Brownian motion 1
Brownian motion
This is a simulation of the Brownian motion of a
big particle (dust particle) that collides with a
large set of smaller particles (molecules of a gas)
which move with different velocities in different
random directions.
This is a simulation of the Brownian motion of 5
particles (yellow) that collide with a large set of
800 particles. The yellow particles leave 5 blue
trails of random motion and one of them has a red
velocity vector.
Brownian motion or pedesis (from Greek: πήδησις Pɛɖeːsɪs
"leaping") is the presumably random moving of particles suspended in
a fluid (a liquid or a gas) resulting from their collision by the
fast-moving atoms or molecules in the gas or liquid. The term
"Brownian motion" can also refer to the mathematical model used to
describe such random movements, which is often called a particle
theory.
In 1827, the botanist Robert Brown, looking through a microscope at
particles found in pollen grains in water, noted that the particles moved
through the water but was not able to determine the mechanisms that
caused this motion. Atoms and molecules had long been theorized as
the constituents of matter, and many decades later, Albert Einstein
published a paper in 1905 that explained in precise detail how the
motion that Brown had observed was a result of the pollen being
moved by individual water molecules. This explanation of Brownian
motion served as definitive confirmation that atoms and molecules
actually exist, and was further verified experimentally by Jean Perrin in
1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his
work on the discontinuous structure of matter" (Einstein had received
the award five years earlier "for his services to theoretical physics"
with specific citation of different research). The direction of the force
of atomic bombardment is constantly changing, and at different times
the particle is hit more on one side than another, leading to the
seemingly random nature of the motion. This transport phenomenon is
named after Robert Brown.
The mathematical model of Brownian motion has numerous real-world
applications. For instance, Stock market fluctuations are often cited,
although Benoit Mandelbrot rejected its applicability to stock price
movements in part because these are discontinuous.
Brownian motion is among the simplest of the continuous-time
stochastic (or probabilistic) processes, and it is a limit of both simpler
and more complicated stochastic processes (see random walk and
Donsker's theorem). This universality is closely related to the universality of the normal distribution. In both cases, it
is often mathematical convenience rather than the accuracy of the models that motivates their use. This is because
Brownian motion, whose time derivative is everywhere infinite, is an idealised approximation to actual random
physical processes, which always have a finite time scale.
Brownian motion 2
Three different views of Brownian motion, with
32 steps, 256 steps, and 2048 steps denoted by
progressively lighter colors
A single realisation of three-dimensional
Brownian motion for times 0 ≤ t ≤ 2
Brownian motion 3
History
Reproduced from the book of Jean Baptiste
Perrin, Les Atomes, three tracings of the motion
of colloidal particles of radius 0.53 µm, as seen
under the microscope, are displayed. Successive
positions every 30 seconds are joined by straight
line segments (the mesh size is 3.2 µm).[1]
The Roman Lucretius's scientific poem "On the Nature of Things" (c.
60 BC) has a remarkable description of Brownian motion of dust
particles. He uses this as a proof of the existence of atoms:
"Observe what happens when sunbeams are admitted into
a building and shed light on its shadowy places. You will
see a multitude of tiny particles mingling in a multitude of
ways... their dancing is an actual indication of underlying
movements of matter that are hidden from our sight... It
originates with the atoms which move of themselves [i.e.,
spontaneously]. Then those small compound bodies that
are least removed from the impetus of the atoms are set in
motion by the impact of their invisible blows and in turn
cannon against slightly larger bodies. So the movement
mounts up from the atoms and gradually emerges to the
level of our senses, so that those bodies are in motion that
we see in sunbeams, moved by blows that remain invisible."
Although the mingling motion of dust particles is caused largely by air currents, the glittering, tumbling motion of
small dust particles is, indeed, caused chiefly by true Brownian dynamics.
Jan Ingenhousz had described the irregular motion of coal dust particles on the surface of alcohol in 1785 —
nevertheless the discovery is often credited to the botanist Robert Brown in 1827. Brown was studying pollen grains
of the plant Clarkia pulchella suspended in water under a microscope when he observed minute particles, ejected by
the pollen grains, executing a jittery motion. By repeating the experiment with particles of inorganic matter he was
able to rule out that the motion was life-related, although its origin was yet to be explained.
