Random walkRandom walkfrom Einstein to the presentfrom Einstein to the present

Thomas SpencerThomas Spencer

School of MathematicsSchool of Mathematics

Leading Scottish BotanistLeading Scottish Botanist

Explored the coast of Explored the coast of

Australia and TasmaniaAustralia and Tasmania

Identified the nucleus ofIdentified the nucleus of

the cellthe cell

Robert Brown (1773-1858)

Around 1827 Brown made a systematic Around 1827 Brown made a systematic study of the “swarming motion” of study of the “swarming motion” of microscopic particles of pollen.microscopic particles of pollen.

This motion is now referred to as This motion is now referred to as Brownian movement. (Brownian motion).Brownian movement. (Brownian motion).

At first, “…I was disposed to believe that At first, “…I was disposed to believe that the minute spherical particles were in the minute spherical particles were in reality elementary units of organic bodies.”reality elementary units of organic bodies.”

Brown then tested plants that had been Brown then tested plants that had been dead for over a century. He remarks on dead for over a century. He remarks on the “vitality retained by these molecules so the “vitality retained by these molecules so long after the death of the plant”long after the death of the plant”

Later he tested: “rocks of all ages … Later he tested: “rocks of all ages … including a fragment of the Sphinx”including a fragment of the Sphinx”

Conclusion: origin of this motion was Conclusion: origin of this motion was physicalphysical, , notnot biological. biological.

Real Brownian MovementReal Brownian Movement

His careful experiments showed that His careful experiments showed that motion was motion was notnot caused by water currents, caused by water currents, light, evaporation or vibration.light, evaporation or vibration.

He could not explain the origin of this He could not explain the origin of this motion. motion.

Many later experiments by others - Many later experiments by others - Inconclusive. But by the late 1800’s the ideaInconclusive. But by the late 1800’s the ideaB-movement was caused by collisons withB-movement was caused by collisons withInvisible molecules gained some acceptance. Invisible molecules gained some acceptance.

Schematic Brownian movementSchematic Brownian movement

Brownian Motion explained in 1905 with the work of Albert Einstein

Title Title : “On the movement of small particles : “On the movement of small particles suspended in a stationary fluid as demanded suspended in a stationary fluid as demanded by the laws of kinetic theory”by the laws of kinetic theory”

MotivationMotivation: To justify the kinetic theory of : To justify the kinetic theory of atoms and molecules – and make atoms and molecules – and make quantitative quantitative predictions predictions

““In this paper it will be shown that according to In this paper it will be shown that according to the laws of molecular-kinetic theory of heat, the laws of molecular-kinetic theory of heat, bodies of a microscopically visible size bodies of a microscopically visible size suspended in a liquid must as a result of suspended in a liquid must as a result of thermal molecular motions, perform motion thermal molecular motions, perform motion visible under a microscope.”visible under a microscope.”

Although the idea of atoms goes back to the Although the idea of atoms goes back to the

Greeks and the kinetic theory to Boltzmann Greeks and the kinetic theory to Boltzmann

and Maxwell, there were many skeptics and and Maxwell, there were many skeptics and

questions:questions:

Were atoms real?Were atoms real?

How many molecules in 18 grams of water ? How many molecules in 18 grams of water ? – Avogadro number– Avogadro number

Einstein’s equations:Einstein’s equations:

B(t)B(t) = position of Brownian particle at time t. = position of Brownian particle at time t.

Distance [B(0), B(t)] =Distance [B(0), B(t)] =

T=temperature, r =radius of particle , T=temperature, r =radius of particle , = viscosity = viscosity

k= Boltzmann constantk= Boltzmann constant

Jean Perrin experimentally Jean Perrin experimentally

verified Einstein’s predictionsverified Einstein’s predictions

In his letter to Einstein:In his letter to Einstein:

““I did not believe it wasI did not believe it was

possible to study Brownian possible to study Brownian

motion with such precision”motion with such precision”

Accurate calculation ofAccurate calculation of

Avogadro numberAvogadro number

What is a random walk?What is a random walk?

Mathematical interlude:Mathematical interlude:

Steps (moves) in all directions are Steps (moves) in all directions are equallyequally likelylikely

(No drift)(No drift)

Each step Each step independentindependent of previous step. of previous step.

How far does an N step Random Walk go?How far does an N step Random Walk go?

