Brownian Motion Problem_Random Walk

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    49RESONANCE August 2005

    GENERAL ARTICLE

    Brownian Motion Problem: Random Walk andBeyond

    Shama Sharma and Vishwamittar

    Shama Sharma is

    currently working as

    lecturer at DAV College,

    Punjab. She is working on

    some problems in

    nonequilibrium statistical

    mechanics.

    Vishwamittar is professor

    in physics at Panjab

    University, Chandigarh

    and his present research

    activities are in

    nonequilibrium statistical

    mechanics. He has written

    articles on teaching of

    physics, history and

    philosophy of physics.

    Keywords

    Brownian motion, Langevins

    theory, random walk, stochas-

    tic processes, non-equilib rium

    statistical mechanics.

    A brief account of developments in the experi-

    mental and theoretical investigations of Brown-

    ian motion is presented. Interestingly, Einstein

    who did not like God's game of playing dice for

    electrons in an atom himself put forward a theory

    of Brownian movement allowing God to play thedice. The vital role played by his random walk

    model in the evolution of non-equilibrium statis-

    tical mechanics and multitude of its applications

    is highlighted. Also included are the basics of

    Langevin's theory for Brownian motion.

    Introduction

    Brownian motion is the temperature-dependent perpet-

    ual, irregular motion of the particles (of linear dimen-

    sion of the order of 106m) immersed in a uid, causedby their continuous bombardment by the surrounding

    molecules of much smaller size (Figure 1). This eectwas rst reported by the Dutch physician, chemist andengineer Jan Ingenhousz (1785), when he found thatne powder of charcoal oating on alcohol surface ex-

    hibited a highly random motion. However, it got thename Brownian motion after Scottish botanist RobertBrown, who in 1828-29 published the results of his ex-tensive studies on the incessant random movement of

    tiny particles like pollen grains, dust and soot suspendedin a uid (Box 1). To begin with when experimentswere performed with pollen immersed in water, it was

    thought that ubiquitous irregular motion was due to thelife of these grains. But this idea was soon discarded asthe observations remained unchanged even when pollenwas subjected to various killing treatments, liquids other

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    GENERAL ARTICLE

    Box 1. Observing the Brownian Motion

    One can observe Brownian movement by looking through an optical microscope with magnification 103

    104 at a well illuminated sample of some colloidal solution (say milk drops put into water) or smoke

    particles in air. Alternately, we can mount a small mirror (area 1 or 2 mm2) on a fine torsion fiber capable

    of rotation about a vertical axis (as in suspension type galvanometer) in a chamber having air pressure of

    about 102 torr and note the movement of the spot of light reflected from the mirror. The trace of the light

    spot is manifestation of angular Brownian motion of the mirror.

    It may be pointed out that movement of dust particles in the air as seen in a beam of light through a window

    is not an example of Brownian motion as these particles are too large and the random collisions with air

    molecules are neither much imbalanced nor strong enough to cause Brownian movement.

    Figure 1. Brownian motion

    of a fine particle immersed

    in a fluid.

    than water were used and ne particles of glass, petri-ed wood, minerals, sand, etc were immersed in variousliquids. Furthermore, possible causes like vibrations ofbuilding or the laboratory table, convection currents inthe uid, coagulation of particles, capillary forces, in-ternal circulation due to uneven evaporation, currentsproduced by illuminating light, etc were considered butwere found to be untenable.

    An early realization of the consequences of the stud-ies on Brownian motion was that it imposes limit onthe precision of measurements by small instruments asthese are under the inuence of random impacts of thesurrounding air.

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    GENERAL ARTICLE

    1These are collected in the book

    Alber t Einstein : Investigations

    on the theory of Brownian

    Movementedited with notes by

    R Furth (Dover Publications,

    New York,1956).

    2 Review article entitled Brown-

    ian Movement and Molecular

    Reality based on his work has

    been transla ted into English by

    F Soddy and pub lished by Tay-

    lor & Francis,London in 1910.

    In 1863, Wienerattributed Brownian

    motion to molecular

    movement of the

    liquid.

    experiments.

