Section 43–5 Molecular diffusion

Fick’s diffusion law / Einstein’s diffusion relation / Boltzmann distribution

 

In this section, Feynman derives Fick’s first law of diffusion, then proceeds with Einstein's diffusion relation, and shows why the Boltzmann distribution is essential for completing the derivation. However, the title “Molecular Diffusion” is overly narrow, describing only a physical mechanism. A more revealing title could be “Einstein’s Diffusion Relation: Balancing Drift and Diffusion,” which better captures the central concept and logical structure. The section’s deeper theme is not merely how molecules spread out, but the profound connection between random diffusive transport and force-induced drift—two seemingly distinct processes linked by the thermal energy. At thermal equilibrium, diffusion and drift are opposing transport tendencies whose balance reveals a fundamental relation between thermal fluctuations and dissipative response, sometimes known as the fluctuation-dissipation theorem.

 

1. Fick’s diffusion law

“In terms of n_a we can express the difference (n₊−n₋) as (n₊−n₋) = (dn_a/dx)Δx = (dn_a/dx)×2l (43.23). Substituting this result in Eq. (43.22) and neglecting the factor of 2, we get J_x=−lvdn_a/dx (43.24). We have found that the flow of special molecules is proportional to the derivative of the density, or to what is sometimes called the ‘gradient’ of the density (Feynman et al., 1963).”

 

Perhaps Feynman could have stated Fick’s first law of diffusion, which describes the net transport of particles through a medium. Specifically, diffusion is the spontaneous spreading of particles—typically in a liquid or gas—moving from regions of higher concentration to regions of lower concentration. In one dimension, Fick’s first law is expressed as J_x = -D(dn/dx), which means that the diffusive flux (J_x) is proportional to the magnitude of the concentration gradient (dn/dx) and the diffusion coefficient (D). The negative sign does not mean the gradient itself is negative; rather, it indicates that the flux moves down from regions of higher concentration to lower concentration. Strictly speaking, Fick’s first law is applied under steady-state conditions, where the concentration remains constant and the flux into and out of each volume element is equal.

 

Note: For simplicity, Feynman’s notation n_a is replaced by n. Some authors prefer the symbol c to represent concentration.

 

A 3-step Derivation of Fick’s First Law of Diffusion

Step 1: Relate Particle Flux to Particle Densities

Consider an imaginary plane perpendicular to the x-axis. The net diffusive flux J_x across the plane is the difference between the particles moving from the left and the right:

J_x = (n_- - n₊)v

where n_- and n₊ are the particle number densities on the left and right sides of the plane, respectively, and v is the average molecular speed. In general, particle flux is defined as the product of the number density and the velocity component normal to the surface.

 

Step 2: Evaluate the Densities One Mean Free Path Away

A molecule crossing the plane typically made its last collision, on average, one mean free path l away. Thus, n₋ and n₊ correspond to the number densities at x - l and x + l, respectively.

Assuming a smooth density gradient and using Taylor’s expansion, the density difference is:

n_- - n₊ » -2l (dn/dx)

Substituting this approximation into the particle flux expression gives:

J_x » - lv(dn/dx)

 

Step 3: Refine the Geometry and Identify the Diffusion Coefficient

The above estimate assumes that all molecules move perpendicular to the plane. In reality, molecules travel in random directions in three-dimensional space. A more rigorous treatment requires a geometric factor of 1/3 for the particle flux moving along the x-axis:

J_x » -(lv/3)(dn/dx)

Comparing this expression with Fick’s first law J_x » -D(dn/dx) allows us to identify the diffusion coefficient: D = lv/3.

 

“All of these refinements can be made; the result of a more careful analysis shows that the right-hand side of Eq. (43.24) should be multiplied by 1/3. So a better answer is J_x = −(lv/3)dn_a/dx (Feynman et al, 1963).”

Feynman briefly notes that a more careful analysis introduces a geometric factor of 1/3, giving the diffusion coefficient D = lv/3. This factor arises from the isotropic three-dimensional molecular motion: only the component of a molecule’s motion along the concentration gradient contributes to the net diffusive flux. However, this is based on a simplified model because it assumes a single molecular speed and isotropic spreading.

