. Although it was formulated and perfected by many scientists over the years, its true origin lies in one of Einstein’s four ground-breaking papers in his miracle year of 1905. The contents of three of those papers are widely known and have even seeped into popular culture: one gave us relativity; another created the most famous formula in science, E=mc2; and the third one truly laid the foundations of quantum mechanics by using light quanta to explain the photoelectric effect. But not many outside of physics have heard of the fourth paper. Yet that fourth paper is his most cited work among them, and, in terms of practical usage and real-life applications, this was perhaps Einstein’s most important paper with applications in everything from physics, chemistry, and biology to finance and economics. This was his paper on Brownian motion.
Robert Brown was a Scottish botanist who lived in the nineteenth century long before Einstein. In 1827, the observant Brown noticed that if he scattered some pollen grains on the surface of a bowl of water, the grains moved about in random and erratic jittery motion, even when the bowl and water are absolutely still with no currents of air to disturb the water surface in the slightest. In case you are thinking it has something to do with pollen grains being “alive,” it turns out that the same thing happens with dust particles as well. If you have access to a microscope, you can try and see this for yourself. The big mystery was, since the water itself was perfectly still, what was causing this motion? There were many speculations, of course, but the key idea needed was an ancient one overlooked by all because very few really believed in it. And that idea, which was first proposed by Democritus of ancient Greece about 2,500 years ago, was that all matter was composed of tiny particles, which he called atoms. Incredible as it may seem today, even at the beginning of the twentieth century when the world was well on its way into the modern technological age, scientists still had not come to terms with the existence of atoms and molecules, which even kids learn about in elementary school today. In fact, Einstein’s solution for Brownian motion is considered to be the final bit of hard evidence that pushed the idea of atoms from being just a good hypothesis to the realm of solid, indisputable scientific fact.
, and note its qualitative similarity with . In the next couple of pages, we will accompany a typical college student along this tortuous path laid out in and get an appreciation of just how the fluctuation-dissipation theorem defines his progress as he tries to catch up on his reading and homework assignments on a Saturday afternoon. He sets out toward the campus library, full of purpose and energy. Halfway there, he runs into a friend, starts chatting, and is easily persuaded to walk with her to grab some coffee. Twenty minutes later, coffee in hand, he finally bids adieu and continues on toward the library, makes it there with no further interruption, finds a secluded corner, opens his books, turns on his laptop, and settles down for a productive afternoon. Then, of course, the phone vibrates, “There’s a party tonight, you wanna go?” He virtuously replies, “I would love to, but dude I have to study, but can you call back after dinner?” The negotiations take up another ten minutes. Now where was he? Ah, page 4, but now he has completely forgotten what he had read, so he starts again from page 1. He settles down and is finally making some progress with the reading, but then he encounters something he does not understand, so he goes online to check it out on Wikipedia. While opening his browser, something catches his eye on Yahoo! news, so after Wiki, he starts reading that, and continues on to read several of the mindless comments at the end of the news article, and then fluidly moves on to some other pieces of gossip about assorted celebrities and reality stars, and before he knows it, another half hour has gone by. Ah, now he is truly sick of reading in general, so he closes his book and decides that he might as well start writing the paper that is due Monday, opens up Microsoft Word, and begins typing. But he has not read the material through, so no wonder he does not get very far with the writing; he must have writer’s block, or so he tells himself. Perhaps walking around a bit will clear the thoughts, and besides, that large coffee is having an effect, he needs to use the restroom anyway. He eventually makes it back to the table and for the next fifteen minutes hammers away furiously at the keyboard and produces an entire page—double spaced. He sits back for a second, and then suddenly realizes that he has not checked his email today, so he logs on, and there are a couple of funny ones from his sister, so he has a good laugh, quietly though—it is the library after all. He shoots off an appreciative reply. Well, while he’s at it, he might as well check his text messages—reading, deleting, and replying takes another fifteen minutes. Back to the paper—he types a few sentences while still thinking about the messages he just read. Pretty soon he finds himself checking out his Facebook news feed. After half an hour of that, he begins to feel some guilt creeping up and tells himself, now it is time to get serious—but he has run out of inspiration for the paper. Why not try something completely different? He reluctantly opens his calculus textbook and stares at the problems for an intense five minutes, but he has no idea where to start, and it is too late to go to the tutoring center. Perhaps he can do the first problem; he tries and thinks he has figured out the first few steps, but it is still too hard. To hell with it, why not take a break; he opens up YouTube, puts on headphones, and just forgets time. Before he knows it, an hour has gone by, and he realizes with alarm that the cafeteria is about to close and if he does not hurry, he might miss dinner. He hurriedly packs everything up and walks toward the cafeteria feeling exhausted and frustrated that after four hours in the library, he only got about half an hour’s worth of work done. All that energy he started with, all dissipated and very little progress to show for it. We can follow the student’s trajectory for the afternoon in the sketch in —isn’t it very similar in spirit to the trajectory of a particle undergoing Brownian motion in ?
. Well, not so with Brownian motion; in fact, we call such motion a random walk, because at each step you can move in any direction, like in . As you can imagine, such motion would not be very efficient, it is like a really drunk guy staggering around looking for the way back to his house from the local bar. In fact, for Brownian motion, it turns out that there is a square-root relation: so in 64 seconds you could cover a net distance of 8 meters on average from your starting point, in 100 steps only 10 meters, and so on, and therefore, as shown in , Brownian motion covers significantly less distance over the same period of time. Unlike ordinary calculus (used to predict trajectories of ordinary motion like that of a thrown baseball), stochastic calculus inherently assumes this square-root dependence on time.
When light (hence “photo”) shines on certain metals, it creates an electric current, because the energy in the light knocks out electrons from the metal. This is used, for example, in automatic doors that open when our presence blocks an invisible beam of light and interrupts the photoelectric current.
The square root is the opposite or inverse of taking a square of a number, which means multiplying a number by itself. Thus, the square of 8 is 8 × 8 = 64, so the square root of 64 is 8. Likewise, 10 × 10 = 100, so the square root of 100 is 10, and so on.