Distance from its starting point = Distance from its starting point =

N N ~ t = time~ t = time

Random WalkRandom Walk

One Dimensional Random Walk

2D Random Walk

3D random walk3D random walk

Basic properties of Random WalkBasic properties of Random Walk

In In 2D, Random walk 2D, Random walk is is RecurrentRecurrent::

It It returnsreturns to its starting point to its starting point infinitelyinfinitely often. often.

In In 3D, 3D, Random walk Random walk is is TransientTransient::

After some time, walk will After some time, walk will Not Not return to its return to its starting pointstarting point..

Fractal dimension = Fractal dimension = 22

This means that This means that : :

In large cube of side In large cube of side L L with with LL33 points points

inside, a random walk visits inside, a random walk visits LL22 points points..

In mathematicsIn mathematics::

Brownian motion = Limit of Random walkBrownian motion = Limit of Random walkwith with infinitesimal,infinitesimal, independentindependent steps. steps.

Defined by Defined by Norbert WienerNorbert Wiener (1920’s) (1920’s)

A Brownian path, B(t), is a A Brownian path, B(t), is a continuouscontinuous function of time t, but it is very irregular.function of time t, but it is very irregular.

Crosses itself Crosses itself infinitelyinfinitely often in often in 22 or or 3D3D

In In 4D4D, two Brownian paths do , two Brownian paths do notnot cross. cross.

..

Louis BachelierLouis Bachelier (1870-1946)(1870-1946)

19001900 Thesis: Thesis: “Theorie de la Speculation”“Theorie de la Speculation”

..

Major new Ideas and results:Major new Ideas and results:

Market fluctuations described in terms ofMarket fluctuations described in terms ofBrownian motionBrownian motion

Brownian Motion has Normal distributionBrownian Motion has Normal distribution

Martingale theory, Chapman-Kolmogorov eqn’sMartingale theory, Chapman-Kolmogorov eqn’s

Bachelier-Wiener ProcessBachelier-Wiener Process..

Self-avoiding Walks or PolymersSelf-avoiding Walks or Polymers

In 1940’s, Paul Flory, chemist, studied longIn 1940’s, Paul Flory, chemist, studied long

chains of monomers – polymers chains of monomers – polymers

Each monomer Each monomer ~~ step of the walk. step of the walk.

Except: monomers Except: monomers cannotcannot occupy the same occupy the same

space – excluded volume effect space – excluded volume effect

Polymer made of 500 monomersPolymer made of 500 monomers

What is the diameter of polymer What is the diameter of polymer made of N monomers?made of N monomers?

Each Each polymerpolymer with with NN monomers monomers equally likelyequally likely

In 2D: Diameter = C NIn 2D: Diameter = C N3/43/4 ? ?Fractal dimension = 4/3 Fractal dimension = 4/3

In 3D: Diameter = C NIn 3D: Diameter = C N , , .6 ?? .6 ??

Above Above 4D4D , polymer , polymer ~~ random walk , random walk , = .5 = .5

Self-Avoiding path with 20,000 stepsSelf-Avoiding path with 20,000 steps

Branched Polymer Branched Polymer

Branched polymer, N = 10,000Branched polymer, N = 10,000

Each branched polymer formed with N edges or Each branched polymer formed with N edges or monomers is assumed to be monomers is assumed to be equally likelyequally likely..

TheoremTheorem (D. Brydges and J. Imbrie, 2002): (D. Brydges and J. Imbrie, 2002): InIn 3D3D

DiameterDiameter of BP of BP = = C C NN = # monomers. = # monomers.

Supersymmetry used to prove dimensional Supersymmetry used to prove dimensional reduction.reduction. Problem is Problem is unsolvedunsolved in 2, 5, 6, 7 dimensions. in 2, 5, 6, 7 dimensions.

SLE revolution in 2 Dimensions: SLE revolution in 2 Dimensions:

Charles LCharles Lööwnerwner, , 1920’s , studied 1920’s , studied 2D2D conformal conformalmappings using differential equations. mappings using differential equations.

SLESLE = = Brownian motionBrownian motion + + LLööwner’s wner’s equationequation

Oded Schramm, Greg Lawler and Wendelin Oded Schramm, Greg Lawler and Wendelin Werner (2000 – present)Werner (2000 – present)

Solved: Solved: manymany problems in the geometry ofproblems in the geometry ofBrownian paths, percolation, loop erased,Brownian paths, percolation, loop erased,Walks, CFT….Walks, CFT….

