The Third Revolution
When
we look at a glass of water or a piece of metal, without realizing it we are
observing an enormous number of atoms and molecules. In one liter of water, for
example, there are about 1025 molecules.
It is customary to define a quantity called Avogadro’s number
(named after the Italian chemist Amedeo Avogadro) as the number of molecules in
a sample of material whose weight in grams is equal to the molecular weight of
the material (for instance, the number of molecules in 
)
we can usually think of macroscopic systems as a continuum, which is not
made up of basic discrete units. On the other hand, when we observe the
microscopic world at length scales of a nanometer or less, the atomistic
description is necessary because atoms and small molecules are about a tenth of
a nanometer in size. At this level the world is actually described in terms of
discrete units – atoms – and not as a continuum.
Statistical physics is the theory behind our understanding why materials (that is, this enormous and complex collection of discrete particles) are found in different states or phases; how they transform from one phase to another; what determines their elastic, thermal, electric or magnetic properties; and how factors such as temperature and pressure affect all of these. For example, statistical physics (along with quantum mechanics) plays a crucial role in the explanation of electrical properties of solids, and thus in the development of semi-conductors and the computer revolution that followed. It is amazing that Albert Einstein when he was only 26 made decisive contributions to each of the three revolutions of modern physics, and that each of his three contributions was published in the same year - the Annus Mirabilis, 1905.
Conflict between Two Approaches
In order to properly appreciate the circumstances in which Einstein’s 1905 paper on Brownian motion appeared, we should understand that at the start of the 20th century it was not yet self-evident that one could describe materials as consisting of atoms or molecules. On the contrary, there was a passionate controversy between two schools of thought.
On one side stood one of the crowning achievements of 19th century physics – thermodynamics – which supplied an exact description of the behavior of macroscopic materials by treating them as a continuum and not as a collection of discrete particles. Thermodynamics is a deterministic theory, which can foretell with certainty how a material will respond to changes of temperature or pressure, when it will undergo a phase transition, and so on. The predictions of thermodynamics were confirmed with great precision by many experiments in the course of the 19th century.
On
the other side were the works of the Austrian Ludwig Boltzmann and the British
James Clerk Maxwell – the fathers of statistical physics and among the
most important scientists of the 19th century. In their works on
“The Molecular Kinetic Theory of Heat” – as
statistical physics was then called – they claimed that the macroscopic
description of materials may be derived as statistical results of the
movement and interrelationships of a huge number of discrete particles, a
number of the order of Avogadro’s number. This theory is not
deterministic, and regards the measured properties of materials, such as
pressure or density, as statistical quantities with a mean, standard
deviation, etc. As early as the 19th century there was mounting and
convincing evidence, particularly in chemistry, that all materials are composed
of discrete particles, molecules, and that these are composed of even more
basic particles, atoms. Still a significant number of scientists at the end of
the 19th and start of the 20th century, led by the
Austrian physicist Ernst Mach and the German chemist Wilhelm Ostwald, clung to
the continuum theory and to thermodynamics as the only correct approach to a
description of the behavior of macroscopic materials. In their view, not only
was it completely impossible to observe atoms, but also since in the description
of material behavior we are only interested in macroscopic characteristics
(such as volume, pressure, density) there is no need or point at all to an
atomic theory. The French physicist Jean Perrin, who was responsible for the
experimental verification of Einstein’s predictions, later remarked in
response to this: “I find it very difficult to understand this point of view,
since what is inaccessible today may become accessible tomorrow… and also
because coherent assumptions on what is still invisible may increase our
understanding of the visible.”
Today we know that the two theories do not conflict and that the laws of thermodynamics are simply the macroscopic result of the laws of statistical physics. It is interesting to note that relics of this old ambiguity still exist. They may be seen, for example, in programs of study and textbooks of undergraduate physics and chemistry courses.
The verification of the atomic picture of matter, and of the statistical theory that describes it, has a history of nearly 200 years. The story begins in a surprising place – neither physics nor chemistry, but rather in botany, 80 years before Einstein’s work….
