Biased Brownian motion - rudi/sola/seminar-  · PDF filedescription of Brownian motion....

Click here to load reader

  • date post

    03-Sep-2018
  • Category

    Documents

  • view

    218
  • download

    0

Embed Size (px)

Transcript of Biased Brownian motion - rudi/sola/seminar-  · PDF filedescription of Brownian motion....

  • SEMINAR

    Biased Brownian motion

    Author: Mitja Stimulak

    Mentor: prof. dr. Rudolf Podgornik

    Ljubljana, January 2010

    Abstract

    This seminar is concerned with Biased Brownian motion. Firstly Brownian motionand its properties are described. Then we follow Einsteins steps towards mathematicaldescription of Brownian motion. Brownian motor and a physical principle that explainsit, Biased Brownian motion, are introduced next. In the second part we turn our at-tention to an experiment with a single Brownian particle. There we briefly describethe setup of the experiment and explain how optical trap works. Particle dynamics isdiscussed with the help of two models one originating from Langevins equation andanother from more general Fokker Planck equation.

  • Contents

    1 Introduction 1

    2 Brownian motion 22.1 Einsteins Explanation of the Brownian Movement - Probability Density Dif-

    fusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Brownian motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Biased Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    3 Periodic forcing of a Brownian particle 73.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Maximum trapping force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Particle dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    3.4.1 Phase-Locked regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4.2 Phase-slip regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4.3 Diffusive regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

    4 Langevin equation-Stochastic Differential Equations 12

    5 Fokker-Planck equation 15

    6 Conclusion 17

    7 References 17

    1 Introduction

    Brownian motion, in some systems can also be described as noise, is a continuous motion ofsmall particles.

    Noise is unavoidable for any system in thermal contact with its surroundings. In tech-nological devices, it is typical to incorporate mechanisms for reducing noise to an absoluteminimum. An alternative approach is emerging, however, in which attempts are made toharness noise for useful purposes.

    To some extend this can be achieved by inducing some additional potential in systemfrom outside. This process is also known as a Biased Brownian motion.

    We are going to study basic principles of Biased Brownian motion. Specifically we aregoing to look into different regimes of moving of a single Brownian particle that is periodicallyforced by an optical trap. We will show that these regimes can be derived and described bymathematical formula.

    1

  • 2 Brownian motion

    First detailed account of Brownian motion was given by eminent botanist Robert Brown in1827 while studying the plant life of the South Seas. He examined aqueous suspensions ofpollen grains of several species under microscope and found out that in all cases the pollengrains were in rapid oscillatory motion.[1]

    It is interesting that he initially thought that the movement was peculiar to the malesexual cells of plants. He quickly corrected himself when he observed that the motion wasexhibited by both organic and inorganic matter in suspension.[1]

    -16

    -14

    -12

    -10

    -8

    -6

    -4

    -2

    0

    2

    4

    -10 -8 -6 -4 -2 0 2 4 6

    1000 steps-1 walk

    Figure 1: Simulation of a 2D random walk of a Brownian particle. Particle started at theposition (0,0) and made one thousand random steps. At each step the step length l and thedirection of moving were randomly chosen.1 < l > 1 and [0, 2]

    Following Browns work there were many years of speculation as to the cause of thephenomenon, before Einstein made conclusive mathematical predictions of a diffusive effectarising from the random thermal motion of particles in suspension. Before Einsteins work,very detailed experimental investigation was made by Louis Georges Gouy a French physicist.His conclusions may be summarised by the following seven points[1]:

    1. The motion is very irregular, composed of translations and rotations, and the trajec-tory appears to have no tangent.2. Two particles appear to move independently, even when they approach one another towithin a distance less than their diameter.3. The smaller the particles, the more active the motion.4. The composition and density of the particles have no effect.5. The less viscous the fluid, the more active the motion.6. The higher the temperature, the more active the motion.7. The motion never ceases.

    2

  • 2.1 Einsteins Explanation of the Brownian Movement - Probabil-ity Density Diffusion Equations

    Einsteins analysis was presented in a series of publications, including his doctoral thesis,that started in 1905 with a paper in the journal Annalen der Physik.[2]

    Einstein derived his expression for the mean-square displacement of a Brownian particleby means of a diffusion equation which he constructed using the following assumptions:[1]

    (i) Each individual particle executes a motion which is independent of the motion of allother particles in the system.(ii) The motion of a particle at one particular instant is independent of the motion of thatparticle at any other instant, provided the time interval is large enough.

