If a great many equal spherical particles were in motion in a perfectly elastic vessel, collisions would take place among the particles, and their velocities would be altered at every collision; so that after a certain time the vis viva [kinetic energy] will be divided among the particles according to some regular law. – James Clerk Maxwell [1]

[Note: the below is an excerpt from Chapter 41, “Maxwell: the rise of statistical mechanics,” in my book Block by Block – The Historical and Theoretical Foundations of Thermodynamics.]

It was really James Clerk Maxwell (1831-1879) who, with his advanced mathematical skills, raised Clausius’s kinetic theory of gases to a much higher level of theoretical power, paving the way for the future contributions of Boltzmann and others to create statistical mechanics. An important step in his work concerned the evolution of his assumptions regarding the distribution of molecular velocities in a gas system.

What did Maxwell believe to be true?

Maxwell deeply believed that while collisions amongst a large number of moving bodies are random, “some regular law” must describe the end result.  Just as each coin flip is random, the distribution between heads and tails after a large number of flips isn’t.  The statistics involved are clear and powerful and show how chaos with small numbers can lead to structure with large numbers.  In the world of physics, it was Maxwell who first led us in this direction.

A leap of faith

The history surrounding Maxwell’s introduction of statistics into physics—“a new epoch in physics” [2]—is very interesting.  When reading his 1860 paper, there’s a moment where he lays out some rather simple-sounding, “these-make-sense” assumptions which had nothing to do with his preceding discussion on collision theory and the conservation of energy, and then bam!, seemingly out of nowhere he states, “Solving this functional equation, we find”[3] which is followed by the Gaussian distribution.  When I first read this, eager to see Maxwell’s rational for his distribution, eager to finally learn its origin, my reaction was, “Say what?”

Did Maxwell see the answer before he started the process?

While Maxwell wrote as if he were following a true deductive process, historians suggested an alternative and more likely version of events, one in which he saw the answer before he started the process.  They suggested that Maxwell knew where he needed to go and just had to figure out how to get there.

In his 1860 paper Maxwell stated, “It appears from this proposition that the velocities are distributed among the particles according to the same law as the errors are distributed among the observations in the theory of the “method of least squares,”’[4] a method that he first learned about in an 1850 review by John Herschel (1792-1871) of Adolphe Quetelet’s  (1796-1874) 1846 book, Letters on the Theory of Probability.  As a student in Edinburgh, Maxwell read Herschel’s review the year it was published and was deeply impressed, writing about it to his friend Lewis Campbell, “…the true logic for this world is the Calculus of Probabilities.”[5] This was an interesting statement to make during a time when determinism held sway.  But, in fact, there was no conflict here.  The elastic collision of spheres is deterministic.  But when multiplied by the reality of millions of spheres, such determinism falls apart, not because it’s no longer there but instead because one simply can’t keep track of it all.  Rather than simply giving up, Maxwell realized that statistics could provide a pathway out based on his belief that nature’s seemingly random behavior at the small scale gives way to a structured behavior at the large scale that could be captured by the statistical laws of large numbers.

We simply don’t know what Maxwell’s exact path of discovery was

In considering this history in greater depth, here’s what we know.  We know that Maxwell not only knew of Herschel’s essay but was likely influenced by it in his 1860 derivation,[6] which would explain the lack of physics in the derivation.  Turning now to what we don’t know, we don’t know what Maxwell’s exact path of discovery was.  He didn’t describe it in his 1860 paper.  So here’s what we believe based on the historical detective work of, among others, Francis Everitt, Stephen Brush and Michael Strevens.

But here’s what we believe happened

Maxwell had no precedent to follow in addressing this problem.  He was on his own.  Statistics had not yet been used in physics.  But his intuition into the physical nature of “many bodies in constant collision” told him that his answer existed in statistics.  Because he was well read in the theories on collisions and energy, he knew that the velocities of the spheres in his mechanical system must be distributed about a mean.  His thought process led him down a path towards a shape that had earlier resonated with him, namely the Gaussian distribution as described in Herschel’s review.  He knew that this was the answer he wanted.[7] He just didn’t have the mathematical means in 1860 to arrive at this answer based on physics.  And he likely didn’t want to wave his hands around and present an answer he believed to be true without a sound deductive process supporting it.  So he relied on Herschel’s derivation by converting assumptions about the distributions of darts on a dartboard to distributions of colliding spheres in a box.

The spread of darts on a dartboard

Read carefully Maxwell’s description of his derivation.[8] He envisioned a group of his moving spheres clustered at the x-y-z origin and then at an initial time (t= 0) allowed them to disperse, almost like allowing a small volume of gas atoms to experience free expansion.  After a given unit of time, the fastest ones would naturally have traveled further than the slower ones and the resulting population density as a function of distance or radius from the origin would reflect the velocity distribution contained in the original system.  The similarity between the distribution of spheres around the origin after a certain time and the distribution of darts around the bulls-eye is telling.

