Brownian Motion

and

The Atomic Theory

Albert Einstein

Annus Mirabilis

Centenary Lecture

Simeon Hellerman

Institute for Advanced Study,

5/20/2005

Founders’ Day

1

1. What phenomenon did Einstein explain?

(a) Random motion of particles in liquid(b) Dependence on temperature(c) Dependence on viscosity of the liquid(d) Dependence on the size of the particles

2. How did he explain it?

(a) Statistical Theory of Heat(b) Ideal Gases(c) Treat Microscopic and Macroscopic Objects

Uniformly

3. Why was it important?

(a) Universal Validity of Statistical Mechanics(b) Reality of Atoms and Molecules

2

Robert Brown’s Discovery

pollengrains

In 1827 Robert Brown, a Scottish botanist and curatorof the British Museum, observed that pollen grainssuspended in water, instead of remaining stationaryor falling downwards, would trace out a randomzig-zagging pattern. This process, which could beobserved easily with a microscope, gained the nameof Brownian motion.

3

1865

MAY 1864

MAY 1865

It is discovered that no matter how long theparticles are left in the liquid in a sealed container,they never come to rest or even slow down. Theexperimenters observed a system of pollen motes in asealed volume of liquid over a period of one year, andBrown’s zig-zagging effect continued undiminished.

4

1870

Pa . sη = 1.5glycerol

.25 10−3ethyl alcohol

η = sPa .

It is discovered that the speed with which thepollen motes disperse was slower in more viscousliquid and faster in less viscous liquid.

5

1900

It is discovered that the zig-zagging gets faster asthe temperature of the water is raised; also, that themotion becomes slower as the size of the particles ismade larger.

6

1905

random behavior temperature ...

statistical theory

...

use the

affected by

!heatof

By 1905, Einstein had begun to think aboutBrownian motion in the context of the young scienceof statistical mechanics.

Statistical mechanics aims to understand thethermal behavior of macroscopic matter in terms ofthe average behavior of microscopic constituents

under the influence of mechanical forces.

7

Heat as energy

The theory originated in an attempt to solve thepuzzle of energy conservation.

(depends on speed)kinetic energy

1 meter

Both are forms of

(depends on position)potential energy

MECHANICAL ENERGY

0 m/s

4.4 m/s

1 meter

0 m/s0 m/s

NOTis conserved hereMECHANICAL ENERGY

8

Development of the theory

Sometimes mechanical energy is conserved;sometimes it is partially or completely converted toheat.

1842-1847: Conservation of Energy

Several scientists (including Mayer, Helmholtz,

and Joule) proposed that a modified law ofconservation of energy would hold for all processes,when heat was appropriately taken into account as aform of energy:

dU = dQ - dW

which means that

mechanical energy + thermal energy

= constant over time

in a closed system. This is the First Law of

Thermodynamics.

9

1850-1898: Kinetic theory of Gases

From studies of the behavior of gases, anunderstanding of thermodynamics began to emerge:

• Matter is made up of microscopic molecularconstituents.

• The molecules of matter are constantly in

motion, even for matter which appears to beat rest. The macroscopic momentum of matter isonly an average momentum of its constituents.

• The speed of the molecular constituents much

higher at ordinary temperatures than the typicalspeed of the matter in bulk.

• Most importantly: thermal energy of macroscopicmatter is identical with the mechanical energy ofits constituents.

This last idea gives a simple explanation forthe First Law of thermodynamics: on molecularscales, there is only mechanical energy, and it isabsolutely conserved!

10

Ideal Gases

Scientists studying gas behavior introduced theconcept of an ideal gas – a gas whose molecules’energy was purely kinetic. An ideal gas is a tinyset of billiard balls, which move in straight lines,occasionally colliding and exchanging momentum.

Ludwig Boltzmann then proved that in the limitof a large number of such billiard balls, the collectionof balls would approach thermal equilibrium – acondition of uniform temperature and density,macroscopically a gas at rest, with thermal energystored as kinetic energy of the billiards.

11

An ideal gas satisfies the law

pV = n R T,

where p = pressure, T = temperature, V is thevolume of the container, and R is the universal

gas constant,

R = 8.315 Joule/degree centigrade.

n is the number of molecules divided by Avogadro’s

number NA, the number of atoms in a gram of

hydrogen.

As a physical theory, the ideal gas law wasextraordinarily successful, correctly describing thebehavior of many actual gases over a wide range oftemperatures and pressures.

12

The trouble with atoms

One weakness of the kinetic theory of gases wasthat it only dealt with ratios of numbers of atomsor molecules. So atoms took on a certain abstract

quality. If you imagined an alternate universe

in which the number of atoms per unit weightwas multiplied by a thousand, the behavior ofgases (and thus actual gases) would be completely

unaffected!

So if the absolute scale of an atom was never

important, should they be considered real at all? Orwere they ultimately like Leibnitz’s infinitesimals –just mathematical devices to model the behavior of

gases, without any robust reality?

This is the question Einstein was to settle

conclusively with his paper on Brownian motion.

To see this, let us set the stage with two moreideas from statistical mechanics.