The boundary of Brownian path in The boundary of Brownian path in 2D 2D has has fractal dimension = 4/3. (LSW 2000) fractal dimension = 4/3. (LSW 2000)

Driving in Manhattan or Driving in Manhattan or Quantum diffusionQuantum diffusion

(Model due to John Cardy and others)(Model due to John Cardy and others)

Description of the model:Description of the model:

Obstructions at street corners appearObstructions at street corners appear

randomlyrandomly with probability = with probability = pp , , 0<p<1 0<p<1

DriverDriver is a is a robotrobot and follows the streets and and follows the streets and

turns turns onlyonly at obstructions. at obstructions.

This model is This model is equivalentequivalent to a model of a to a model of a

quantumquantum electron in 2D, interacting with electron in 2D, interacting with Impurities (obstructions).Impurities (obstructions).

p=0.5

p=0.25

Conjectures (for Conjectures (for pp >0): >0): All paths eventually form a loop.All paths eventually form a loop. (known for (known for pp>1/2)>1/2)

Electron is trapped – Electron is trapped – no conductionno conduction..

If obstructions are rare, loops are If obstructions are rare, loops are extremelyextremely long. long.

Paths behave like random walks for a very long time: Paths behave like random walks for a very long time:

Thus electron Thus electron diffusesdiffuses for a long time before it is for a long time before it istrappedtrapped. .

CommentsComments

If If pp ==1/101/10 the average length of the loop the average length of the loop ~ ~ 10104040

Numerical computations are not reliable for Numerical computations are not reliable for pp <1/4<1/4..

Most paths are too long for modern computers to Most paths are too long for modern computers to check whether a path eventually loops back.check whether a path eventually loops back.

In In 3D,3D, expectexpect that most paths do not close – that most paths do not close – motion is diffusive - motion is diffusive - like random walklike random walk..

Breathing and Brownian MotionBreathing and Brownian Motion

Lung surface has a complicated fractal structure.Lung surface has a complicated fractal structure.

Think of Think of oxygen oxygen molecules moving aboutmolecules moving about

through collisions like a through collisions like a Brownian pathBrownian path..

What is the What is the optimaloptimal shape of a surface for it to shape of a surface for it to

absorb oxygen most efficiently?absorb oxygen most efficiently?

Where would a Brownian molecule most likely Where would a Brownian molecule most likely

strike a surface? (Harmonic measure)strike a surface? (Harmonic measure)

If you make the surface too rough, If you make the surface too rough,

(fractal dimension too high), the Brownian (fractal dimension too high), the Brownian

paths will be unable to hit most of the surface. paths will be unable to hit most of the surface.

Jean BourgainJean Bourgain - -

Fractal dimension of a surface that BM can Fractal dimension of a surface that BM can

hit cannot be too close to 3.hit cannot be too close to 3.

Tom WolffTom Wolff - showed that there - showed that there existexist

surfaces of fractal dimension bigger than 2 surfaces of fractal dimension bigger than 2

which are accessible to a Brownian path.which are accessible to a Brownian path.

Conjecture:Conjecture:

The largest fractal dimension of a surface The largest fractal dimension of a surface

accessible to Brownian path is 2.5accessible to Brownian path is 2.5

(Peter Jones).(Peter Jones).

AcknowledgementsAcknowledgements

Thanks to: Joel Lebowitz and Michael LossThanks to: Joel Lebowitz and Michael Loss

And to: Thomas Uphill and Michelle And to: Thomas Uphill and Michelle

HugueninHuguenin

a c

b (t)

via Conformal map

a b c

B(at)

LÖWNER

Some conventional wisdom:Some conventional wisdom:

ccertain ertain quantum field theoriesquantum field theories are equivalent to are equivalent to a gas of Brownian paths in 4D.a gas of Brownian paths in 4D.

InteractionInteraction occurs when the paths occurs when the paths intersectintersect. .

No interaction in 4D ? ? .No interaction in 4D ? ? .

Model is not interacting unless embedded inModel is not interacting unless embedded in

non-abelian gauge theorynon-abelian gauge theory