Brown’s Pollen Dance
At the beginning of the 19th century the science of botany flourished, and many expeditions, particularly British, reached all the corners of the earth and returned with an enormous range of new species of plants. In 1827 Robert Brown, a globally renowned British botanist, looked through his microscope at a suspension of pollen in water. To his surprise the pollen grains were not at rest but moved without cease in a random, infinite dance. At first Brown thought that the grains were alive. However, with skill and care he attempted to observe particles of similar size (a thousandth of a millimeter – a micrometer) made of inorganic material such as dust or soot. These particles too danced at random. It was clear therefore that this movement, henceforward called Brownian motion, was a physical rather than biological phenomenon.
Brownian motion remained without sufficient explanation for several decades. In 1877 the Belgian scientist Joseph Delsaux was first to put forward the explanation that it is related to the discrete particle character of the liquid that surrounds the grains, that is, to the thermal motion of the water molecules and their collisions with the grains. The first to attempt thoroughly to test this idea was the French physicist Louis Gouy in 1888. Gouy showed that the rate of Brownian motion is inversely proportional to the viscosity of the liquid in which the grains are suspended. However, it was Albert Einstein who first formulated a complete theory of the phenomenon. Not only that, Einstein understood its importance as an experimental touchstone, measurable with a simple microscope, for the validity of statistical physics.
The microscopic world and the mesoscopic world
It is important to bear in mind the
relevant dimensions. Brown observed particles the size of micrometers using an
optical microscope. Micrometer-sized particles are 10,000 times as large as
atoms; it is of course impossible to see atoms through an optical microscope.
Micron and sub-micron particles, down to single nanometers, belong to the
mesoscopic world – a world that contains hundreds of billions (but not
Avogadro’s number) of atoms and molecules. Nanoscopic and mesoscopic
physics today are at the forefront of scientific research. Colloidal scientist
Wolfgang Ostwald (son of Wilhelm Otswald) said as long as 100 years ago that
“the mesoscopic world is the world of neglected dimensions.”
Einstein’s Paper on Brownian Motion
In
1905, the same year in which Einstein published his well known papers on
relativity theory and on the quantum theory of light (the photoelectric
effect), another paper of his appeared titled: “On the Movement of
Small Particles Suspended in a Stationary Liquid Demanded by the Molecular
Kinetic Theory of Heat”. At the beginning of the paper Einstein
remarks with characteristic caution that possibly the motion that he discusses
is identical to Brownian motion. But in addition to the explanation of Brownian
motion Einstein was completely aware of the real and profound significance of
his work: “If the
movement discussed here can actually be observed (together with the laws
relating to it that one would expect to find), then classical thermodynamics
can no longer be looked upon as applicable with precision to bodies even of
dimensions [as large as those] distinguishable in a microscope: an exact
determination of actual atomic dimensions is then possible. On the other hand,
had the prediction of this movement proved to be incorrect, a weighty argument
would be provided against the molecular-kinetic conception of heat.”
Inasmuch as it was not possible to see atoms in Einstein’s day – they are simply too small and too speedy – could it be possible to infer their existence by means of the laws of statistical physics for particles that could be seen? Einstein assumed that a Brownian particle in a liquid behaves like a very big atom in equilibrium with the surrounding liquid. The liquid can be thought of as a collection of much smaller particles, moving at random and colliding constantly with each other as well as with the larger particle. The large particle moves with Brownian or diffusive motion within the liquid medium as a result of frequent collisions with large groups of particles of the liquid. Diffusive processes in continuous media (as for example with a solution of water and salt) were known to scientists in the 19th century. Einstein showed in his paper that it was possible to demonstrate a precise connection between such diffusive processes on the macroscopic scale and random processes of discrete particles on the microscopic scale (see box).
Random
Walks and Brownian Motion
In order to explain Brownian motion
envision a drunken man staggering right and left along a straight line. At each
time period 
he decides to
take one step of length a to the right or left at random. The question is: what is the
characteristic distance the man has staggered after N steps? With each step the location of the drunk changes by 
. After N steps his
location will be: 
, but this is a statistical number, which can vary between 
and 
. When the number of steps N is very large, the average distance written 
will be zero,
because neither right nor left is a preferred direction. From the statistical
average of the square of the sum
,
we obtain the standard deviation, which
is a statistical measure of the characteristic distance the drunk has passed
after N steps of time,
.
