Einstein and Brownian MotionorHow I spent My Spring Break(not in Fort Lauderdale)C. JuiUndergraduate SeminarMarch 25, 2004I will make available this presentation athttp://www.physics.utah.edu/~jui/brownian
AcknowledgmentsMy thanks to:Sid Rudolph, the director of the ACCESSprogram, which is designed to integratewomen into science, mathematics, andengineering careers.Gernot Laicher, the director of ElementaryLaboratory, who also prepared the micro-sphere suspension and took the videosequence of Brownian motion.
Acknowledgments (continued)Also thanks to:Lynn Monroe and Dr. Wilson of theKen-A-vision Company who loaned usthe T-1252 microscope which we used tomake the measurement of Avogadro’sNumber from Brownian Motion
Outline• What is Brownian Motion• The phenomenon 1827-1000• Einstein’s paper of 1905• Langevin’s Complete Derivation• My Science Fair Project (How I spent mySpring Break)• Epilogue
What is Brownian Motion?• #1 answer from Google (Dept. of Statistics)http://galton.uchicago.edu/~lalley/Courses/313/WienerProcess.pdf
Other answersFrom computer Science at Worcester Polytechnic Inst.http://davis.wpi.edu/~matt/courses/fractals/brownian.htmlBrownian Motion is a line that will jump up and down a random amount andsimular to the "How Long is the Coast of Britain?" problem as you zoom in onthe function you will discover similar patterns to the larger function.The two images above are examples of Brownian Motion. The first being afunction over time. Where as t increases the function jumps up or down avarying degree. The second is the result of applying Brownian Motion to the xy-plane. You simply replace the values in random line that moves around the page.
Electrical Engineering• A less commonly referred to color of noiseis brown noise. This is supposed simulateBrownian motion a kind of randommotion that shifts in steady increments.Brown noise decreases in power by 6 dBper octave.
The source of Confusion• There are two meanings of the termBrownian motion:– the physical phenomenon that minute particlesimmersed/suspended in a fluid will experiencea random movement– the mathematical models used to describe thephysical phenomenon.• Quite frequently people fail to make thisdistinction
Robert Brown• Robert Brown’smain claim to fameis his discovery ofthe cell nucleuswhen looking atcells from orchidsunder hismicroscope20 orchid epidermal cellsshowing nuclei (and 3 stomata)seen under Brown’s originalmicroscope preserved by theLinnean Society London
Brown’s Microscope• And Brownian motion of milk globules in waterseen under Robert Brown’s microscope
Brown’s Observations• At first Brown suspected that he might have beenseeing locomotion of pollen grains (I.e. they movebecause they are alive)• Brown then observed the same random motion forinorganic particles…thereby showing that themotion is physical in origin and not biological.• Word of caution for the Mars Explorationprogram: Lesson to be learned here from Brown’scareful experimentation.
1827-1900• Desaulx (1877):– "In my way of thinking the phenomenon is aresult of thermal molecular motion (of theparticles) in the liquid environment”• G.L. Gouy (1889):– observed that the "Brownian" movementappeared more rapid for smaller particles
F. M. Exner (1900)• F.M. Exner (1900)– First to make quantitative studies of thedependence of Brownian motion on particlesize and temperature– Confirmed Gouy’s observation of increasedmotion for smaller particles– Also observed increased motion at elevatedtemperatures
Louis Bachelier (1870-1946)• Ph.D Thesis (1900): "Théoriede la Spéculation" Annales delEcole normale superiure• Inspired by Brownian motionhe introduced the idea of“random-walk” to model theprice of what is now called abarrier option (an option whichdepends on whether the shareprice crosses a barrier).
Louis Bachelier (continued)• The “random-walk” model is formally known as“Wiener (stochastic) process” and often referredto as “Brownian Motion”• This work foreshadowed the famous 1973 paper:Black F and Scholes M (1973) “The Pricing ofOptions and Corporate Liabilities” Journal ofPolitical Economy 81 637-59• Bachelier is acknowledged (after 1960) as theinventor of Mathematical Finance (andspecifically of Option Pricing Theory)
Black and Scholes• Fischer Black died in 1995• Myron Scholes shared the1997 Nobel Prize ineconomics with RobertMerton– New Method forCalculating the prizeofderivatives
Albert Einstein• Worked out a quantitativedescription of Brownianmotion based on theMolecular-Kinetic Theoryof Heat• Published as the third of 3famous three 1905 papers• Awarded the Nobel Prizein 1921 in part for this.
