Life's Ratchet: How Molecular Machines Extract Order from Chaos (14 page)

When Brown discovered the random motion of pollen grains, he wondered if their motion had something to do with the fact that pollen grains were alive. He decided to study suspended dust particles of similar size. They, too, performed the same strange dance. Brown concluded that the random motion was not due to the pollen’s being alive, but was due to
some inherent motion in matter. The solution to the mystery came almost a hundred years later, when Albert Einstein proved that the random movement of much smaller particles caused the jittery dance of the pollen or dust grains. Dust and pollen grains are jostled around by random collisions with countless atoms. A pollen grain moves because of small temporary imbalances between the number of atoms hitting from one direction and the number hitting from the other. Einstein suggested experiments to prove his theory of Brownian motion. Using the most sophisticated micro scopes of the early 1900s, the French physicist and Nobel laureate Jean Perrin (1870–1942) used Einstein’s theory to prove once and for all that atoms exist and are always in motion. Boltzmann, just two years after his tragic death, was vindicated.

The tiny scale of atoms and molecules is dominated by continuous motion. Scientists call this continuous motion of atoms and molecules
thermal motion
. Thermal motion does not mean gently floating atoms: At room temperature, air molecules reach speeds in excess of the fastest jet airplane! If we were reduced to the size of a molecule, we would be bombarded by a
molecular storm
—a storm so fierce, it would make a hurricane look like a breeze. Yet, despite their stupendous speeds, molecules in the air do not get very far, because they frequently collide with each other. When this happens, the colliding molecules bounce like tiny billiard balls. The jittery dance that Brown observed and that Einstein explained is the result of this underlying tempest of colliding atoms.

Now we can return to the falling moon rock. If energy is supposed to be conserved, what happened to the rock’s kinetic energy at impact? Where did the kinetic energy of the rock disappear to?

Atoms cannot move as freely in a solid as they do in a gas or liquid. But the atoms in a solid are still moving; they oscillate at high frequencies about a central position, like the head on a bobblehead doll. Physicists often think of solids as collections of atoms connected by springs, all wobbling about, while on average staying at particular positions. When the rock hit the ground, the kinetic energy of the rock did not disappear. Instead, the kinetic energy of the rock transferred to atoms in the rock and
in the ground, making the atoms wobble more vigorously. As a result, the rock and the ground got warmer. The kinetic energy of the rock was converted into thermal energy or heat. Energy remained conserved.

The conservation of energy, coupled with the fact that heat is a type of energy, is called the
first law of thermodynamics
(we will meet the second law shortly). Thermodynamics is the science that deals with thermal energy (the word comes from the Greek
therme
, meaning “heat,” and
dynamis
, meaning “power”) and is the macroscopic “sister science” of statistical mechanics. Thermodynamics is what emerges when we average the random motions of atoms using the tools of statistical mechanics.

Now that we have solved the mystery of where the energy of the falling rock went on impact, let me ask a dumb question (science has advanced by asking a lot of these): Why don’t rocks extract heat from the ground and jump up spontaneously? This would not violate energy conservation. The rock could take heat from the ground, making the ground cooler, and turn the extracted heat into kinetic energy. Yet, we never see this happen. Rocks don’t spontaneously jump off the ground. Why not?

After our rock hit the ground, the atoms in the rock and the ground started to shake more violently. Both the rock and the ground became warmer. Atoms in solids are attached to other atoms, so if one atom shakes, neighboring atoms will soon shake as well. When an atom excites its neighbors, the atom loses some energy, which its neighbors gain. In turn, the atom’s neighbors excite
their
neighbors, and the extra energy provided by the rock’s impact is soon randomly distributed among an astronomical number of atoms in the rock and in the ground.

In the story of the late-night robber, the robber stole my money, he spent some of it, and the people who received money from him spent their money, too. Imagine that the robber stole a thousand pennies instead of a ten-dollar bill. After a while, one thousand people could potentially each have a penny of my money. It would be highly unlikely that my pennies would be spontaneously reunited, as this would require one thousand people (probably unacquainted) to go to the same merchant at the same time to spend their pennies. Similarly, it would be impossible for all the atoms in our rock to spontaneously concentrate their energy to make the rock jump. If we can believe that it is close to impossible for one thousand pennies to be spontaneously reunited, consider the immense
number of atoms in the rock (something like a trillion trillion atoms!). This giant number of atoms would have to simultaneously push in the same direction for the rock to jump up from the ground. Yet, there is no master choreographer that tells the atoms in which direction to shake. They all shake randomly.

Consider the argument I have just made. I did not say that it is
impossible
for a rock to extract heat from the ground and spontaneously jump up from the ground. The word
impossible
has no place in science. Instead I have made a probabilistic argument: While it is not impossible for the rock to jump up by itself, it is extremely unlikely. Remember, we are dealing with statistical mechanics, so every statement is probabilistic in nature. This is quite different from the physics we learn in school: There is supposed to be only one correct answer, and all others are wrong. When I drop a rock, I know it will fall and not rise. But in reality, it
could
rise—it is just highly improbable, and nobody has yet seen it happen or likely ever will.

Impact and friction readily turn kinetic energy into heat, but heat does not easily revert back to kinetic energy. Different types of energy are not always interchangeable. The law of energy conservation tells us that we cannot create or destroy energy, but it does not tell us if a particular type of energy can be converted to some other type. What makes some types of energy more convertible than others?

