Section 40–4 The distribution of molecular speeds

 Molecular collisions / Independent of direction / Relativistic effects

 

This section can be examined from the perspectives of molecular collisions, directional independence, and relativistic effects. Strictly speaking, Feynman is not explaining the distribution of molecular speeds, but the distribution of molecular velocities. For example, Figure 40-5 depicts a velocity distribution function with a Gaussian, bell-shaped form, which differs from the speed distribution function (chi distribution). However, the section could be titled “The Maxwell–Boltzmann distribution,” which refers broadly to the probability governing molecular motion in an ideal gas at thermal equilibrium.

 

1. Molecular collisions:

“Now we return to the question about the neglect of collisions: Why does it not make any difference? We could have pursued the same argument, not with a finite height h, but with an infinitesimal height h, which is so small that there would be no room for collisions between 0 and h. But that was not necessary: the argument is evidently based on an analysis of the energies involved, the conservation of energy, and in the collisions that occur there is an exchange of energies among the molecules. However, we do not really care whether we follow the same molecule if energy is merely exchanged with another molecule. So it turns out that even if the problem is analyzed more carefully (and it is more difficult, naturally, to do a rigorous job), it still makes no difference in the result (Feynman et al., 1963).”

 

Feynman emphasizes that neglecting collisions in deriving the molecular velocity distribution does not alter the final result, because what matters is the total molecular energy, not which molecule carries it. Even though collisions continuously redistribute energy among molecules, total energy is conserved, so the statistical distribution of velocities remains unchanged. In Illustrations of the dynamical theory of gases, Maxwell (1860) writes: “the mean distance travelled over by a particle between consecutive collisions, = 1/447000th of an inch, and each particle makes 8,077,200,000 collisions per second (p. 32).” This shows that Maxwell did not simply ignore collisions; rather, he derived the mean free path—the average distance of a molecule traveled between collisions—without tracking individual trajectories. However, Maxwell did not elaborate on how collisions drive gases toward thermal equilibrium.

In contrast, Boltzmann placed collisions at the heart of his model: they enable the transfer of energy among molecules, gradually reshaping the distribution of energies and velocities until the equilibrium distribution is established is reached. This is related to Boltzmann’s H-theorem, which shows that when collisions between molecules are allowed, such distributions tend to irreversibly seek towards the minimum value of H. Molecular collisions are therefore essential for explaining not just the form of the distribution, but also its dynamic emergence. Feynman’s discussion, which downplays collisions, is better seen as a pedagogical simplification to focus on intuition and the final distribution. However, it is worthwhile to recognize the distinct contributions of Maxwell and Boltzmann in shaping our understanding of the molecular velocity distribution.

 

Note: In Further Studies on the Thermal Equilibrium of Gas Molecules, Boltzmann (1872) writes “This is essentially the result already obtained in another way by Maxwell: once this velocity distribution has been reached, it will not be disturbed by collisions (p. 263)……. As a result of collisions, many molecules will acquire larger velocities and others will come to have smaller velocities, until finally a distribution of velocities among the molecules is established such that it is not changed by further collisions. In this final distribution, in general all possible velocities from zero up to a very large velocity will occur (p. 265).” It shows that Boltzmann explicitly identified collisions as the mechanism by which gases approach thermal equilibrium.

 

2. Independent of direction:

“So far we have, of course, only the distribution of the velocities “vertically.” We might want to ask, what is the probability that a molecule is moving in another direction? Of course these distributions are connected, and one can obtain the complete distribution from the one we have, because the complete distribution depends only on the square of the magnitude of the velocity, not upon the z-component. It must be something that is independent of direction, and there is only one function involved, the probability of different magnitudes (Feynman et al., 1963).”

 

In Illustrations of the Dynamical Theory of Gases, Maxwell (1860) assumed “the existence of the velocity x does not in any way affect that of the velocities y or z, since these are all at right angles to each other and independent (p. 22).” This independence assumption (or intuition) allowed him to factorize the probability distribution function. However, his contemporaries were uneasy with his approach: while isotropy could be accepted as a consequence of spatial symmetry, statistical independence of perpendicular components seemed a less obvious claim (Brush, 1976, pp. 177–179). In On the Dynamical Theory of Gases, Maxwell (1867) responded to their concerns by providing a more rigorous justification, but it was not a direct use of Newton’s laws of motion. With this statistical proof, Maxwell strengthened the foundations of kinetic theory despite its limitations of applicability.