    To actually develop the diffusion equation we introduce a time interval , which is largeenough so the second assumption is valid, but small compared to the observable time intervals.Now lets suppose that there are particles in a liquid per unit volume at x at time t. Aftera time has elapsed we consider a volume element of the same size at point x.

    The probability of a particle entering from neighbouring element to x is a function of adistance between x and x, = x x and . We denote this probability by = (, ).Since the particle must come from some volume element, the density at time is [1]

    (x, t+ ) =

    (x+ , t)(, )d (1)

    Now positive and negative displacements are equiprobable. We have an unbiassed randomwalk. Thus the function (, ) satisfies (, ) = (, ).We now suppose that is verysmall so that we can expand the left hand side of Eq. (1) in powers of .

    (x, t+ ) = (x, t) +

    t+O( 2) (2)

    Furthermore, we develop (x+ , t) in powers of small displacement , so obtaining

    (x+ , t) = (x, t) + (x, t)

    x+

    2

    2!

    2(x, t)

    x2+ ... (3)

    The integral equation (1) then becomes:

    +

    t=

    (, )d +

    x

    (, )d +2

    x2

    2

    2!(, )d + ... (4)

    Since (, ) is a probability density function and (, ) = (, ) we must have [1]

    (, )d = 1

    3

  • (, )d = = 0

    and

    2(, )d = 2,

    where the overbar denotes the mean over displacement for each particle. We suppose thatall higher order terms such as 4 are at least of the order 2. Hence Eq. (4) simplifies to

    t=2

    x2

    2

    2!(, )d (5)

    If we set

    D 1

    2

    2!(, )d =

    2

    2(6)

    then Eq. (5) can be written as

    t= D

    2

    x2, (7)

    We developed the diffusion equation in one dimension. Conditions on the solution are

    (x, 0) = (x) (8)

    Since is a probability distribution we also have

    (x, t) = 1 (9)

    This is well known problem and solution to it in one dimension is

    (x, t) =1

    4Dtex

    2/4Dt (10)

    Einstein determined the translational diffusion coefficient D in terms of molecular qualitiesas follows.Let us suppose that the Brownian particles are in a field of force F(x) (the gravitational filedof earth); then the Maxwell-Boltzmann distribution for the configuration of the particlesmust set in.

    = 0eU (11)

    If the force is constant, as would be true when gravity acts on the Brownian particles, thepotential energy is

    U = Fx , (12)

    where we suppose that the particles are so few that their mutual interaction may be neglected(like an atmosphere of Brownian particles). Now the velocity of a particle that is inequilibrium under the action of the applied force and viscosity is given by Stokess law:

    =F

    6a(13)

    4

  • Number of particles crossing unit area in unit time (current density of particles) is then

    J =F

    6a(14)

    For a normal diffusion process, particles cannot be created or destroyed, which means theflux of particles into one region must be the sum of particle flux flowing out of the surroundingregions. This can be captured mathematically by the continuity equation.

    t+ divJ = 0 (15)

    We combine equations (Eq.7) and (Eq.15) integrate over x and we get

    Dx

    =F

    6a(16)

    From equations (Eq.11) and (Eq.12) we get

    1

    x= F

    kBT(17)

    Comparing equations (Eq.16) and (Eq.17) we obtain

    D =kBT

    6a(18)

    or by definition2

    2 D = kBT

    6a(19)

    We developed Einsteins formula for the translational diffusion coefficient. One can seethat 2 .

    T is the temperature, is the viscosity of the liquid and a is the size of the particle. Thisequation implies that large particles would diffuse more gradually than molecules, makingthem easier to measure. Moreover, unlike a ballistic particle such as a billiard ball, thedisplacement of a Brownian particle would not increase linearly with time but with thesquare root of time.[3]

    Experimental observation confirmed the numerical accuracy of Einsteins theory. Thismeans that we understand Brownian motion just as a consequence of the same thermal mo-tion that causes a gas to exert a pressure on the container that confines it. We unde