Section 43–3 The drift speed

S-molecules / Drift velocity / Mobility

 

In this section, Feynman introduces the concepts of S-molecules, drift velocity, and mobility (electrical and generalized). Although the section title is “The drift speed,” Feynman later shifts to using “drift velocity” in the text. This transition aligns with the standard physics term because the phenomenon is inherently directional, arising from an asymmetric external force. Conversely, "drift speed" is simply the scalar magnitude of that vector, a term best reserved for contexts where directional orientation is irrelevant. 

 

1. S-molecules

“We shall refer to the “majority” molecules as the “background” molecules, and we shall call the molecules which are different from the background molecules “special” molecules or, for short, the S-molecules (Feynman et al, 1963).”

 

The term S‑molecules (special molecules) serves as a shorthand for identifying the particle under study, distinguishing it from background molecules without lengthy descriptions. This abstraction unifies the treatment of diffusion of gases, centrifugal separation, and ionic conduction, under a framework, providing an intuitive explanation of drift velocity. However, the term is nonstandard in the broader literature, where expressions such as test particles or tagged particles could be used. Its use may confuse readers and inadvertently suggest that S‑molecules possess unique properties, but it is merely an analytical description. Furthermore, the model assumes a dilute system in which the S-molecules neither disturb the background molecules nor interact with one another, which is unlikely to happen in the real world. Here’s a fun twist, in the Audio Recordings of this lecture, Feynman did not use the term “S‑molecules”, but relied instead on phrases like “this thing,” “heavy object,” or simply “it.” He did, however, explicitly introduce the “fresh start” assumption—the idea that molecules do not remember their acceleration after each collision.

 

The limitations of the “fresh start” assumption can be illustrated with a bowling ball analogy: a heavy particle moving through a sea of lighter molecules behaves like a bowling ball rolling through bowling pins—each impact alters its motion only slightly. The assumption of a "fresh start" after every collision is unrealistic because the particle’s momentum remains highly correlated from one collision to the next; a single collision is usually insufficient to erase the memory of its prior motion. Instead of adopting t as the mean time between consecutive collisions, we can reframe it as the momentum relaxation time—the characteristic time over which cumulative collisions dissipate the particle’s initial directed momentum. This approach preserves the structure of the simple formula while incorporating the cumulative effects of many collisions, making the model both intuitive and more realistic. A rigorous treatment of this process is provided by the Boltzmann equation, Langevin equation, or Green-Kubo relations.

 

“We shall not work out the details, but just state that the result is equivalent to replacing τ, the average collision time, by a new—and longer—τ which corresponds to the average ‘forgetting time,’ i.e., the average time to forget its forward momentum (Feynman et al., 1963).”

 

It is worthwhile to clarify the distinction between the mean collision time and the momentum relaxation time (Feynman’s "forgetting time"). The mean collision time is a kinematic quantity that measures the average time between consecutive collisions and characterizes the frequency of collision events. By contrast, the momentum relaxation time is a transport quantity that measures the average time required for cumulative collisions to “erase” a particle’s directed momentum and establish a new equilibrium. The two timescales are identical if each collision completely randomizes the particle’s momentum. In real gases, liquids, and plasmas, momentum is lost gradually through a succession of small-angle deflections. Consequently, the momentum relaxation time is typically longer than the mean collision time, serving as the appropriate parameter for describing drift velocity and mobility.

 

2. Drift velocity

“We say that there is a drift, superposed on its random motion. We would like to know what the speed of its drift is—its drift velocity—due to the force F (Feynman et al, 1963).”