The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the
method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD
thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets.
Albert Einstein (in one of his 1905 papers) and Marian Smoluchowski (1906) brought the solution of the problem to
the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules.
Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste
Perrin in 1908.
Einstein's theory
There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian
particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while
the second part consists in relating the diffusion coefficient to measurable physical quantities.[citation needed] In this
way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular
weight in grams, of a gas.[citation needed] In accordance to Avogadro's law this volume is the same for all ideal gases,
which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is
referred to as Avogadro's number, and the determination of this number is tantamount to the knowledge of the mass
of an atom since the latter is obtained by dividing the mass of a mole of the gas by Avogadro's number.
Brownian motion 4
The characteristic bell-shaped curves of the diffusion of Brownian particles. The
distribution begins as a Dirac delta function, indicating that all the particles are
located at the origin at time t=0, and for increasing times they become flatter and
flatter until the distribution becomes uniform in the asymptotic time limit.
The first part of Einstein's argument was to
determine how far a Brownian particle
travels in a given time interval.[citation needed]
Classical mechanics is unable to determine
this distance because of the enormous
number of bombardments a Brownian
particle will undergo, roughly of the order of
1021 collisions per second. Thus Einstein
was led to consider the collective motion of
Brownian particles.[citation needed] He showed
that if ρ(x, t) is the density of Brownian
particles at point x at time t, then ρ satisfies
the diffusion equation:
where D is the mass diffusivity.
Assuming that all the particles start from the
origin at the initial time t=0, the diffusion equation has the solution
This expression allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that
the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is,
however, non-vanishing, being given by
This expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression
Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its
square root.[2] His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the
"single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the
time it takes a Brownian particle to reach a given point.
The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the
mean squared displacement of a particle in a given time interval. This result enables the experimental determination
of Avogadro's number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being
established between opposing forces. The beauty of his argument is that the final result does not depend upon which
forces are involved in setting up the dynamic equilibrium. In his original treatment, Einstein considered an osmotic
pressure experiment, but the same conclusion can be reached in other ways. Consider, for instance, particles
suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts
to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle
acquires a downward speed of v = μmg, where m is the mass of the particle, g is the acceleration due to gravity, and μ
is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius r
is , where η is the dynamic viscosity of the fluid. In a state of dynamic equilibrium, the particles are
distributed according to the barometric distribution
where ρ−ρ0 is the difference in density of particles separated by a height difference of h, kB is Boltzmann's constant
(namely, the ratio of the universal gas constant, R, to Avogadro's number, N), and T is the absolute temperature. It is
Avogadro's number that is to be determined.
Brownian motion 5
The equilibrium distribution for particles of
gamboge shows the tendency for granules to
move to regions of lower concentration when
affected by gravity.
Dynamic equilibrium is established because the more that particles are
pulled down by gravity, the greater is the tendency for the particles to
migrate to regions of lower concentration. The flux is given by Fick's
law,
where J = ρv. Introducing the formula for ρ, we find that
In a state of dynamical equilibrium, this speed must also be equal to v
= μmg. Notice that both expressions for v are proportional to mg,
reflecting how the derivation is independent of the type of forces
considered. Equating these two expressions yields a formula for the
diffusivity:
Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of
Boltzmann's constant as kB = R / N, and the fourth equality follows from Stokes' formula for the mobility. By
measuring the mean squared displacement over a time interval along with the universal gas constant R, the
temperature T, the viscosity η, and the particle radius r, Avogadro's number N can be determined.
The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J.
Thomson[3] in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the
velocity generated by a concentration gradient given by Fick's law and the velocity due to the variation of the partial
pressure caused when ions are set in motion "gives us a method of determining Avogadro's Constant which is
independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".