So far, we have encountered three types of energy: gravitational energy, kinetic energy, and heat (or thermal energy). Each type of energy is associated with certain properties of a system (
system
is physicist lingo for a situation containing objects, energies, and forces). Gravitational energy is completely determined by the height of the object above the ground. Similarly for kinetic energy, the only parameter needed is the speed of the object. However, to completely describe the state associated with thermal energy, we need to know the speeds and locations of all the atoms contained in our system—that is, we would need an astronomical amount of information to fully describe the state of a system that contains thermal energy. Because this is not a realistic proposition, physicists use average values instead. For example, the temperature of a gas is given by the average
kinetic energy of the atoms multiplied by a constant. Individually, the atoms in a gas can have different kinetic energies. Since temperature is an average, this tells us little how kinetic energy is distributed among all the atoms of the gas.

Imagine a gas consisting of just five atoms. All five atoms have the same kinetic energy of 40 meV (meV stands for milli-electron volt, a very small energy unit used by physicists). Then the
average
energy of the atoms is 40 meV as well. Now think of a different situation. Another gas also consists of five atoms, but two of them have 5 meV each, one atom has 10 meV, another has 50 meV, and the final atom has 130 meV of kinetic energy. Now, the average kinetic energy is (5 + 5 + 10 + 50 + 130)/5 = 200/5 = 40 meV, or the same kinetic energy as our first example. But clearly the situation is not the same. In one case, all the atoms have the same energy, while in the other case, the atoms have very different energies. The difference is the distribution of energy among the atoms. An everyday example of the difference between the average and the distribution of items is household income. The average income in the United States was $50,233 in 2007. But we know that some families scrape by on much less than this, while for others, $50,000 is mere pocket change. The interesting story lies in the distribution, not in the average.

Now, we are ready to measure how convertible energy is. An important property for convertibility is the distribution of energy among all the atoms in a system. All atoms in a rock have about the same height above ground, and therefore, the same gravitational energy. The distribution of gravitational energy in this case is quite simple and can be accurately described by the height of the rock above the ground. But the distribution of thermal energy among all the atoms is much more difficult to describe. Each atom has different energy and vibrates randomly at its own pace. All we know about the rock’s thermal energy is its average, given by the temperature, but we know very little about the energy of each atom.

When physicists talk about the state of a system, they distinguish between macrostates and microstates. The macrostate of a rock can be described
by all the things we know about the rock: its height above the ground, its speed, its temperature, and so forth. The microstate is the exact state of all the parts of the rock, that is, the distribution of speeds and positions of all of its atoms—information we do
not
know. As you can imagine, there are a huge number of microstates compatible with any observed macrostate. Atoms in a rock can wiggle in many different ways, but on average, the rock would still have the same temperature, similar to our example above with the five atoms. Knowing macroscopic parameters, such as temperature, tells us little about the particular micro state of the system.

All this talk of microstates allows us to zero in on a mysterious, yet powerful quantity: entropy. By one definition, entropy is the amount of unknown information about a system or, in other words, the amount of information we would need to fully describe the microstate. Think back to the robber. When I still had my ten dollars, the microstate was very easy to describe: The ten dollars was in my pocket. After the crook stole my money and spent it, the money spread through many hands. The macrostate stayed the same (the amount of money was still ten dollars), but the microstate became more and more unknown (even to the robber). As the robber gave other people part of my money, they in turn spent their money (and so on). The entropy, as it were, of my money increased.

Here is another example: Think of the room of a teenager. The macrostate of his room may be stated as follows: It contains sixty-seven pieces of clothing, twenty-three books, a desk, a chair, a lamp, a laptop, and miscellaneous junk. To describe the microstate, we need to know where all these items are located. There are only a few arrangements (microstates) compatible with a tidy room: clothes in the closet, grouped by long-sleeve shirts, short-sleeve shirts, slacks, jeans, and so on; books on the shelves, sorted alphabetically by author; and so forth. However, there are almost unlimited ways the room could be messy: jeans on top of the computer, stat-mech books on the floor, shirts strewn across the bed. The entropy of a tidy room is much lower (there are fewer possible tidy rooms) than the entropy of a messy room. Now you may ask yourself the same question I ask myself daily: Why is my room always a mess?

This question (not exactly in this form) occupied physicists in the 1800s, and their answer was the second law of thermodynamics. The actual question they addressed is the one we are trying to answer as well: Why are
some types of energy more useful than others, specifically, why can some types of energy be converted, while others appear difficult to convert, thus making them useless? Moreover, useful energy eventually turns into useless energy. These unfortunate facts can be understood from our tidy-versus-messy room example. Most of us leave items in random places—places where the items don’t belong. Keeping a room tidy requires a lot of work, but without this work, the room inevitably becomes messy. You may wonder, Why does the random placement of items about the room not lead to a tidy room? Why does it always end up messy? Why can’t you randomly put your books back in alphabetical order on the bookshelf ? This is where our microstates come in: If you leave items lying about in random places, you are more likely to end up with a messy rather than a tidy room because there are many ways (microstates) corresponding to a messy situation and only a few corresponding to a tidy one. In a sense, by leaving items in random places, you randomly pick a microstate from all possible microstates, and because there are more microstates that are messy (and fewer tidy states), chances are high you picked one of the messy states.

Coming back to energy, few microstates are compatible with a certain amount of gravitational energy. This is obvious, as the only relevant parameter is the height of each atom above the ground. Gravitational energy is a low-entropy energy; it is the equivalent of a tidy room. The situation for heat is quite different. Heat is like a messy room; there are so many possibilities for energy distribution among atoms at a particular temperature. Heat is high-entropy energy.