 

A comprehensive definition of Drift velocity

Drift velocity is the average, directed velocity acquired by charged particles (such as electrons or ions) when subjected to an external electric field—a slow, systematic drift superimposed upon their much larger, random thermal motion (typically averages to zero in the absence of a field). Microscopically, it arises from the acceleration by the applied force (typically F= qE) during the brief intervals between collisions and molecular force from each collision. Macroscopically, the drift velocity is directly proportional to the applied electric field and is related to the electric current density by J = nqv, where n is the charge carrier density. Although the random thermal speeds of particles are typically orders of magnitude larger than the drift velocity, it is this tiny net motion that is responsible for macroscopic electrical conduction, diffusion, and other transport phenomena. In electrical circuits, this allows the drift velocity to bridge the microscopic molecular collisions to the measurable electrical currents in response to applied fields.

 

“So between the two collisions it has, on the average, a velocity one-half of the final velocity, so the mean drift velocity is ½ Fτ/m. (Wrong!) This result is wrong and the result in Eq. (43.13) is right, although the arguments may sound equally satisfactory…… Short times occur more often but make less contribution to the drift velocity because they have less chance “to really get going.” If one takes proper account of the distribution of free times between collisions, one can show that there should not be the factor ½ that was obtained from the second argument. The error was made in trying to relate by a simple argument the average final velocity to the average velocity itself (Feynman et al., 1963).”

 

Feynman warns that the drift velocity cannot be obtained by simply halving the final velocity, but he only mentions the “distribution of free times” without showing the calculation. A more instructive approach uses the exponential distribution of free-flight times. Although the mean time between collisions is the constant t, individual free-flight times t are random and vary from one collision to another. They follow the probability density function p(t) = (1/t)e^{-t/t}, which arises from the assumption that collisions occur randomly and independently—a Poisson process (Feller, 1968). Immediately after a collision, the probability that the next collision occurs between t and t + dt is p(t)dt. During this interval, the particle accelerates under a constant force F, gaining an additional velocity (F/m)t. Averaging over all possible free-flight times gives the drift velocity:

Thus, the drift velocity is determined by the mean of the exponential free-flight distribution—not by assuming that every molecule accelerates for the same duration. However, while this approximation is valid for dilute gases, it becomes less reliable in dense fluids or systems with correlated collisions.

 

Note: In the Audio Recordings (26 min: 55 sec) of this lecture, Feynman mentions that the ½ factor error could be due to the textbook of Resnick and Halliday, or others.

 

Alternatively, Feynman could have clarified that a molecule’s initial velocity immediately after a collision is not zero. More precisely, after each collision, a molecule moves with a random (initial) velocity v₀, which is typically several orders of magnitude larger than the drift velocity. Between successive collisions, the external force F imparts a small systematic velocity increment Dv = (F/m)t where t is the time elapsed since the last collision. The molecule’s actual (instantaneous) velocity is therefore: v = v₀ + (F/m)t. The crucial assumption is that the post-collision initial velocities are isotropically distributed (equally likely in any direction), meaning their statistical average is zero (<v₀> = 0). This does not imply that each molecule has zero initial velocity after a collision; rather, the random directions of many molecules cancel statistically. Thus, the drift velocity arises solely from the small, nonzero average produced by the force-induced term. The macroscopic drift represents a small directional bias superimposed on the much larger random thermal motion.

 

3. Mobility

“If in an electrical problem the force is written as the charge times the electric field, F=qE, then the constant of proportionality between the velocity and the electric field E is usually called the “mobility.” In spite of the possibility of some confusion, we shall use the term mobility for the ratio of the drift velocity to the force for any force. We write v_drift = μF (43.14). in general, and we shall call μ the mobility (Feynman et al., 1963).”

 

Perhaps Feynman could have distinguished between electrical mobility and generalized mobility. His discussion begins with an electrical problem in which the force on a charged particle is electrical (F = qE). In this context, the proportionality constant between the drift velocity and the electric field is the electrical mobility, typically denoted as m_e, giving the equation v_d = m_eE. Electrical mobility is a physical property that quantifies how fast a charged particle drifts through a medium under an applied electric field. Mathematically, it is defined as the ratio of the drift velocity to the electric field strength: m_e = v_d/E. Experimentally, it can be measured by several techniques, including electrical mobility analyzers (EMAs) or Hall-effect measurements. Electrical mobility is a key transport parameter in semiconductor and ionized gases, and it plays a central role in the design of electronic devices such as transistors, diodes, and solar cells.

