUSION: The jaggedness must give way to a nice, smooth path when the snapshot time reach

small enough a value. This smooth motion must show the particle to move with its thermal speed vMSo our task is to discover the natural snapshot times, i.e., the natural time and space scales of

Brownian motion, call them tthermalandLthermal,that mark the transition between jagged motionand smooth motion. For convenience, call the time and space scales tthandLth.

QUESTION 3: Show thatvs = 6D / ts .

r1 r2r3

r8

rN

rN-1

Start

End

FIGURE 3: the effect of

shortening the snapshot

time by 1/4.


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V. BEFORE CONTINUING, AN IMPORTANT STATISTICAL PROPERTYSimple case: coin flips. Flip a coin N times. Do we get exactly N/2 H and N/2 T? Of course

not. But very likely (with about a 95% probability) well get something in this range:

N 2 N H, N m 2 N T .

These equations show that the deviation (or in our context, the fluctuation) from equal H and T

GROWS asNincreases, even though the fractional deviation orprecision, 2 N /N = 2 / N ,

decreases. People usually focus on how small N is compared toNwhenNgets very large, forgetti

that N is getting larger too!

For Brownian motion, the key thing is the growth in thefluctuation in the impact of molecule

on the particle. Most of the molecular impacts cancel outfor every impact driving the particle in odirection, there is very likely another driving it equally in the opposite direction. But the balance is n

perfectmost is not all, and the resulting imbalance grows over time as impacts accumulate on tparticle. Put another way: as time increases, fluctuation effects grow, until after a time, fluctuations

cause a particles initial direction of motion to change.

In most circumstances, the change in direction of the particle occurs very gradually comparedthe rate of molecular impacts. In the words of Edward Nelson in his essential bookDynamical Theor

of Brownian Motion,1

It is not correct to think simply that the jiggles in a Brownian trajectory are due to kicksfrom molecules. Brownian motion is unbelievably gentle. Each collision has an entirely

negligible effect on the position of the Brownian particle, and it is only fluctuations in theaccumulation of an enormous number of very slight changes in the particle's velocity that

give the trajectory its irregular appearance.

So a realistic picture emerges if snapshots are taken very fast. Here is what the original jagged

diagram might look like:

1A copy of Nelsons book can be obtained on line at by pasting the following into your browser addr

window: http://www.math.princeton.edu/~nelson/books/bmotion.pdf

FIGURE 4: The particles motion is

smooth if observed continuously, and on

average, it travels with its thermal speed

along its continuous path:

vM= 3k

BT/M (=0.00487 m/s for our

typical BP). It takes about a distanceLth

for the path direction to change

significantly.

r1

r=

displacementafterNste s

rN-1

r8

rN

initial direction

new directionLthermal

traversedin time tth

Actual Brownian particle path


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On average, the particle in the above diagram drifts a distanceLthermal, shown above, in an almstraight, just gradually curving, path (gradually because fluctuations constantly operate) before its

direction is substantially changed from what it started with.

CONCLUSION: When observed frequently enough, the motion of the Brownian particle no longer

appears jagged--we observe smooth motionalso called ballistic Brownian motion, known to exisince 1905 but not observed experimentally in an indirect way until 1989,2

and not directly until 1995For sufficiently rapid snapshots, the particle is observed on average to travel with its thermal

speed in a direction that changes very slowly.

VI. THE TIME SCALE tth AND THE ASSOCIATED SPACE SCALE,Lth

A simple way to estimate what the time scale for Brownian motion is to determine the snapsh

period for which vS just equal to the thermal speed: vs =6D

ts

, so forvs =3kBT

M, we want

6D

ts

=

3kBT

M. Letting tth be this special thermal motion value of ts that represents the transitio

from a jagged snapshot path where the particle appears to move slower than vM, to a smooth one whe

the particle moves with tM, we get:

tth =6D

vM2

=

6DM

3kBT.

Experimentally,D = 4.39x10-13

m2/s, so forM= 5.24x10

-16kg and T= 300 K:

tth =1.11x10"7

s .

Note that tthis very small. Ballistic Brownian motion can only be observed fortSis somewhat below

so it is no wonder that this motion wasnt directly observed until 1995!