 

It is important to recognize that isotropy (independence of direction) and the statistical independence of velocity components are distinct concepts. Isotropy means all directions of motion are equally probable in an ideal gas and the distribution is rotationally invariant. Statistical independence means the probability distribution for one component of velocity (e.g., v_x) is independent of the distributions for the perpendicular components (v_y, v_z). Currently, the Maxwell–Boltzmann distribution is derived from the canonical ensemble, where the Gaussian form of the velocity distribution leads to both isotropy and statistical independence as results rather than assumptions. In a sense, Maxwell effectively inverted the modern reasoning by assuming statistical independence right at the beginning. However, neither directional independence nor statistical independence are universally valid features of all velocity distributions.

 

Note: Maxwell’s derivation of the distribution of molecular velocities rested on two key assumptions: (1) Isotropy (independent of direction) and (2) Statistical Independence (Brush, 1976). Maxwell later recognized that the second assumption was precarious. The assumption of isotropy does not necessarily imply the statistical independence of the variables along different directions (Walstad, 2013). To be precise, the Maxwell–Boltzmann distribution was derived under the assumptions of an ideal gas with no external force fields or significant gravitational effects.

 

Of course these distributions are connected, and one can obtain the complete distribution from the one we have, because the complete distribution depends only on the square of the magnitude of the velocity, not upon the z-component (Feynman et al., 1963).”

 

The word connected used by Feynman is potentially misleading, since in an ideal gas at equilibrium the velocity components are statistically independent. One possible clarification is that the connection is mathematical rather than physical. Mathematically, the Gaussian form of the full distribution f(v_x, v_y, v_z) implies the probability law for any single component (e.g., v_z), so the “connection” follows from probability theory and symmetry. Physically, however, the velocity components are uncorrelated—random collisions do not link v_x and v_z, e.g., and no force would continuously connect them together. Recognizing this distinction prevents the potential misconception that the distribution of  molecular velocities arises from some inherent physical linkage between the velocity components, rather than from statistical symmetry.

 

3. Relativistic effects:

“Since velocity and momentum are proportional, we may say that the distribution of momenta is also proportional to e^−K.E./kT per unit momentum range. It turns out that this theorem is true in relativity too, if it is in terms of momentum, while if it is in velocity it is not, so it is best to learn it in momentum instead of in velocity: f(p)dp=Ce^−K.E./kTdp (Feynman et al., 1963).”

 

Feynman emphasizes that the Maxwell–Boltzmann distribution can be expressed in terms of momentum rather than velocity. In the non-relativistic case, momentum and velocity are proportional p = mv, so the probability distributions in either variable are equivalent, with the distribution proportional to e^−K.E./kT. In the relativistic case, the proportionality between momentum and velocity breaks down, so the velocity distribution cannot be represented by a simple exponential form. However, expressing the distribution in terms of momentum, f(p)dp=Ce^−K.E./kTdp, still preserves the exponential form even relativistically. Therefore, Feynman suggests that the distribution in terms of momentum is more fundamental and general, allowing it to be applied beyond classical speeds. This approach highlights that momentum-based distributions provide a consistent description of thermal equilibrium for both classical and relativistic particles.

 

Feynman’s statement requires careful interpretation in the context of special relativity, where momentum and velocity are no longer simply proportional and directional effects become significant. The correct relativistic generalization is the Maxwell–Jüttner distribution. Unlike the Maxwell–Boltzmann distribution, which is isotropic in the rest frame of the gas, the Maxwell–Jüttner distribution exhibits apparent directional dependence when viewed from a moving frame because of relativistic transformations. In this respect, Feynman’s explanation oversimplifies the relativistic case and risks leaving the impression that the classical Maxwell–Boltzmann form remains valid in special relativity. However, the Maxwell–Jüttner distribution reduces to the Maxwell–Boltzmann distribution in the non-relativistic limit, thereby unifying the description of thermal equilibrium for both classical and relativistic gases.

 

“Of course these distributions are connected, and one can obtain the complete distribution from the one we have, because the complete distribution depends only on the square of the magnitude of the velocity, not upon the z-component. (Feynman et al., 1963).”

 

Feynman’s claim that the full velocity distribution can be built from one component by assuming independence of directions overlooks a key caution raised by Maxwell himself: the assumption that velocity components along x, y, and z are statistically independent is not self-evident. Maxwell introduced this hypothesis in his 1860 paper Illustrations of the Dynamical Theory of Gases, but explicitly remarked that its validity was “dubious,” since collisions or hidden correlations might couple motion in different directions. While in the classical, nonrelativistic regime this assumption works and leads to the Maxwell–Boltzmann form, in special relativity it fails outright: the finite speed limit c ties the velocity components together, so they cannot be treated as independent Gaussians. Thus, both Maxwell’s original caution and relativistic constraints suggest that Feynman’s simplified statement about independence of directions is not generally correct.