 

Note: In his Lectures on Computation, Feynman (1996) explains: “Drawing on engineering practice, we can write the drift velocity v_d in terms of the mobility m of the charge carriers as v_d = m_eE, E is the electric field across the drain/source (p. 225).”

 

When Feynman introduces the general equation v_d = mF, the quantity m does not represent electrical mobility but a generalized (or mechanical) mobility. This transport coefficient relates the drift velocity to any applied force—electrical, gravitational, or chemical. Under an electric field, where F = qE, this equation becomes v_d = mF = m(qE). Comparing this with the definition of electrical mobility, v_d = μ_eE, gives: μ_e = qm. Thus, a particle’s electrical mobility is simply its generalized mobility multiplied by its charge. The generalized mobility is also the quantity that appears in the Einstein diffusion relation, D = mkT, where D is the diffusion coefficient. Expressed in terms of the electrical mobility, the relation becomes D = μ_ekT/q. In summary, the generalized mobility is the more general charge-independent transport coefficient, while the electrical mobility is a special case that describes a particle’s response specifically to an electric field.

 

Key Takeaways (In Feynman’s Tone):

1. The "Special Molecule" Idea

Suppose you've a gas with an enormous number of molecules, all flying about, colliding with one another, and changing their directions every instant. Trying to follow every molecule would be hopeless. Instead, we single out one molecule—the special molecule, or S-molecule—and ask only what happens to it. Maybe it’s heavier than the others, maybe it carries a charge, or maybe it’s being pushed by an electric force while the background molecules simply collide with it. By focusing on this one molecule, we can understand the basic ideas behind many transport phenomena—diffusion, electrical conduction, and sedimentation. It's not the whole truth—it's an approach to get started.

 

2. Where the Drift Comes From

Now consider what happens to the S-molecule. Between collisions it is accelerated by an applied force, but every collision sends it off again in a new random direction. The collisions randomize the motion; the external force biases the direction.

The result is a tiny average velocity—the drift velocity—superimposed on the molecule’s much larger random thermal motion. You can barely see it, but it's this small average motion that produces measurable phenomena such as electric current, diffusion, and sedimentation.

 

3. Mobility: How Easily a Molecule Drifts

How fast does the molecule drift for a given force? That depends on the mobility, which is how readily the molecule moves through a medium. Here's the catch: you have to be careful what you mean by "mobility."

• If you define it as the drift velocity per unit force, we'll call it the generalized mobility—it works for any force, electric or others: m = v_d /F .
• If you define it as drift velocity per unit electric field, we call that the electrical mobility: m_e = v_d /E.

They're simply related: μ_e = qm. Thus, the electrical mobility is just the generalized mobility multiplied by the charge. So if you know one, you know the other. It's not two different physical properties—it's the same mechanism, just calibrated for different kinds of forces.

 

The Moral of the Lesson: From Drift Velocity to Cholesterol Transport

Cholesterol does not circulate through the bloodstream as isolated molecules. Because it is a hydrophobic lipid, it must be packaged into spherical vehicles known as lipoproteins—such as low-density lipoproteins (LDL) and high-density lipoproteins (HDL). From a physicist’s perspective, these lipoproteins can be viewed as Feynman’s "special molecules" (S-molecules) moving through a complex fluid environment crowded with background molecules. Just like the S-molecule, each lipoprotein undergoes continual collisions with surrounding water molecules, plasma proteins, and blood cells, while external forces—whether drag, gravity or electrical—can affect its motion.

 

One laboratory technique that illustrates this idea is electrophoresis, in which an electric field causes charged molecules to migrate through a supporting medium. Because different types of lipoproteins possess distinct surface charges arising from their apolipoproteins, they experience different electrical forces. After a brief period of acceleration, the electrical force is balanced by the viscous drag of the medium, causing each molecule to reach a constant drift velocity. The ratio of this drift velocity to the applied electric field is called the electrophoretic mobility. Although electrophoresis is used mainly for specialized lipoprotein analysis rather than routine cholesterol testing, it provides a real-world example of Feynman’s discussion of drift velocity and mobility.