IMPORTANT NOTE: tth is much bigger (1014

times bigger!) than the time between collisions of wa

molecules with the BP, which is about 10-21

s!4

The space scale associated with tth is:

Lth = vMtth ,

which for our example (vM= 0.0048 m/s, tth = 1.11x10-7

s) is:

Lth = smallest random step size = 5.41x10-10

m,

2D.A. Weitz, et al., "Nondiffusive Brownian Motion Studied by Diffusing-Wave Spectroscopy,"Phys. Rev. Lett., 63 (16

1747-1750 (1989).3B. Lukic et al, Direct Observation of Nondiffusive Motion of a Brownian Particle,Phys.

Rev. Lett. 95, 160601 (2005).4

S. Chandrasekhar, Stochastic Problems in Physics and Astronomy,Rev. Mod. Phys., 15 (1), Janu

1943 (p. 23)


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about 50 H-atom radii. (Remember: for significantly smaller snapshot times, the steps are no longerrandom.)

VII. STANDARD VALUES FOR TIME AND SPACE SCALES

Suppose particle is traveling with its thermal average speed to the right:

vM =3kBT

M

With no fluctuations, how long does it take for friction to bring it to a stop, or more precisely, to brinits speed down significantly? This time can be found by using Stokes friction coefficient, which is

appropriate for most ordinary fluids like water and air (but fails for low-density gases): "# $6%&R,

Newtons 2nd

Law gives:

Mdv

dt= "6#$Rv .

Solve forv :

v = vMe"t/#,

" =M

6#$R.

Example: for our typical Brownian particle (density equal to waters,R = 0.5x10-6

m, M= 5.24x10

kg, in water at 300K):

" = 5.60x10-8

s.

(Note that this " is half the value of our original tS calculated above to be 1.11x10-7

s. No problem: tare just different measures of the snapshot time or time scale below which the motion starts to smoot

out.)

The space scale associated with " can be taken as how far it would travel with thermal speedin time ":

" = vM#.

For our typical Brownian particle,"=2.73x10-10 m,

about half of what we got forLth.

Since " is only about 10-8

s for our typical Brownian particle, and the motion is observed to b

smooth only for snapshots much smaller than this, we can appreciate why it took so long to observeballistic Brownian motion!

CAUTION#1:

A time scale marks the division between two types of behavior, in our case between smooth

Brownian particle paths fortS


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likewise for using " orLth for the space scale. In other words, the time scale is not valuable for itsexactness but for telling us the approximate conditions under which a transition between jagged and

smooth motion occurs.

CAUTION #2:

Note that, in Brownian diffusion, a Brownian particle is constantly subject to friction as its pais being gradually changed by accumulating fluctuations. In molecular diffusion, on the other hand, molecule flies freely between collisions and its direction of motion is changed abruptly with each

collision. Nelson is concerned more with molecular than Brownian diffusion, and indeed his discusssurrounding Eq. (4.11) is for the latter case, not for Brownian diffusion.

VIII. TWO INTERPRETATIONS OF "

Remember what " is: if the snapshop period is >>

", the particles path appears jagged; if tS


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A more general form of Einsteins relation can be obtained that applies to molecules as well as toBrownian particles. A Brownian particle dragged through a viscous medium due to an externally

imposed forcefreaches a terminal or drift speed ofvdrift = f /(6"#R) , i.e., 6"#R = f /vdrift, so:

Df = kBTvdrift.

This more general form of Einsteins relation applies to molecular diffusion, because even though the

concept of viscosity does not directly apply to such motion, terminal or drift speed under an externallapplied force does.

Another common way of showing the Einstein relation is for an electric fieldEputting a forcef = qE

charged particles immersed in a fluid and pulling them through the fluid. In terms of the mobility= vdrift/E, Einsteins relation reads

D /= kBT/q .

Answers to Questions

QUESTION 1: What happened toi=1

N

" ri# rjj$i

N

" in going from (B) to (C)?