Einstein’s 1905 papers1. On a Heuristic Point of View on the Creationand Conversion of Light (Photo-Electric Effect)http://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/eins2. On the Electrodynamics of Moving Bodies(Theory of Special Relativity)http://www.fourmilab.ch/etexts/einstein/specrel/www/3. Investigation on the Theory of the BrownianMovementhttp://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/einspdf
Historical Context• Einstein’s analysis of Brownian Motionand the subsequent experimentalverification by Jean Perrin provided 1st“smoking gun” evidence for the Molecular-Kinetic Theory of Heat• Kinetic Theory is highly controversialaround 1900…scene of epic battles betweenits proponents and its detractors
Molecular-Kinetic Theory• All matter are made of molecules (or atoms)• Gases are made of freely moving molecules• U (internal energy) = mechanical energy of theindividual molecules• Average internal energy of any system:〈U〉=nkT/2, n = no. of degrees of freedom• Boltzmann: Entropy S=klogW where W=no. ofmicroscopic states corresponding to a givenmacroscopic state
Ludwig Boltzmann (1844-1906)Committedsuicide in 1906.Some think thiswas because ofthe viciousattacks hereceived fromthe ScientificEstablishmentof the Day forhis advocacy ofKinetic Theory Boltzmann’s tombstonein Vienna
Einstein’s Paper• In hindsight Einstein’s paper of 1905 onBrownian Motion takes a more circuitous routethan necessary.• He opted for physical arguments instead ofmathematical solutions• I will give you the highlights of the paper ratherthan the full derivations• We will come back to a full but shorterderivation of Paul Langevin (1908)
• Einstein reviews the Law of OsmoticPressure discovered by J. van’t Hoffwho won the Nobel Prize inChemistry for this in 1901In a dilute solution:Section 1: Osmotic PressureνπNRT=π = osmotic pressureν = solute concentrationN = Avogadro’s numberR = gas constantT = absolute temperature
Section 1 (continued)• Einstein also argues that from the pointof view of the Kinetic Theory the Lawof Osmotic Pressure should applyequally to suspension of small particles
Section 2• Einstein derives the Law of OsmoticPressure as a natural consequence ofStatistical Mechanics– The law minimizes the Helmholtz Free Energywith entropy calculated following Boltzmann’sprescription
Section 3: Diffusion• Using Statistical Mechanics (minimizing freeenergy) Einstein shows that a particle (insuspension) in a concentration gradient (in x) willexperience a force K given (in magnitude) byxNRTxK∂∂=∂∂=νπν• This force will start a flow of particles against thegradient.
Diffusion (continued)• Assuming a steady state flow (in a constantgradient and in a viscous medium) the particleswill reach terminal velocity ofaKVTπη6=Here π = 3.1415.. η = viscosity of fluid mediuma = radius of spherical particles executing Stokesflow and experiencing a resistive force ofVaFR πη6−=
Diffusion (continued)• The resulting flux of particles is then given byaKxDπηνν6=∂∂≡ΦResulting in a definite prediction for the diffusionconstant D given byaNRTDπη61=This result a prediction of Kinetic Theory canbe checked experimentally in Brownian Motion!
Section 4: Random Walk• Einstein then analyzes the Brownian Motionof particles suspended in water as a 1-drandom walk process.• Unaware of the work of Bachelier hisversion of random walk was very elementary• He was able to show with his own analysisthat this random walk problem is identical tothe 1-d diffusion problem
Random Walk (continued)• The 1-d diffusion equation is22),(),(xtxfDttxf∂∂=∂∂tDtxDftxf)4/exp(4),(20 −=π• This equation has the Green’s Function (integralkernel) given by• Which is then the expected concentration ofparticles as a function of time where all startedfrom the origin.
Section 5: Average x2• Taking the initial position of each particle to be itsorigin then the average x2is then given bytaNRTDtx ==πη3122• Einstein finishes the paper by suggesting that thisdiffusion constant D can be measured by followingthe motion of small spheres under a microscope• From the diffusion constant and the knownquantities R η and a one can determineAvogadro’s number N
Jean Perrin (1870-1942)• Using ultra-microscope JeanPerrin began quantitative studies ofBrownian motion in 1908• Experimentally verified Einstein’sequation for Brownian Motion• Measured Avogadro’s number tobe N = 6.5-6.9x1023• From related work he was the firstto estimated the size of watermolecules• Awarded Nobel Prize in 1926
Paul Langevin (1872-1946)• Most known for:– Developed the statisticalmechanics treatment ofparamagnetism– work on neutron moderationcontributing to the success to thefirst nuclear reactor• The Langevin Equation andthe techniques for solving suchproblems is widely used ineconometrics
η = viscosity of waterFext is a “random” force on the particleLangevin Equation• In 1908 Paul Langevin developed a more directderivation based on a stochastic (differential)equation of motion. We start with Newton’s 2ndLaw (Langevin Equation):extFrr+−=dtdadtdM πη622
• We now take the dot (scalar product) of theequation of motion by r :Scalar product by r( )extFrrrrr •+•−=•dtdadtdM πη622• Next we re-express the above equation interms of r2instead.