This is an important result. It means
that the distance in a random process such as diffusion grows only in
proportion to the square root of the number of steps, or in other words by the
square root of the time. Despite the considerable simplification, the fact is
that particles undergoing Brownian motion in liquid (see figure 1) behave
exactly as the random walker. In order to illustrate the difference between
diffusion and motion at a constant speed (see figure 2), we take a
micrometer-sized particle. After a million steps, each one micrometer in
length, the characteristic distance such a particle will move in Brownian
motion will be 
, only one millimeter, whereas a particle moving at a
constant velocity for a million steps in the same direction will advance a distance
of 
, one meter, that is, a distance a thousand times longer
Figure 1. Example of a random Brownian walk
of 100 steps in the plane, drawn from a computer simulation. The blue arrow
connects the initial point to the final point. Note how short the arrow is
relative to the length of the entire path. If the length of a single step is a,
the entire path length is 100a. In contrast to this,
if we were to draw many processes such as that of the figure and work out the
average arrow length, we would find a length of only 10a.
Figure 2. The average velocity of a particle is
the ratio between the distance X0
and the time t required
to travel this distance. In the figure the red line shows constant velocity as
a function of time. For a Brownian particle, however, the distance grows as 
,
and so we find an average velocity proportional to 
,
as shown by the blue curve. This means that as we measure average velocity over
smaller and smaller time periods, the velocity will grow enormously. This
experimental fact had confounded investigators of the Brownian motion before
Einstein’s theory was published.
Einstein
showed that a micrometric particle buffeted at random by particles in the
surrounding liquid will fulfill a statistical law according to which the
distance the particle passes will increase only as the square root of the time,
(see box).
The coefficient in this relation, D, is called the diffusion coefficient.
For a one micron particle suspended in water, 
. This means that in one second the particle advances a
characteristic distance of 
- a
distance similar to its diameter.
But
how to calculate the diffusion constant of the particle? Here too Einstein made
a decisive contribution, when he demonstrated that the diffusion constant
depends on the relationship between the temperature T and the
coefficient of friction between particle and liquid,
:
This
elegant result is to this day known as the Einstein relation. The
coefficient of friction depends on the
size of the Brownian particle and on the viscosity coefficient of the liquid,
but surprisingly the velocity of the particle does not appear in this relation.
The coefficient in the Einstein relation contains two constants: the gas
constant R and Avogadro’s number 
. The Einstein relation therefore provided a direct
method of measuring Avogadro’s number by means of observations of the
motion of Brownian particles, as will be shown in the next section.
Perrin’s Experimental Verification of the Atomic Picture
Since at the start of the 20th century it was not possible to “see” atoms and molecules, scientists had to infer their existence from physical phenomena, which could be measured at macroscopic or mesoscopic length scales, but whose explanation was inextricably connected to “atomic reality” – that is, to the huge number of discrete basic units that make up matter.
Even
before Einstein there were experimental estimates of Avogadro’s number.
But after Einstein’s work had appeared in was measured to great precision. Of these experiments
the most noteworthy were those of French scientist Jean Perrin. These proved to
be the ultimate experimental verification of Einstein’s theory of
Brownian motion, as well as proof of the existence of atoms and molecules (see
box). Perrin received the 1926 Nobel Prize in physics for this work.
As Brownian motion cannot be directly measured at atomic scales, Perrin used a suspension of powder grains produced from the gum (gamboge) tree, spherical in shape and of micrometric size. Such a suspension can be produced in a liquid and the grains are then magnified by an optical microscope for observation. Perrin and his colleagues performed a precise analysis of the particle trajectories, and thus succeeded in verifying experimentally the ½ power law of Brownian motion – that is, the fact that the characteristic distance a particle goes through grows in direct proportion to the square root of the time.
In
addition – this was the crowning achievement of these revolutionary
experiments – Perrin succeeded in obtaining an accurate estimate of
Avogadro’s number, 
, which was in amazing accordance with
estimates that had been made from completely different physical phenomena:
thermal radiation of heated bodies, Rayleigh scattering of sunlight by the
atmosphere (the effect that makes the sky look blue), and radioactivity.