In physical systems, more rigorous frameworks, such as the Chapman–Enskog theory, yield transport coefficients that depends on the temperature and complex intermolecular potentials. For example, the Enskog-type corrections refine the diffusion coefficients as:

where D₀ is a constant, n is the number density, g(s) is the radial distribution function evaluated at the molecular contact diameter (s), and W(T) is a collision integral that characterizes the temperature-dependent interaction potential.

 

Note: In the standard elementary derivation, the one-way particle flux crossing a reference plane in the +x direction is written as nv/6 instead of nv/3 (Reif, 1965), because only one-sixth of the molecules—those with velocity components in the +x direction—contribute equally in the three orthogonal dimensions (±x, ±y, ±z). The six faces of a die serve as a useful analogy for these six equally probable directions.

 

2. Einstein’s diffusion relation

“We find that D, the diffusion coefficient, is just kT times μ, the mobility coefficient: D=μkT (43.31) (Feynman et al, 1963).”

 

A Condensed 3-Step Derivation of Einstein’s Diffusion Relation

This section could be confusing because Feynman uses nearly 20 equations to develop Fick’s first law of diffusion and then derive Einstein’s diffusion relation. To make this development more accessible, the derivation is reduced to three crucial steps:

 

Step 1. Two Opposing Fluxes

Particle motion in this context arises from two competing mechanisms:

Diffusion flux (J_diffusion): Random thermal motion causes particles to spread from regions of higher concentration to lower concentration. According to Fick’s First Law of Diffusion, the particle flux is proportional to the density gradient:

J_diffusion = -D(dn/dx)

Drift flux (J_drift): An external force (e.g., gravity) causes a systematic drift velocity v_d = mF, where m is generalized mobility. The resulting particle flux is:

J_drift = nv_d = nmF

 

Step 2. Zero Net Flux

At thermodynamic equilibrium, the diffusive and drift fluxes exactly balance:

J_diffusion + J_drift = 0

Substituting the expressions for both fluxes gives:

-D(dn/dx) + nmF = 0 Þ D(dn/dx) = nmF

The balance equation links the diffusion coefficient D to the mobility m and density gradient.

 

Step 3. Apply the Boltzmann distribution

At thermal equilibrium, the particle density follows the Boltzmann distribution:

n = n₀e^-U/kT

Differentiating this expression with respect to x and using F = -dU/dx gives

dn/dx = nF/kT (see equation 43.38)

By comparing the two expressions for dn/dx, we have nmF/D = nF/kT

By canceling out the density and force, we get Einstein’s diffusion relation:

D = mkT

 

Note: By balancing drift and diffusion, Feynman demonstrates that mobility (m) and diffusion (D) are not independent properties—they are two sides of the same coin, linked directly by thermal energy (kT).

 

“We now adjust the force F so that the drift current due to F just balances the diffusion, so that there is no net flow of our special molecules. We have J_x+J_drift = 0, or Ddn_a/dx = n_aμF (43.35) (Feynman et al., 1963).”

 

Reconciling Feynman’s Remark: Does Diffusion Stop at Equilibrium?

In the audio recording (48 min: 45 sec) of his lecture, Feynman remarks that it is possible to adjust a force so that there is “no diffusion” to achieve equilibrium—an intriguing comment that was omitted from the published text. This seems paradoxical because Feynman also states: “The current J_x (=J_diffusion) and the density gradient dn_a/dx can be measured by macroscopic observations.” However, the apparent paradox could be resolved by distinguishing between two different definitions of diffusion.

 

1. The Operational view: “No Diffusion”

The “no diffusion” view is defensible if diffusion is defined operationally as the observable net spreading or redistribution of particles. At thermodynamic equilibrium, the particle-density distribution is stationary and the net particle flux vanishes (J_net = 0). There is therefore no observable net transport or sustained redistribution of particles. In principle, this condition can be verified experimentally by monitoring the number of particles crossing a specified surface—e.g., counting particles in successive microscope images. In this operational sense, one may say that there is no diffusion at equilibrium because there is no observable net transport from one region to another.