13

Thermal Equilibrium

The most basic axiom of thermodynamics

states that two systems interacting with one anotherwill eventually approach the same temperature aftera sufficiently long time has passed. Boltzmannshowed that this principle could be derived fromstatistical mechanics under broad conditions.

Principle of Equipartition

This principle refers to a gas in which themolecules have internal degrees of freedom –internal structures which can move, oscillate, orvibrate according to mechanical forces within themolecule.

The principle of equipartition states that for a gasin thermal equilibrium, the average energy stored ineach available degree of freedom will be equal tothe overall kinetic energy of the molecule’s center

of mass.

With these two ideas, we can understandEinstein’s theory of Brownian motion.

14

� Einstein’s Idea : Brownian motion comes fromcollision of macroscopic particles with microscopicones.

Pollen

Is it true?How do you make it quantitative?

What are the observable consequences?

15

������������������������������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������������������������������

pollen

Einstein asked himself, “What happens if you treatlarge, complicated objects like grains of pollen

as if they were large, round gas molecules withan enormous mass and many, many degrees of

freedom? “ That is, treat the dissolved pollen like agas of extremely complicated spherical molecules

– in thermal equilibrium with the ambient water.With only this input, he could make a quantitative

prediction about the average motion of the grains.

16

A calculation

Let us consider the motion for the pollen grain inthe x direction, under the influence of the surroundingwater. We are going to write an equation for the timederivative of the quantity xx. Its time derivativewill have two pieces – a macrocopic piece, which weobtain an expression for by thinking of the sphere as aclassical object moving in continuous water, anda microscopic piece, which we obtain by thinking ofthe sphere as a complicated molecule, in thermal

equilibrium with the water.

d

dt[xx] = x2 + xx

In order to evaluate these quantities withconfidence, average the equation over a time scaleof many molecular collisions, so that the energy ofthe molecule can be continuous trajectory.

17

Microscopic term

Since the sphere is in thermal equilibrium withthe water, the average kinetic energy of its center

of mass must be 12RT/NA, in accordance with the

principle of equipartition.

So the first term on the right hand side is givenby

x2 = RT/(mNA)

where m is the mass of the pollen grain.

Here we have just used the ideal gas law, theequipartition principle, and the classical formula

for the kinetic energy: E = 12mv2.

18

Macroscopic term

To evaluate the second term, xx, we switch pointsof view and think of the pollen grain as a macroscopic

sphere, travelling at some speed through a viscousliquid.

By 1905, there was a well-known hydrodynamic

formula for the force on a sphere due to viscous

drag from a fluid. For a sphere of radius r,the force is proportional to the velocity of thesphere and the viscosity of the fluid, and given byF = mx = −6πηrx. So the macroscopic term inthe time derivative of xx is

xx = −6πηr

mxx

Here η is the dynamical viscosity coefficient.

19

Solving for the sphere’s motion

So we have the following differential equation for xx:

ddt

[xx] = RTmNA

− 6πηrm

[xx]

to which the solution (assuming x = 0 at t = 0) is:

xx = 1

2

ddt

[x2] RT6πηrNA

(

1 − exp{

− tt0

})

with t0 defined as the characteristic time scale

t0 ≡ m6πηr

.

Remember, we have used the equipartition

formula for x2, so this equation holds only in anaverage sense. Without knowledge of the state ofevery water molecule in the system, the actualbehavior of the grains appears random rather thandeterministic.

20

Features of the solution

At times longer than t0, we can drop the exponentialterm, to obtain

∆(t) ≡√

x2 + y2 + z2 =√

3|x| ∼√

RTπηr NA

√t

The middle equality is again an average, since motionsof the particle in the three spatial dimensions shouldbe statistically uncorrelated. We have obtained anexpression for the average distance from the origin

due to Brownian motion after a time t.

Note this behavior has the following desirablefeatures.

• The motion increases with temperature T ;

• it decreases with increasing viscosity;

• it also decreases with the size of the particles.

All these behaviors agree with the previouslyobserved behavior of Brownian motion.

21

What we learn from this

Notice that the magnitude of the motion directly

senses the discreteness of the molecules in the liquid;that is, it is inversely proportional to Avogadro’snumber NA.

In the approximation where the number ofmolecules in a gram of water is infinite, Brownianmotion does not occur at all! Einstein’s directdemonstration of the finiteness of NA disposedpermanently of the empiricists’ objection to themolecular theory of heat – that molecules were merelymathematical book-keeping devices.

22

Conclusion:

= means

HAPPY ANNIVERSARY, ALBERT!

23

Figure 1: Random trajectory of a pollen mote movingby Brownian motion in a liquid at room temperature.

24

Pollen

Figure 2: In the statistical theory of liquids, watermolecules move with random speeds in randomdirections. If a macroscopic object like a pollen moteis suspended in the water, some of the water moleculeswill occasionally hit it, and change its momentum.

25

pollengrains

Figure 3: Random trajectories of three pollen motesmoving by Brownian motion in water at roomtemperature.

26

Figure 4: A second random trajectory of a pollenmote moving by Brownian motion in a liquid at roomtemperature.

27

28