    In 1863, Wiener attributed Brownian motion to mole-cular movement of the liquid and this viewpoint wassupported by Delsaux (1877-80) and Gouy (1888-95).On the basis of a series of experiments Gouy convinc-ingly ruled out the exterior factors as causes of Brown-ian motion and argued in favour of contribution of thesurrounding uid. He also discussed the connection be-tween Brownian motion and Carnot's principle and there-by brought out the statistical nature of the laws of ther-modynamics. Such ideas put Brownian motion at theheart of the then prevalent controversial views about

    philosophy of science. Then, in 1900 Bachelier obtaineda diusion equation for random processes and thus, atheory of Brownian motion in his PhD thesis on stockmarket uctuation. Unfortunately, this work was notrecognized by the scientic community, including hissupervisor Poincare, because it was in the eld of eco-nomics and it did not involve any of the relevant physicalaspects.

    Eventually, Einstein in a number of research publica-

    tions beginning in 1905 put forward an acceptable modelfor Brownian motion1 . His approach is known as `ran-dom walk' or `drunkards walk' formalism and uses the

    uctuations in molecular collisions as the cause of Brown-ian movement. His work was followed by an almost sim-ilar expression for the time dependence of displacementof the Brownian particle by Smoluchowski in 1906, who

    started with a probability function for describing themotion of the particle. In 1908, Langevin gave a phe-nomenological theory of Brownian motion and obtained

    essentially the same formulae for displacement of theparticle. Their results were experimentally veri ed byPerrin (1908 and onwards)2 using precise measurementson sedimentation in colloidal suspensions to determine

    the Avogadro's number. Later on, more accurate val-ues of this constant and that of Boltzmann constant kwere found out by many workers by performing similar

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    GENERAL ARTICLE

    The correctness ofthe random walk

    model and

    Langevins theory

    made a very strong

    case in favour of

    molecular kinetic

    model of matter.

    The techniquesdeveloped for the

    theory of Brownian

    motion form

    cornerstones for

    investigating a variety

    of phenomena.

    The correctness of the random walk model and Lange-vin's theory made a very strong case in favour of mole-cular kinetic model of matter and unleashed a wave of

    activity for a systematic development of dynamical the-ory of Brownian motion by Fokker, Planck, Uhlenbeck,Ornstein and several other scientists. As such, during

    the last 100 years, concerted eorts by a galaxy of physi-cists, mathematicians, chemists, etc have not only pro-vided a proper mathematical foundation to the physical

    theory but have also led to extensive diverse applicationsof the techniques developed.

    The two basic ingredients of Einstein's theory were: (i)the movement of the Brownian particle is a consequenceof continuous impacts of the randomly moving surround-ing molecules of the uid (the `noise'); (ii) these impactscan be described only probabilistically so that time evo-lution of the particle under observation is also proba-bilistic in nature. Phenomena of this type are referred toas stochastic processes in mathematics and these are es-sentially non-equilibrium or irreversible processes. Thus,Einstein's theory laid the foundation of stochastic mod-eling of natural phenomena and formed the basis ofdevelopment of non-equilibrium statistical mechanics,wherein generally, the Brownian particle is replaced bya collective property of a macroscopic system. Con-sequently, the techniques developed for the theory ofBrownian motion form cornerstones for investigating avariety of phenomena (in dierent branches of scienceand engineering) that have their origin in the eect ofnumerous unpredictable and may be unobservable eventswhose individual contribution to the observed feature is

    negligible, but collective impact is observed in the formof rather rapidly varying stochastic forces and dampingeect (see Box2).

    The main mathematical aspect of the early theories ofBrownian motion was the `central limit theorem', whicham- ounted to the assumption that for a suciently large

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    GENERAL ARTICLE

    Box 2. Brownian Motors

    One of the topics of current research activity employing these techniques is referred to as Brownian motors.

    This phrase is used for the systems rectifying the inescapable thermal noise to produce unidirectional

    current of particles in the presence of asymmetric potentials. Their study is relevant for understanding the

    physical aspects involved in the movement of motor pro