 

2. The Transport-Theory view: “Diffusion Persists”

The “diffusion persists” view focuses on the individual transport mechanisms that contribute to the observed flux. Here, diffusion is defined as the spreading of particles driven by random thermal motion, where a density gradient supports a diffusive flux J_diffusion = -D(dn/dx). At thermodynamic equilibrium, the diffusion flux is exactly opposed by a force-induced drift flux J_drift = nmF, such that J_diffusion + J_drift = 0. Thus, the absence of net transport does not imply that each transport contribution has vanished. This is analogous to a book resting on a table: a downward gravitational force and an upward normal force continue to act even though the net force is zero. This view analyzes the underlying transport mechanisms and leads directly to Einstein’s diffusion relation.

 

Bridging the Two perspectives: Measurement vs. Mechanism

The apparent disagreement can be framed as a philosophical distinction between an operational view and a mechanistic view of diffusion—a distinction rooted in the difference between what is measured and how the measurement is interpreted. The operational view defines diffusion in terms of its observable macroscopic consequences, such as net particle flux or redistribution. The mechanistic view focuses on the underlying physical process: continuous random thermal motion. Although Feynman explains that diffusion can be measured macroscopically via the density gradient, an experiment can only measure the net flux—it cannot directly isolate the “diffusive” and “drift” fluxes. Thus, Feynman’s explanation reveals a subtle ambiguity in the word diffusion: the experimentally measured quantity is the net flux, while the transport theory decomposes that flux into diffusive and drift components.

 

Expanding the Definition: Normal versus Anomalous Diffusion

There is an additional complication when we are considering the precise definition of diffusion in the context of Brownian motion. The classical theory describes a transport regime where the mean-square displacement scales linearly with time, which underlies the familiar Einstein relation. More generally, if the mean-square displacement scales as <r²> µ t^a, then a = 1 corresponds to normal diffusion, whereas a ¹ 1 indicates anomalous diffusion (e.g., subdiffusion for a < 1, superdiffusion for a > 1). In  the anomalous case, a constant diffusion coefficient defined via the Einstein relation is no longer adequate; the effective diffusivity may depend on time or length scale, reflecting the complex underlying stochastic process. One might argue that Feynman could have problematized the definition of diffusion explicitly, or critiqued contemporary definitions of diffusion.

Source: Classification of stochastic processes by convolutional neural networks - IOPscience

 

Conclusion: A Semantic Distinction with Physical Significance

The apparent conflict is partly semantic because it depends on how one defines diffusion. However, the physical distinction is important: macroscopically, there is no observable diffusion as net spreading because the diffusive and drift fluxes exactly balance; microscopically, random thermal motion persists, and a diffusive flux contribution can still be mathematically identified. Thus, we may conclude that diffusion persists as an underlying microscopic tendency, while there is no macroscopic net redistribution of particles at thermodynamic equilibrium. Feynman’s phrase “no diffusion” is therefore best understood as referring to the absence of observable net transport, rather than the disappearance of microscopic random molecular motion.

 

Note: For clarity, Feynman’s equation J_x + J_drift = 0 is rewritten here as J_diffusion + J_drift = 0. Because this equilibrium balance is central to the derivation, it would be reasonable to assign it a numbered equation.

 

3. Boltzmann distribution

“[This is just exactly Eq. (40.2), from which we deduced e^−U/kT in the first place, so we have come in a circle] (Feynman et al., 1963).”

Addressing the “Circular Logic” Critique

The editors’ inserted comment—“…we have come in a circle”—might lead some readers (e.g., philosophers) to suspect that Feynman’s derivation is logically circular, as if he assumed what he set out to prove. A more appropriate interpretation, however, is that the argument functions as a consistency check: it shows that the Boltzmann distribution is compatible with the dynamical balance between diffusive and drift fluxes at thermodynamical equilibrium. Furthermore, the Boltzmann distribution is not a fixed, universal formula; its precise form depends on the physical constraints and boundary conditions of a given system. Historically, Einstein did not anchor his argument explicitly in terms of Boltzmann distribution. Instead, he referred to distribution of random errors, linking Brownian motion to the statistics of random walk. In Feynman’s derivation, the Boltzmann distribution is valuable not merely as an exponential function that simplifies the algebra, but because it connects the microscopic nature of thermal motion with the macroscopic equilibrium states that emerge from that randomness.

