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., vx) is independent of
the distributions for the perpendicular components (vy, vz). 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 approach by assuming statistical independence
right at the beginning. However, directional independence and statistical
independence are not 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 thermal 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(vx, vy,
vz) implies the probability law for any single component (e.g., vz), so the
“connection” follows from probability theory and symmetry. Physically, the
velocity components are uncorrelated—random collisions do not somehow connect vx, vy,
and vz of moving molecules together by some unknown
forces or influences. 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 does not hold, so the velocity distribution cannot
remain as a simple form. However, expressing the distribution in terms of
momentum, f(p)dp=Ce−K.E./kTdp, preserves the
exponential form even at relativistic speeds. Therefore, Feynman suggests that
the distribution in terms of momentum is more fundamental, which is applicable
beyond classical speeds. This approach highlights that momentum-based
distributions provide a consistent description of thermal equilibrium for particles
moving at classical and relativistic speeds.
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.
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 vx and
vx + dvx, vy and vy
+ dvy, as well as vz and vz
+ dvz. 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
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Source: Maxwell–Boltzmann distribution | tec-science |
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Source: Derivation of the Maxwell-Boltzmann distribution function | tec-science |
Historical Note:
In his 1867 paper On
the Dynamical Theory of Gases, Maxwell reflected on his 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.