By 1909 these results had brought even the greatest skeptics to accept the validity of the atomic picture, and put to an end to one of the most loaded disputes in the history of physics.
Measurement
of Avogadro’s Number
The Einstein relation provides a method of directly measuring
Avogadro’s number, and that is what Perrin did. The diffusion constant D is calculated
from the measurement of the relationship between the distance X0 that the Brownian particle passes, and the duration of its motion t. The friction constant of a spherical
particle, with known diameter and moving in a liquid of known viscosity, may be
calculated by use of hydrodynamic theory (Stokes’ formula). The gas
constant R was known from thermodynamic measurements. Therefore it is plain to
see that at a given temperature T Avogadro’s number may be
calculated by plugging in all the other quantities to the Einstein relation. In
1908 Perrin found a value of 
for Avogadro’s number, and by 1914 the experimental value matched
today’s value of 
with a four-digit precision.
Before moving on to the far-reaching implications of
Einstein’s paper on contemporary research let us sum up its three central
contributions:
(*) A conclusive explanation of Brownian motion. The explanation was based on analysis of random processes, and showed that the distance passed by a Brownian particle grows with the square root of the time. This revolutionary description of random motion paved the way for an entire field of scientific research in analysis of stochastic processes and signals.
(*) Verification of the atomic picture. Einstein’s model required treatment of the liquid surrounding the particle not as a continuous medium but as made up of molecules. What is more, Einstein showed that Avogadro’s number can be directly measured from characteristics of Brownian motion.
(*) The relationship between random motion and friction. In his paper Einstein proved the existence of a deep relationship (the Einstein relation) between the diffusion constant and the temperature and friction coefficient. This relationship between random motion (a microscopic phenomenon) and friction (a macroscopic characteristic of matter) laid the groundwork for generalization and applications in many systems, and became one of the cornerstones of statistical physics.
100 Years After Einstein: Implications and Current
Applications
Although one hundred years have gone by since Einstein and his colleagues did this work, random processes are still being investigated intensively in a surprising variety of areas of science and technology, and not only in the natural sciences. As examples we have chosen to summarize here a number of applications.
Polymers: Polymers are long flexible molecules that are found in nature as polysaccharides, DNA, proteins and more. Since the start of the 20th century polymers that are by-products of the petrochemical industry have been the basis of all the plastic products so characteristic of our times. Every polymer is made up of base-units (monomers) chemically joined. When a polymer molecule is dissolved in liquid, it is characterized by great flexibility, and any spatial conformation it forms appears to be a random walk (figure 3). This walk resembles Brownian motion, but differs from it in one important characteristic: the walk cannot intersect itself (“self-avoiding walk”), since two different monomers cannot occupy the same space.
Figure 3. A schematic description of a
polyethylene molecule containing 50 base-units (monomers). Such polymers may
contain a far larger number of monomers. When they are dissolved in a dilute
solution, the statistics of their possible conformations corresponds to that of
a self-avoiding random walk.
Statistical analysis of “walks” of polymer chains teaches a great deal about polymers, both for single chains and for polymer material consisting of a great number of chains. We point out two important characteristics:
(*) As a polymer chain can be very long (thousands and even hundreds of thousands of base-units) and flexible, the enormous variety of possible chain conformations (its entropy) dictates to a large extent its material properties. This quality distinguishes polymers from other materials. For instance, we are used to the fact that solid material such as metal becomes less rigid when heated, and so a metal spring under a load, for example, will lengthen when heated (see figure 4a). On the other hand, if we replace the metal spring with a rubber band made of polymeric material, to our surprise we find that the rubber band shrinks when heated (figure 4b) because the higher temperature makes it harder to restrict the polymer conformations (reduce its entropy) by stretching.
Figure 4. (a) Metal spring under a load.
When heated the rigidity of the spring lessens and the weight causes it to
lengthen. (b) Polymer rubber band under the identical stress. When heated it
becomes rigid and contracts.
(*)
As shown in figure
(a) (b)
Figure 5. (a) Example of a self-avoiding random
walk of 100 steps, drawn from a computer simulation. The statistics of such
processes describes the different conformations of polymer molecules in
solution (figure 3). The blue arrow joins the starting point to the end point.