 

“We have shown that Eq. (43.31), which gives the diffusion current in terms of the mobility, has the correct coefficient and is very generally true. Mobility and diffusion are intimately connected. This relation was first deduced by Einstein (Feynman et al., 1963).”

 

In his 1905 paper on Brownian motion, Einstein did not express the diffusion relation in the modern form D = mkT. He did not formulate his theory in terms of a generalized “mobility” coefficient (m)—he used the symbol m for the mass of the suspended particle. Instead, Einstein derived the relation by balancing the osmotic force against the Stokes drag force on spherical particles suspended in a liquid, obtaining D = (RT/N)(1/6pkP). In this equation, R is the gas constant, N is the Avogadro’s constant, P is the radius of the suspended particle, and k is the coefficient of viscosity of the liquid—not the Boltzmann constant. According to Stokes’ law, the drag force is given by F = 6pkPv_d. From this, the mechanical mobility (m) can be identified as the inverse of the drag coefficient: m = v_d/F = 1/6pkP. Modern physics textbooks (including Feynman’s lectures) substitute the mobility coefficient m and Boltzmann’s constant (k = R/N) to rewrite Einstein’s relation in its generalized form D = mkT. Thus, the modern form is best considered not as the equation Einstein originally wrote, but as a reformulation of the relation he deduced.

 

The Complex History of the Diffusion Relation

William Sutherland had already presented his results on the diffusion coefficient in June 1904 at the meeting of the Australian Association for the Advancement of Science held in Dunedin, New Zealand. Independently, Marian Smoluchowski arrived at a similar diffusion relation around the same time, employing a random walk framework. Einstein subsequently published his derivation in 1905 using a different but closely related theoretical approach. Thus, the diffusion relation has a more complex history than the familiar label “Einstein relation” suggests. To properly acknowledge these independent contributions, it is sometimes referred to as the Sutherland-Einstein relation or Smoluchowski-Einstein relation.

 

Key Takeaways:

1. Fick’s First Law of Diffusion: Diffusion originates from random thermal motion and molecular collisions, rather than being caused by the density gradient (which determines the direction and magnitude of net flow). According to Fick’s diffusion law, the diffusion flux is directly proportional to the magnitude of density gradient and the diffusion coefficient. Furthermore, the geometrical factor of 1/3 arises from three-dimensional angular averaging, rather than serving as an arbitrary numerical correction.

2. Einstein’s diffusion relation: Thermodynamic equilibrium is a dynamic steady-state, not a static cessation of microscopic motion. Einstein’s diffusion relation can be derived by establishing an equilibrium condition, where the diffusive flux and the drift flux cancel exactly. This directly links the diffusion coefficient to the generalized mobility and thermal energy.

3. We need not simply criticize Feynman for saying that it is possible to adjust a force so that there is “no diffusion” and thereby achieve equilibrium. The more useful approach is to distinguish between two levels of descriptions. From an operational perspective, diffusion may be measured by observable net transport or spreading. From a mechanistic perspective, random thermal morion continues indefinitely. Thus, the apparent ambiguity is a matter of how diffusion is defined and at what level it is being measured, rather than a contradiction in the underlying physics.

 

The Moral of the Lesson:

Einstein’s diffusion relation provides a useful framework for understanding how molecular size, mobility, thermal motion, and transport properties are interconnected. Although this relation is not a routine quality-control equation used in food manufacturing, its underlying principles help explain how molecular transport changes during processes such as food fermentation and how such changes can be investigated using appropriate analytical methods.

 

1. Yogurt Production and Protein Mobility

During yogurt fermentation, starter cultures (e.g., Lactobacillus bulgaricus) acidify the milk and contribute to the proteolysis of milk proteins, producing smaller peptides and free amino acids. Based on Einstein’s relation, smaller solutes generally diffuse faster because their hydrodynamic drag is lower. This provides a useful conceptual link between fermentation-induced proteolysis and molecular transport.