 

Difference between distribution of molecular velocities and molecular speeds:

The distribution of molecular velocities gives the probability that a molecule has a specific velocity vector, i.e., that its components lie between v_x and v_{x +} dv_x, v_y and v_{y +} dv_y, and v_z and v_{z +} dv_z. By contrast, the distribution of molecular speeds gives the probability that a molecule has a certain speed, regardless of direction. In this case, the probability of finding a molecule with speed between v and v + dv is obtained by summing over all velocity vectors whose magnitude is v. Geometrically, this corresponds to integrating over the spherical shell of radius  in velocity space. Thus, the velocity distribution is directional and vector-based, while the speed distribution is scalar and direction-independent.

 

Analogy: To see the difference between the distribution of molecular velocities and the distribution of molecular speeds, imagine throwing darts at a flat target. The velocity distribution is like asking for the probability of a dart landing in a tiny square at some coordinates (x, y) on the target—it tracks direction as well as magnitude. The speed distribution, by contrast, is like asking for the probability that a dart lands at a given distance from the bullseye, regardless of angle; what matters is the radius, not the coordinates. There is only one point lies at the bullseye (speed = 0), but entire rings of points exist at larger radii (higher speeds), so higher speeds could be more likely. The velocity components follow Gaussian (bell-curve) distributions, while the speed follows a chi distribution that rises from zero, peaks, and then tails off asymmetrically.

 

Historical Note:

In his 1867 paper On the Dynamical Theory of Gases, Maxwell reflected on earlier mistakes: “I also gave a theory of diffusion of gases, which I now know to be erroneous, and there were several errors in my theory of the conduction of heat in gases which M. Clausius has pointed out in an elaborate memoir on that subject (p. 51).” Later in the same paper, he addressed a key assumption in his derivation: “1 have given an investigation of this case, founded on the assumption that the probability of a. molecule having a velocity resolved parallel to x lying between given limits is not in any way affected by the knowledge that the molecule has a given velocity resolved parallel to y. As this assumption may appear precarious, I shall now determine the form of the function in a different manner (p. 62).” His remark reveals an early awareness that the independence of velocity components was not self-evident but required justification. The fact that he attempted to resolve the potential problem within the same work shows both his humility and scientific thoroughness.

 

Key Takeaway:

The behavior of gas molecules can be described only statistically, not deterministically, with equilibrium distributions emerging from symmetry, randomness, and conservation laws. Maxwell’s work shows that assumptions such as isotropy and statistical independence—when carefully justified—lead naturally to the Gaussian form of the velocity distribution (the Maxwell–Boltzmann distribution), which successfully explains macroscopic properties like pressure and temperature. His treatment also highlights the importance of making assumptions explicit, testing their validity, and, where possible, providing multiple lines of justification, as he did with the independence of velocity components. Importantly, the Maxwell–Boltzmann distribution is not obvious—without Maxwell’s insight, one might wrongly expect velocities to be uniformly distributed, missing the statistical order in molecular chaos.

 

The Moral of the lesson: Maxwell’s willingness to acknowledge weaknesses in his earlier paper, and to replace them with a more rigorous derivation, reflects both intellectual honesty and scientific integrity.

 

Review Questions:

1. Should molecular collisions be neglected when deriving the distribution of molecular velocities, and what role do they play in reaching equilibrium?

2. Are molecular velocities truly independent of direction in the context of the Maxwell–Boltzmann distribution?

3. Does Maxwell’s distribution of molecular velocities remain the same or valid at relativistic speeds if it is expressed in terms of momentum?

 

References

Boltzmann, L. (1872). Further studies on the thermal equilibrium of gas molecules. In The kinetic theory of gases: an anthology of classic papers with historical commentary (pp. 262-349).

Brush, S. G. (1976). The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century. Amsterdam: North-Holland.

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.

Maxwell, J. C. (1860). V. Illustrations of the dynamical theory of gases.—Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 19(124), 19-32, 281–291.

Maxwell, J. C. (1867). On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London, 157, 49–88.

Walstad, A. (2013). On deriving the Maxwellian velocity distribution. American Journal of Physics, 81(7), 555-557.