Change of Variable to r2( ) ( )( )•+=•=⇒•=•=⇒•==2222222222222dtddtddtddtddtrddtddtddtrdrrrrrrrrrrrrr( )( )22222222121dtddtrddtddtrddtdrrrrr−=•=•
• Rearranging the equation and denoting ||dr/dt||2=V2• Next we take the average over a largenumber of particles (“ensemble average”denoted by 〈 〉 ) and using u ≡ 〈r2〉• The equation of motion now becomes:Ensemble Average( ) ( ) ( )extFrr•+−=−dtrdadtdMdtrdM22222321πη
• The last term on the right vanishes becauseFext is a “random” force not correlated withthe position of the particle.• By Equi-partition Theorem we have〈½MV2〉 = nkT/2 (a constant!) where n isthe number of spatial dimensions involvedand k is again the Boltzmann Constant.The Physics!!!extFr •+=+ 222212321MVdtduadtudM πη0PHYSICS!!!
⇒ A 2ndorder linear inhomogenousODE:Solving the Differential Equationκτ=+dtdudtud 122aM πητ 6/= 22/2 VMnkT ==κwhere and• Using MAPLE to solve this :( ) 212)/exp( CtCtru +−−=≡ ττκτ
1. 〈r2〉 = 0 at t = 0• Assuming initial position of eachparticle to be its origin.2. d〈r2〉/dt = 0 at t = 0• At very small t (I.e. before any subsequentcollisions) we have ri2= (Vit)2where Viis thevelocity the ithparticle inherited from aprevious collision.⇒ 〈r2〉 = 〈V2〉 t2⇒ d〈r2〉/dt |t=0 = 0Initial Conditions
Applying Initial Conditions• We arrive at the solutionκτκτττ−=⇒=+==⇒=+−===110212210200CCdtrdCCCCrtt[ ]τττκτ −−+= )/exp(2ttr
Langevin: t << τ case• Expanding the exponential in Taylor seriesto 2ndorder in t. Note 0thand 1storder termscancel with the two other terms.2222221...211tVtttttr=≈−−+−+ →<<κττττκττ
Langevin: t >> τ case• Taking the other extreme which is the case ofinterest( ) ( )tankTtaMMnkTtttr=×=≈− →>>πηπηκττκττ3622• Which is the same as Einstein’s result but with anextra factor of n (Note Einstein’s derivations werefor a 1-d problem)
My Science Fair Project(How I Spent my Spring Break)• We are setting up a Brownian motion experimentfor UGS 1430 (ACCESS summer program) andfor PHYCS 2019/2029• Will use inexpensive Digital microscopes with a100X objective• Use 1 µm diameter (3% uniformity) polystyrene• Did a “fun” run using an even cheaper digitalscope with a 40X objective
The Tool• Used a Ken-A-VisionT-1252 DigitalMicroscope loaned tous by the company• Up to 40X objectve• USB interface forvideo capture
Data Analysis• Followed 14 particles for 80seconds• digitized x and y position every10 seconds using Free package“DataPoint”http://www.stchas.edu/faculty/gcarlson/physics/datapoint.htm• Raw data for particle #9 shownto the right (x, y coordinates inpixels)t(s) x y0 238 41410 246 40220 247 39630 246 39740 250 40550 238 40360 228 41470 227 40080 225 39790 241 409100 234 408110 236 408120 238 410
Data Analysis (continued)• Some “bulk flow” was observed: 〈x〉 and〈y〉 were non-zero and changed steadilywith time. For pure Brownian motion theseshould be constant AND zero• To account for the flow we used σx2=〈x2〉-〈x〉2and σy2= 〈y2〉-〈y〉2instead of just 〈x2〉 and〈y2〉 in the analysis
The answer• Assuming 70F temp (and associated viscosity)we get NA= 5.1x1023
Epilogue• Brownian motion is a topic that touchesmany different disciplines• Einstein’s contribution was to use Brownianmotion as a vehicle to prove the Molecular-Kinetic Theory of Heat– Often misunderstood by non-physicists• Brownian motion can be investigatedexperimentally for less than $500!!!