Forum: An exploration of life’s randomness

Professor of physics Don Lemons recently published An Introduction to Stochastic Processes in Physics (Johns Hopkins University Press, 2002). It is the only random variables textbook that has been written for upper-level college physics students and includes problems at the end of each of the 10 chapters.

In the following interview, Lemons conversed via e-mail with two Bethel alumni—Christopher Graber and David L. Kaufman—and with Luke Schmidt, a current Bethel student, about the nature and meaning of randomness.

Graber is a second-year medical student at the University of Kansas Medical Center in Kansas City, Kan. He graduated in 2001 with a bachelor’s degree in biology, chemistry, math and physics. For his math and physics senior seminar, Graber helped edit the first eight chapters of An Introduction to Stochastic Processes in Physics.

Kaufman, a 1998 graduate in physics and mathematics, is a Ph.D. candidate in operations research at the University of Michigan in Ann Arbor. During his senior year at Bethel, Kaufman worked with Lemons on modeling the Brownian motion of a charged particle in a magnetic field.

A senior chemistry and physics major, Schmidt recently helped research and verify a stochastic process model of a randomly lying cord. Of particular interest to the model, is the squared distance between the ends of a dropped cord. The results were published in “The Shape of a Randomly Lying Cord” by Lemons and T.C. Lipscombe in the American Journal of Physics, Vol. 70, No. 6, June 2002.

Lemons: Do random processes—i.e., quantities that change randomly as a function of time or as a function of some parameter that plays the role of time—enter into your current work, study or teaching?

Graber: Randomness is everywhere within the human body; it could be thought of in questions such as “why did I get this disease?” or “what are my chances for survival?”When I study the systems of the body, my textbooks show diagrams that consist of a single molecule interacting with a single protein; but, I know that this is not the case in the actual body. Molecules do not line up on some Ford-like assembly line to be processed through a computer-driven machine. ... Rather, all of these molecules float around in our cells and in our bodies and randomly come into contact with everything else around them.It’s almost a miracle to think that a body’s basic process works on stochastic principles. A normal, healthy human doesn’t have to worry about all of this randomness because it seems to regulate itself just fine.

Lemons: Your comments suggest that the human body can be thought of as a system that is undergoing a random process? This seems like it might especially be true when looking at relatively microscopic parts of the body—its randomness becomes more noticeable.

Graber: Thanks for that clarification, Don. I did mean that randomness within the human body occurs microscopically—within the cells of the body. Once you reach the cellular level and upwards in complexity—tissues, organs, organ systems, organisms—the system acts in a more organized manner. It is at the subcellular level that the randomness dominates. In most cases, substances simply flow from areas of higher density to areas of lower density. For example, compounds like water don’t “decide” to enter a cell. Rather, if there are more water molecules outside the cell than inside, the odds are good that more water will wander into the cell than out of the cell. It is not a one-way path because individual water molecules will move both ways, but the end result is that there is now more water in the cell. This applies not just to water, but also to ions, gases, vitamins and other nutrients.

Lemons: This is certainly consistent with the way I think about randomness in physical systems—it is most evident in very small systems. This is because at the molecular level we have chaos—”molecular chaos” as Ludwig Boltzmann put it at the end of the 19th century. Different molecules are moving in different directions with different speeds. They are very unpredictable.
In the example you gave, the water molecule near the boundary of a cell is a part of this molecular chaos. I know that sometimes it also becomes useful to describe macroscopic systems with random variables.

Kaufman: In my studies of operations research, randomness is incorporated into models whenever variability is a necessary aspect of the underlying problem, which for me happens to be most of the time. … For example, besides the microscopic randomness inside healing patients, there is plenty of macroscopic randomness in hospital systems. Take arrivals at the emergency room. Patients seem to get sick and hurt in uncertain ways. Once a doctor sees a patient, the amount of time spent in the ER varies—based on the type of injury, ability of the doctor, etc. What’s really bad for the patient is the time spent waiting just to see a doctor. Simple mathematical queuing models can describe what goes on. The most basic of results, the P-K (Pollaczek-Khintchine) formula, gives a nice description of how the expected number of patients and their expected delay grows with the ratio of the service rate to the arrival rate. The formula captures how things get worse as the variance in service time increases. Mathematical models that capture variability can aid capacity decisions such as staffing. If ER “customers” arrived with the same type of problem every hour on the hour, 24 per day, then only one specialist would be needed at a time for a low average waiting time. If 24 bleeding people arrived all in one hour of the day, then having only one doctor on hand would be a huge problem.

Lemons: One of the interesting things about studying physics is that the methods one applies to
fundamental and simple physical systems also can be used to model other systems—systems never dreamed of by the original workers in the field. I am, in particular, fascinated by the fact that people are using random process theory to model processes like ER room schedules in the field of operations research. I assume stochastic operations research models have been successful, or at least, useful. Some purist physicists would dismiss this kind of work as being merely phenomenological or merely descriptive without exposing fundamental causes. The success of these models, while merely phenomenological in one sense, are fundamental in another because they expose something about the human condition—we can’t know it all and we can quantify our ignorance! That proposition has got to be at least as fundamental as Newtonian dynamics.

Schmidt: Perhaps these phenomenological models are not widely accepted as fundamental due to their relatively recent introduction to physics. Newton’s Principia was published in the late 1600s; I do not think it was successful overnight. Newton’s work involved simple systems compared to the systems we have been talking about. For example, the two-body problem in celestial mechanics involves modeling a very small number of parts. I think this is where the work in randomness may branch to eventually—how do we predict what will happen in a random situation involving only a few cases? Another example, flipping a coin: Over the long run we know the average ratio of heads to tails is 50 percent. But we have no way of predicting what the next flip will be. I think that right now this is the limit to the models—short-term prediction as opposed to long-term averages. But, perhaps, the short-term is irrelevant as long as the original problem is solved.

Lemons: I think that the relative newness of random processes as a mathematical tool may play some role in physicists’ suspicion or ignorance of them. Einstein and Langevin initiated the study of stochastic processes in 1905 and 1907, respectively. That makes stochastic process theory much younger than Newtonian mechanics, but older by about two decades than quantum mechanics. But I also think that some physicists have a more fundamental critique. It goes like this: Fundamental physics means the study of fundamental forces and particles. Solid state physics, plasma physics, statistical mechanics and acoustics, etc., are not, in this sense, fundamental. Furthermore, if one studies fundamental particles and their forces, quantum mechanics and relativity suffice. One doesn’t need the mathematics of stochastic processes.

Kaufman: In a “Talk of the Nation” interview on National Public Radio a few months ago, I heard Stephen Wolfram taking questions about his book, A New Kind of Science. He mentioned the goal of fundamental physics as opposed to other branches of science. It seems that fundamental physicists want to expose the most fundamental building blocks that will allow them to capture the complete universe. On the other hand, the goal of Newton was to model certain phenomenon, e.g., the motion of the planets, using mathematical formulas without necessarily asking about all the underlying mechanisms. Newton was able to represent the motion of planets by abstractly modeling them as homogenous spheres of mass and he didn’t suggest why they moved. And yes, purists would dismiss this type of work as being phenomenological—not fundamental in the strict sense since it does not describe fundamental particles and forces. Does this mean the mathematics of Newtonian mechanics are not important? No. Again, I think it is a matter of scope. The questions we are trying to answer, the motivation for building models, determine what level of approximation we are willing to accept. ... I totally agree that we cannot know it all. Even if a physicist learned all the fundamental building blocks and rules of the universe, I doubt we would have the ability to extrapolate to the complex systems of the universe.