ANSWER: First, note thati=1

N

" ri #rjj$i

N

" =i=1

N

" ri #rjj$i

N

" , which you should be able to prove easily

remembering that the average is obtained by doing the experiment a large number of times and summ

the results of all experiments for each step. If you still feel uneasy about this point, read on.In concise terms, fortSsufficiently large, consecutive steps occur randomly, so the dot produc

between any two steps in anN-step experiment is sure to be the negative of that dot product in some

other run of the experiment.Heres a long-winded expansion of the concise answer. Consider two consecutive steps riand

ri+1. (For concreteness, imagine that i = 123, so were looking at the 123rd and 124th steps of an

experiment of, say,N =1000 steps.) These two steps, one occurring right after the other, have the be

chance of having a dot product that is non-zero because occurring consecutively, there might not bemuch change between them (in which case we say they are correlated). But there will be a random

change between them if we assume that the snapshot time tS is assumed to be large enough for manyrandom pushes from the fluid to influence the motion of the particle. To see this, first note that the

average ri" ri+1 is constructed by repeating the experiment ofNsteps many, many times, say a billi

billion (1018

) times. A large tSmeans that in each run of the experiment there are many mini-stepsalong every i

thand i+1

ststep as suggested by Figure 3 in Section IV (which shows only 4 mini-steps

each big step but there could be many more). These mini-steps, as shown in that figure, are in randodirections, and so the same is true of the ith and i+1st steps. Because of this randomness from step to

step, for every experiment where there is some angle, say ", between these two steps, there will be anexperiment where the angle will be " #, and the dot product for" # is the negative of the dot

product for". For example, the angle might be " for the 123rd and 124th steps in the 1,563,255th runthe experiment and "+# for these same steps in the 92,569,759

thrun of the experiment. The averag

for the ith

and i+1st

steps (e.g., the 123rd

and 124th

steps) is formed by summing over these particularsteps for all billion billion experimental runs, and cancellation is sure to occur pair by pair. Of course


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there is nothing special about the 123rd

and 124th

steps, so cancellation occurs not just for the dot proof these two steps, but for the i = 258

thand j = 875

thsteps or for the dot products of any other pair of

steps (except whenj = i, in which case we get ri2). Consequently,

i=1

N

" ri# rjj$i

N

" = 0.

QUESTION 2: Explain why ri

2= L

tS

2 for all steps if in any one experiment different steps have

different lengths.

ANSWER: The following chart of step lengths (just lengths, not directions!) shows what might beexpected of, say, three consecutive steps, starting with the 1

st, in a billion repetitive experiments (eac

experiment starting, e.g., with each particle moving with its thermal speed), and each experimentconsisting ofNsteps, whereNcan be anything.

Experiment

#

1ststep 2nd 3rd . . . ith

. . . Nthstep

1 . . . . . .

2 . . . . . .

3 . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

109

. . . . . .

Average

length

. . . . . .

Do enough experiments (a billion is quite sufficient), and randomness from experiment to experimen

insures that each step will go through the same range of lengths and frequencies of occurrence oflengths, as every other step. Therefore, the average after 10

9experiments will be the same for each s

as shown. Please note: Only four steps are shown in each experiment above. The four shown for eacone certainly dont average to the same thing. One has to average over a large number of experimen

get the same result for each one.

QUESTION 3: Show that vs = 6D / ts .

ANSWER: The average step speed is vS = LS / tS. But D = 6LS2/ tS, so eliminatingLSimmediately l

to the result.

QUESTION 4: Show that D = vM2

".

ANSWER: This is shown simply by using D = 6LS2/ tS, or its close equivalent D = 6"

2/#. Since

" = vM#, we can substitute for" and the desired result is immediately obtained.

QUESTION 5:Under what condition is Eq. (E) precise, giving the correct expression forD? And ho

does violation of that condition arise in our calculation of the Incorrect version ofD? (Hint: look athe transition from Eq. (B) to Eq. (C).)

ANSWER: As suggested at several points in the discussion and in the answer to Question 1, successi

steps are random so long as tSis sufficiently large. This condition on tS is necessary, because iftS is t


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small, the snapshot path is smoothrapid successive snapshots trace out a slowly-varying path. Thimeans that there is not enough time from snapshot to snapshot for consecutive steps to be random. In

other words, if the particle is traveling along thex axis, two successive snapshots will show a step, castep i, along that axis. Then step i+1, which is the straight line between the second and a third snaps

will show the particle still traveling very close to thex axis, because not enough time has passed for

fluctuations to accumulate and change the particles direction of motion significantly. (This isessentially ballistic Brownian motion.) Therefore, ri" r

i+1will not be zero. In fact, this average will

start going to zero only as tSincreases significantly beyond tthor".

SNAPSHOTS OF BROWNIAN MOTION: A PRIMER

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