Note once again how short the arrow is relative to the entire path length. (b)
The normal Brownian process of figure 1 is presented alongside the
self-avoiding walk, at the same scale. Comparison of the two walks clearly
demonstrates how self-avoidance considerably “swells up” the space
covered by the random walk.
Anomalous Random Processes: Einstein’s work on Brownian motion opened up an entire new area of probability theory, which involves stochastic (random) processes. In the past few decades much research in this area focuses on stochastic processes, which give rise to anomalous diffusion, that is, distances that grow at a greater pace (super-diffusion) or a slower pace (sub-diffusion) than the square root of the time. Super-diffusion can be seen, for example, when the steps of the drunkard’s walk are not uniform but may be very large. Such random processes, called Lévy flights (named after the French engineer and mathematician Paul-Pierre Lévy) have been observed for example in the way that various birds fly in their search for food. Other examples of anomalous random processes have been found in a variety of phenomena such as weather fluctuations, water permeation through rocks, and heartbeat irregularities.
“To See One Molecule”: Einstein and Perrin were obliged to deduce the existence and motion of single molecules indirectly through the motion of a particle 10,000 times as large. In the past two decades science has made a huge leap in the field of microscopy, and today it is possible to directly observe molecules and even single atoms. Experiments at the level of a single molecule are at the forefront of today’s research in chemistry, physics, materials science and biology. There are various methods of observing processes at a molecular level. One of these is to chemically attach a “marker” to a molecule to make it easier to see (for example, a fluorescent tag marker).
Figure
6 shows results of recent research at
Figure 6. Random walk of one lipid molecule
in a cell membrane. The molecule executes Brownian planar motion in a delimited
domain for a few milliseconds, then passes on to a nearby domain of the cell
membrane. The colors show the different domains. The horizontal line shows a
length of one micrometer. The experiment was performed by A. Kusumi and
colleagues at
“Noisy Life”: Biology is one of the areas in which the importance of fluctuations and random processes is particularly significant. Living processes require operation of defined mechanical and chemical tasks on one hand, and on the other hand flexibility, dynamism and ability for fine adjustment and error correction. Biological systems therefore face a dilemma with regard to fluctuations and random noise. They must be robust enough so that their operation will not be disturbed by extreme disorder. At the same time they must be capable of exploiting the disorder for their own purposes, for example in order to transfer various molecules in and out of the cell. As a result the energies of interactions in biology are not significantly larger than that of the thermal noise. What is more, biological tissue is not rigid as metal or cement… One can therefore say that life processes take place on the verge of thermal noise. For instance electric currents in live cells (brain, heart, etc.) are carried by means of diffusion (that is to say random processes) of ions in an electric field through selective ionic channels located in the cell membranes. It is thus clear that since the appearance of Einstein’s paper we have gradually developed an understanding of how characteristics and functions are defined with relation to microscopic random processes, and this is of central importance for our understanding of the fundamental processes of nature.
An illuminating example from recent times is in regard to enzyme action – the impressive efficiency increase of a biochemical process by means of the appropriate molecule (enzyme). Experiments in enzyme activity have been performed for dozens of years. These are macroscopic experiments, which teach us about the average activity of an enormous number of enzyme molecules. They are analogous for our purposes to an experiment, which observes diffusion of a huge number of particles in a macroscopic suspension. In recent years, as we have noted, scientists have achieved the ability to observe a single molecule, and in this case the activity of a single enzyme molecule. Such experiments, by the same analogy, are equivalent to Perrin or Brown observing one single grain. It turns out that a single enzyme molecule is located in an extremely noisy environment, and its chemical activity takes place in a sort of random process of on/off switching.
Another example that has been the center of much recent scientific effort is molecular engines – molecular clusters, which perform various mechanical operations necessary for the functioning of the living cell. On one hand, these are indeed “engines” in the sense that they use “fuel” (chemical energy in the form of ATP molecules) in order to perform mechanical work. On the other hand, unlike macroscopic engines, they operate in an environment full of random fluctuations. Moreover, it seems that they even exploit the noise, by inserting asymmetry into the random process, in order to perform their operation more efficiently.