 

Note on Density: The terms “low‑density” and “high‑density” originated from the separation of these molecules via an analytical ultracentrifugation. In a centrifuge, lipoproteins are segregated by their density—not by electrophoretic mobility.

 

The "Drift Velocity" Analogy for Atherosclerosis

The concept of drift velocity also provides a useful analogy for understanding atherosclerosis. LDL molecules do not simply “flow” passively through the bloodstream; rather, they undergo continual random motion while being transported by the flowing blood. The probability that a LDL molecule interacts with the arterial wall increases with its concentration and the condition of the vessel wall.  Consider a mechanical analogy: a healthy artery has a smooth endothelial lining (like a new road), while a damaged one has a rough, permeable surface where molecules can become trapped. Over time, as LDL molecules “drift” through the bloodstream, a higher concentration of them—akin to high traffic density—leads to more interactions with the artery wall. When too many LDL molecules accumulate in a damaged artery wall, they can become trapped and oxidized, triggering an inflammatory immune response that contributes to atherosclerosis. For this reason, many scientists consider the number of LDL molecules to be a better indicator of cardiovascular risk than the total amount of cholesterol—but LDL molecules can be further classified into several subclasses with differing biological properties.

 

The Science of Temporary LDL Increase After Intense Exercise

Intense exercise can cause a temporary increase in measured LDL cholesterol levels. This short term elevation is generally attributed to two physiological mechanisms rather than to any harmful effect of exercise. 

Hemoconcentration: Strenuous exercise, such as long-distance running, results in substantial fluid loss through sweating and respiration. The reduction in plasma volume temporarily increases the concentration of the blood, meaning that about the same number of LDL molecules remain within a smaller volume. Consequently, a lipid panel may show a transient increase in measured LDL concentration. In one study, total plasma cholesterol rose by 24% immediately post-exercise, with LDL-C increased by about 20% (Krum et al., 1991).

 

These changes are short-term physiological responses to intense exercise rather than evidence that exercise is harmful. As hydration is restored and the body recovers, lipid concentrations generally return to their usual baseline. More importantly, the long-term effects of regular physical activity maintain healthy lipid metabolism and reduce cardiovascular risk.

 

Summary and Takeaways

Distinguishing short-term transport phenomena from long-term equilibrium behavior is important in both physics and medicine. The short-term “rise” in LDL after intense exercise is a normal physiological response—not a sign that exercise is harming your heart. When you are well-rested and hydrated, regular exercise has a proven beneficial effect on cholesterol profiles, which is why doctors recommend it for heart health.

To ensure clinical accuracy and eliminate transient variables, cholesterol levels should be measured under consistent conditions—well-rested, fully hydrated, and following a standard 12-hour fast. Thus, doctors should advise the public to avoid strenuous or exhaustive exercise for 12-24 hours prior to a blood test. This precaution ensures that temporary transport dynamics do not obscure the patient’s true, steady-state lipid profile.

 

Review Questions:

1. Why did Feynman introduce the concept of the S-molecule, and what is the purpose of the "fresh start" assumption?

2. Why is it incorrect to assume that the average drift velocity is simply one-half of the final velocity between collisions?

3. How would you distinguish between generalized (mechanical) mobility and electrical mobility?

 

References

Feller, W. (1968). An Introduction to Probability Theory and Its Applications (Vol. 1, 3rd ed.). Wiley.

Feynman, R. P. (1996). Feynman lectures on computation. Reading, Massachusetts: Addison-Wesley.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Krum, H., Conway, E. L., & Howes, L. G. (1991). Acute effects of exercise on plasma lipids, noradrenaline levels and plasma volume. Clinical and experimental pharmacology and physiology, 18(10), 697-701.