An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in
1888[4] in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the
frictional force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the
latter was given by Stokes's law. He writes for the diffusion coefficient k′, where is the osmotic
pressure and k is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's
formula for the viscosity. Introducing the ideal gas law per unit volume for the osmotic pressure, the formula
becomes identical to that of Einstein's.[5] The use of Stokes's law in Nernst's case, as well as in Einstein and
Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in
comparison with the mean free path.[6]
At first the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906
and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who
found displacements 3 times greater than Einstein's formula predicted.[7] But Einstein's predictions were finally
confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909. The confirmation of
Einstein's theory constituted empirical progress for the kinetic theory of heat. In essence, Einstein showed that the
motion can be predicted directly from the kinetic model of thermal equilibrium. The importance of the theory lay in
the fact that it confirmed the kinetic theory's account of the second law of thermodynamics as being an essentially
statistical law.[8]
Brownian motion 6
Intuitive metaphor
Consider a large balloon of 100 metres in diameter. Imagine this large balloon in a football stadium. The balloon is
so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at
different times and in different directions with the motions being completely random. In the end, the balloon is
pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We
might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting
equivalent amounts of force. In this case, the forces exerted towards the left and the right are imbalanced in favor of
the left; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random
motion of the balloon. If we look at this situation from far above, so that we cannot see the supporters, we see the
large balloon as a small object animated by erratic movement.
Brownian motion model of the trajectory of a particle of dye in water.
Consider the particles emitted by Brown's pollen grain moving randomly in water: we know that a water molecule is
about 0.1 by 0.2 nm in size, whereas the particles which Brown observed were of the order of a few micrometres in
size (these are not to be confused with the actual pollen particle which is about 100 micrometres). So a particle from
the pollen may be likened to the balloon, and the water molecules to the fans, except that in this case the balloon is
surrounded by fans. The Brownian motion of a particle in a liquid is thus due to the instantaneous imbalance in the
combined forces exerted by collisions of the particle with the much smaller liquid molecules (which are in random
thermal motion) surrounding it.
An animation of the Brownian motion concept [9] is available as a Java applet.
Theory
Smoluchowski model
Smoluchowski's theory of Brownian motion starts from the same premise as that of Einstein and derives the same
probability distribution ρ(x, t) for the displacement of a Brownian particle along the x in time t. He therefore gets the
same expression for the mean squared displacement: . However, when he relates it to a particle of mass m
moving at a velocity u which is the result of a frictional force governed by Stokes's law, he finds
where μ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy with
the thermal energy RT/N, the expression for the mean squared displacement is 64/27 times that found by Einstein.
The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: "The numerical
Brownian motion 7
coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."
Smoluchowski[10] attempts to answer the question of why a Brownian particle should be displaced by bombardments
of smaller particles when the probabilities for striking it in the forward and rear directions are equal. In order to do
so, he uses, unknowingly, the ballot theorem, first proved by W.A. Whitworth in 1887. The ballot theorem states that
if a candidate A scores m votes and candidate B scores n−m that the probability throughout the counting that A will
have more votes than B is
no matter how large the total number of votes n may be. In other words, if one candidate has an edge on the other
candidate he will tend to keep that edge even though there is nothing favoring either candidate on a ballot extraction.
If the probability of m gains and n−m losses follows a binomial distribution,
with equal a priori probabilities of 1/2, the mean total gain is
If n is large enough so that Stirling's approximation can be used in the form
then the expected total gain will be[citation needed]
showing that it increases as the square root of the total population.
Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a
speed u. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity
transmitted to the latter will be mu/M. This ratio is of the order of 10−7 cm/s. But we also have to take into
consideration that in a gas there will be more than 1016 collisions in a second, and even greater in a liquid where we
expect that there will be 1020 collision in one second. Some of these collisions will tend to accelerate the Brownian
particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the
order of 108 to 1010 collisions in one second, then velocity of the Brownian particle may be anywhere between 10 to
1000 cm/s. Thus, even though there are equal probabilities for forward and backward collisions there will be a net
tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.
These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian
particle, U, which depends on the collisions that tend to accelerate and decelerate it. The larger U is, the greater will
be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could
a such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of
energy applies, the kinetic energy of the Brownian particle, , will be equal, on the average, to the kinetic
energy of the surrounding fluid particle, .
In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion. The
model assumes collisions with M ≫ m where M is the test particle's mass and m the mass of one of the individual
particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is
equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision
always imparts the same magnitude of ΔV. If NR is the number of collisions from the right and NL the number of
collisions from the left then after N col