Random Processes in Economics and Communication: Numerous phenomena in our environment involve statistical fluctuations reminiscent of random processes. Particular examples of this are fluctuations of currency rates and stock prices as well as various economic indices; also weather changes over various periods of time. Figure 7 shows changes in the Dow Jones index of leading stocks in the American stock market. The “time step” arbitrarily chosen here is one month, and the drawing shows the “random walk” of stock values from one month to another, beginning from September 1st, 1999 and up to September 1st, 2005. The similarity to the drunkard’s random walk discussed earlier is plain. The drunkard’s step to the right is equivalent to a rise in the index, and a step to the left – to a drop (or vice versa).
Figure 7. “Random Walk” of the
Dow Jones index. Each point represents the index value at the beginning of the
month, starting from September 1st 1999 and up until September 1st
2005. Financial analysis and investment groups today use a simulation technique
called Brownian Dynamics to analyze such “walks”.
Obviously
processes such as that of the Dow Jones index are far more complex than simple
Brownian motion and far from being completely random. Nevertheless, as early as
the beginning of the 20th century, five years before the appearance
of Einstein’s paper, French scientist Louis Bachelier pointed out the
possibility of analyzing rate changes in the financial market as random
processes. Seventy years later the Americans Fischer Black, Myron Scholes and
Robert Merton presented a model analyzing stock rates as a random walk called Geometric
Brownian Motion. This model has found many applications; in 1997 (after the
death of Black) it brought Scholes and Merton the 1997 Nobel Prize in
Economics. Investment companies and financial analysts of recent years have
increasingly been using analytical tools inspired by the motion of
Brown’s particles, and based on a technique of computer simulation called
Brownian Dynamics.
The theory of random motion and signaling has many technological applications as well. A relevant example from the world of optical communications is a phenomenon called PMD (Polarization Mode Dispersion). When a signal advances through a fiber optic, it accumulates distortion as a result of unavoidable slight defects in the fiber structure. This buildup of distortion is a random process, which obeys the same mathematical laws Einstein formulated for Brownian motion.
Epilogue
In his revolutionary article of 1905 Einstein showed how microscopic processes (motion of molecules in a fluid) can be deduced from mesoscopic processes (motion of a grain of pollen). Thus he resolved the controversy concerning the correctness of the atomic picture and the validity of statistical physics. In recent decades we have witnessed more and more research in the opposite direction, “back to the pollen”, so to speak: how to use our increasing knowledge of random processes at the molecular level to draw conclusions and make predictions about the behavior of large complex systems such as the living cell, the organism, the atmosphere, or the stock market.
Although Einstein’s third revolution is less well known to the public, it
had and still has far-reaching implications for science and for many
applications in everyday life. Therefore, from a perspective of a hundred years
later, we can say this revolution actually had a wider influence than the other
two revolutions Einstein created. With time it has brought forth a new world
view in which randomness has a central role. We conclude with a quotation on a
similar theme from Mark Haw’s article that appeared in January
The authors would like to thank Gil Ariel, Noah Brosh, Daniel Harries, Shlomo Havlin, Eytan Katzav, Yossi Klafter, Jacob Klein, Judy Kupferman, Valerie Parsegian, Shimon Reich, Michael Schick, Zeev Schuss, and Mark Shtaif for their assistance and their helpful comments.
Further Reading
1. Einstein, A.,
Investigations on the Theory of Brownian Movement,
2. Pais, M., Subtle is the Lord: The Science and the Life of
Albert Einstein,
3. Stachel, J.
"Einstein's Miraculous Year",
4. Lindley, D., Boltzmann's Atom: The Great Debate
that Launched a Revolution in Physics, Free Press, 2001. [For
further reading on the statistical physics revolution and the surrounding
controversy.]
5. http://www.phy.ntnu.edu.tw/java/gas2D/gas2D.html
Java realization
showing a Brownian particle in a medium of small particles colliding with it
constantly. The number of colliding particles and their relative size may be
adjusted.
6. http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html
Another Java
realization.
7. http://nobelprize.org/physics/laureates/1926/index.html
Nobel physics prize
site for 1926. Includes Perrin’s biography and his Nobel lecture
summarizing the proof of the existence of the atomic world.