 

2. Greek Yogurt vs. Greek-Style Yogurt

Traditional Greek yogurt is produced by fermenting milk and straining away much of the whey—a process that concentrates the milk solids and proteins to create a naturally thick texture. However, the term “Greek-style” does not inherently imply an inferior yogurt. Greek-style yogurts can achieve a comparable texture and nutritional profile using milk-protein concentrates, milk solids, starches, gelatin, or stabilizers, and many have an excellent nutritional profile. A more meaningful comparison relies on the nutrition and ingredient label: protein, carbohydrates, sugars, fat, sodium, and calcium content. For a simple choice, plain, unsweetened Greek yogurt with a relatively high protein content and minimal additives remains a reasonable option.

 

3. Kefir versus Yogurt

Kefir is produced using kefir grains, which contains a more diverse strains of bacteria and yeasts than typical yogurt. However, claiming that kefir is better for gut health is an oversimplification. The health effects of fermented foods depend on the specific product, microbial strains, individual physiology, and overall diet (including prebiotics). Greek yogurt is advantageous when a concentrated, high-protein food with a thick texture is preferred, whereas kefir provides a drinkable fermented food with a more diverse  microbial community. They are best viewed as complementary rather than competing probiotics.

 

4. Smoluchowski and Transport Theory

Einstein’s and Smoluchowski’s treatments of Brownian motion were closely related but differed in their mathematical formulations and assumptions. Smoluchowski, a professor of physics at the University of Lemberg (now Lviv, Ukraine), died in 1917 during a dysentery epidemic. Dysentery is an intestinal infection causing severe diarrhea (often containing blood or mucus), abdominal cramps, and fever. The primary measures for preventing such infection are safe food preparation, clean drinking water, proper sanitation, and good hygiene. Modern probiotics such as yogurt and kefir contribute to a healthy dietary pattern and support a balanced gut microbiome, but they should not be regarded as substitutes for basic hygiene, sanitation, or medical care.

 

5. A Notable Quotation

While Einstein developed his theory of translational Brownian motion in 1905, he expanded his theory in 1906 to include rotational Brownian motion. Although Einstein initially doubted the feasibility of measuring rotational Brownian motion, his analysis showed that thermal fluctuations should produce not only translational random motion, but also the rotational random motion of suspended particles. Jean Baptiste Perrin later experimentally verified Einstein’s theory of Brownian motion, helping to prove the physical reality of atoms.

In his Nobel Lecture, Perrin reflected on a broader philosophical lesson between the observable and the invisible: “Lastly, and doubtless always, but particularly at the end of the last century, certain scholars considered that since the appearances on our scale were finally the only important ones for us, there was no point in seeking what might exist in an inaccessible domain. I find it very difficult to understand this point of view since what is inaccessible today may become accessible tomorrow (as has happened by the invention of the microscope), and also because coherent assumptions on what is still invisible may increase our understanding of the visible.” Perrin’s quotation provides an especially fitting conclusion to the discussion of diffusion, Brownian motion, and transport theory.

 

Review Questions:

1. Why does the kinetic-theory expression for the diffusion coefficient contain the factor 1/3? Explain why it should not be interpreted as meaning that only one-third of the molecules participate in diffusion.

2. What does diffusion mean in the context of Einstein’s relation? Explain whether a diffusive flux can be present while the net flux is zero at equilibrium when it is exactly balanced by an opposing drift flux.

3. Why does Feynman invoke the Boltzmann distribution in deriving Einstein’s diffusion relation? Explain why should this use of the Boltzmann distribution be regarded as an equilibrium constraint or consistency condition?

 

References:

 

Einstein, A. (1905). On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat. Annalen der Physik, 17, 549–560.

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.

Fick, A. (1855). On Liquid Diffusion. Philosophical Magazine, Vol. 10, pp. 30–39.

Perrin, J. B. (1926). Discontinuous structure of matter. In Nobel Lectures in Physics (Vol. 2). World Scientific.

Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw-Hill.

Spiechowicz, J., Marchenko, I. G., Hänggi, P., & Łuczka, J. (2022). Diffusion coefficient of a Brownian particle in equilibrium and nonequilibrium: Einstein model and beyond. Entropy, 25(1), 42.

Sutherland, W. (1905). LXXV. A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 9(54), 781-785.

Sutherland, W. (1904). The measurement of large molecular masses. In Report of the 10th Meeting of the Australasian Association for the Advancement of Science, Dunedin (pp. 117-121).