Allowing velocity to be variable for each of the x, y and z directions

Likely realizing his need to anchor his derivation in sound physics, Maxwell revisited this problem in his 1867 paper, this time using more elaborate mathematics based on collision theory and conservation of energy while also addressing an important problem with one of his assumptions.  In his 1860 paper, he assumed that the velocity distribution of his elastic spheres was independent of direction.  However, in reality, such is not the case.  For example, the inherent resistance to flow in a gas is caused by a velocity gradient.  In this non-equilibrium transport system, the perpendicular motion of slower moving molecules into the faster moving stream creates an effective resistance to flow as quantified by viscosity.  Direction clearly must be accounted for in this scenario.  Maxwell addressed this by modifying this critical assumption to allow the velocity to be variable for each of the x, y and z directions.

Including changes with time

Included in Maxwell’s mathematics was use of the calculus term, d/dt.  Maxwell began considering how the velocity distribution would evolve over time, which took him in a theoretical research direction that would later inspire Boltzmann.  The fact that the distribution changes over time by itself is interesting, but it’s the direction of change that’s more interesting.  As will be discussed later, the concept of entropy’s increase to a maximum over time in an isolated system is, at the molecular level, the concept of the velocity distribution changing over time towards the one of maximum probability, which is the Gaussian distribution.  Velocity distribution isn’t the only determinant of entropy in an ideal gas system; location is also important.   But the complexities associated with the changing velocity distribution, complexities that Maxwell began to address and that Boltzmann spent his lifetime addressing in his famed H-theorem, were the ultimate key bridge between the entropy of classical thermodynamics and the entropy of statistical mechanics.  It all began in a rough form in Maxwell’s 1867 paper.[9]

The role of intuition

Regarding his distribution, Maxwell didn’t know if he was correct or not [10] and didn’t have a ready experiment in mind to find out.  But it made intuitive sense to him, and later to Boltzmann, and the subsequent development of the kinetic theory of gases and transport property predictions based on this theory and the further development of statistical mechanics and its subsequent validation showed no contradictory information.  Nothing pointed to failure.  But still, without a validating experiment, his distribution would have to remain more hypothesis than fact.  This would change.

References

[1] Maxwell, James Clerk. 2003a. “Illustrations of the Dynamical Theory of Gases.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 148–71. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub., p. 152.

[2] Everitt, C. W. F. 1975. James Clerk Maxwell: Physicist and Natural Philosopher. DSB Editions. New York: Scribner, p. 134

[3] Maxwell, p. 153.

[4] Maxwell, p. 154.

[5] Brush, Stephen G. 1983. Statistical Physics and the Atomic Theory of Matter from Boyle and Newton to Landau and Onsager. Princeton, N.J: Princeton University Press, p. 59.

[6] Brush, Stephen G. 1986a. The Kind of Motion We Call Heat:: A History of the Kinetic Theory of Gases in the 19th Century. Book 1: Physics and the Atomists. 3rd print. North-Holland Personal Library,p. 185; Everitt, C. W. F. 1970. “Maxwell, James Clerk.” In Dictionary of Scientific Biography (New York 1970-1990), New York, p. 219; Everitt, C. W. F. 1975. James Clerk Maxwell: Physicist and Natural Philosopher. DSB Editions. New York: Scribner, pp. 136-7; Strevens, Michael. 2013. Tychomancy: Inferring Probability from Causal Structure. Cambridge, Massachusetts: Harvard University Press, pp. 15-17.

[7] In an interesting aside, this scenario played out twice in history.  The Gaussian distribution and its variations are ones that occur in nature.  Maxwell recognized this when considering the distribution of molecular velocities and Planck recognized this when considering the shape of the black-body radiation curve as we discussed previously.  The shape of this Gaussian-based curve, arising from natural phenomena, resonated with both physicists but perhaps for different reasons.  For Maxwell, it made physical sense whereas for Planck, it provided him, at least initially, with a mathematical form that accurately characterized the BBR data.

[8] Maxwell, p. 153.

[9] Of interest is the thought that since Maxwell’s distribution is correct, the hypotheses on which it was based must in some sense be justifiable.  See  (Everitt, 1975) pp. 137-138 and also Kac, M. 1939. “On a Characterization of the Normal Distribution.” American Journal of Mathematics 61 (3) (July): 726–28, for a mathematical proof.

[10] Darrigol, Olivier. 2018. Atoms, Mechanics, and Probability: Ludwig Boltzmann’s Statistico-Mechanical Writings–an Exegesis. First edition. Oxford, United Kingdom: Oxford University Press, p. 365.  “as late as 1895, Maxwell’s derivation of his distribution law could still be deeply misunderstood and … the assumptions needed in this derivation were just beginning to be understood.”

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