Boltzmann on the Continuum

The Horse's Mouth

"As is well known, the integral is nothing more than a symbolic notation for the sum of infinitely many infinitesimal terms. The symbolic notation of the integral calculus has the advantage of such great brevity that in most cases it would only lead to useless complications to write out the integral first as a sum of p terms and then let p become large. In spite of this, there are cases in which the latter method — on account of is generality, and especially on account of its greater perspicuousness, in that it allows the various solutions of a problem to appear — should not be completely rejected." - Boltzmann (1872), Further Studies on the Thermal Equilibrium of Gas Molecules, translated by Brush in Kinetic Theory.

"We first assume that each molecule is capable of taking only a certain finite quantity of speed, e.g. the speeds

\begin{align} 0, \frac{1}{q}, \frac{2}{q}, \frac{3}{q} \cdots \frac{p}{q} \end{align}

where p and q are arbitrary finite numbers. By the collision of two molecules, an exchange of speeds between the two colliding molecules takes place, such that, however, after the collision, each of the two molecules always has one of the above speeds, either 0, or 1/q, or 2/q etc. until p/q. This fiction admittedly corresponds to no realisable mechanical problem, but to a problem that is mathematically much easier to treat, and which immediately changes into the problem to be solved if one lets p and q grow to infinity.

If this method of treating the problem seems at first sight to be very abstract, nevertheless it leads, for such problems, in the fastest way to the goal, and if one considers that all that is infinite in nature is no other than what is meant by a limiting procedure, then one can hardly interpret the infinite variety of speeds that each molecule is capable of taking on as anything other than the limiting case that occurs when each molecule can take on ever more speeds." - Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht, 1877, pp. 166-7 in Wissenschaftliche Abhandlungen II, my translation.

"It is evident that these formulae do not, with calculation, lead to the goal of getting an approximate formula for the case where p and n have finite values, for this case should in practice have hardly any importance. Instead, they merely give the formulae which, when p and n grow infinitely, certainly lead to the correct boundary values.

However, it will perhaps contribute to perspicuousness, if we first show for a few special cases, that for moderately large values of p and n, the established formulae connect to the truth to at least some degree, so that they are not without value as approximate formulae." - Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht, 1877, p. 182 in Wissenschaftliche Abhandlungen II, my translation.

"While earlier it was necessary to have the condition that in the entire container there are many molecules of each type, so it is now necessary that in a small volume of space, in which the external forces do not substantially vary in either magnitude or direction, there must exist exceedingly many molecules (an assumption that in any case is made for every problem in gas theory as long as external forces come into play). For our method of solution always assumes that the states of very many molecules can be considered as equally interchangeable in the sense that the distribution over states is not altered if the molecules exchange states with one another. The probability of the distribution of states is then determined through the number of complexions that emerge from the distribution of states in question." - Boltzmann, Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht, 1877, p. 202 in Wissenschaftliche Abhandlungen II, my translation.

"Anybody who has studied mechanics will remember the difficulty he had in understanding the proof that motion during a very short time can be regarded as uniform and rectilinear and the forces during such a time as constant. These difficulties reside in the fact that these proofs are simply not true.

We have made analytical functions into a representation of the facts of experience. That these functions are differentiable cannot be taken as proof that empirically given functions are equally so, since the number of conceivable undifferentiable functions is just as infinitely great as that of differentiable ones." - Boltzmann, Lectures on the Principles of Mechanics (1897), as translated in Theoretical Physics and Philosophical Problems, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 233.

"For my feeling there is still a certain lack of clarity in the differential coefficients with respect to time. Except for the few cases where one can find an analytic function that has exactly the prescribed differential coefficients with respect to time, then in order to set up a numerical picture one will always have to imagine time as divided into a finite number of parts before one proceeds to the limit. Perhaps our formulae are only very closely approximate expressions for average values that can be constructed from much finer elements and are not strictly speaking differentiable." - Boltzmann, Lectures on the Principles of Mechanics (1897), as translated in Theoretical Physics and Philosophical Problems, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 243-4.

"we shall begin from the fundamental assumption of a large though finite number of material points. It is usually said that differential equations avoid a picture that starts from a finite number, but that again is an illusion. Differential equations require just as atomism does an initial idea of a large finite number of numerical values and points in the manifold, that is positions in the manifold of numbers. Only afterwards is it maintained that the picture never represents phenomena exactly but merely approximates to them more and more the greater we choose the number of these points from the manifold and the smaller the distance between them. Yet here again it seems to me that so far we cannot exclude the possibility that for a certain very large number of points the picture will best represent phenomena and that for greater numbers still it becomes less accurate again, so that atoms do exist in large but finite numbers.

The qualitative laws of natural phenomena and their quantitative relations under very simple circumstances, for example the conditions of equilibrium of a heavy parallelipiped of edges in the ratio 1:2:3, can of course be pictured in the mind without starting from a very large finite number of elements. However, as soon as one wants to specify the quantitative laws for complicated conditions one always must start from differential equations, that is first imagine a large finite number of points in the manifold, in short one must think atomistically, and this is not altered by the fact that afterwards we can increase the number of imagined points and so come arbitrarily close to the continuum without ever reaching it." - Boltzmann (1897), Lectures on the Principles of Mechanics, as translated in Theoretical Physics and Philosophical Problems, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 227-8.

"People have put forward as ideal the mere setting up of partial differential equations and prediction of phenomena from them. However, they too are nothing more than rules for constructing alien mental pictures, namely of series of numbers. Partial differential equations require the construction of collections of numbers representing a manifold of several dimensions. If we remember the meaning of their symbolism they are nothing more than the demand to imagine very many points of such manifolds… and, using certain rules, constantly to derive from them new points of the manifold, to imagine, as it were, a progressive movement of the points in the manifold.

Thus if we go to the bottom of it, Maxwell's electromagnetic equations in their Hertzian form likewise contain hypothetical features added to experience, which are fashioned, as always, by transferring the laws we have observed in finite bodies to fictitious elements of our own making. These equations, like all partial differential equations of mathematical physics, …are likewise only inexact schematic pictures for definite areas of fact, even though the pictures are pieced together from elements that are somewhat different from the atoms to which we are accustomed. The justification of these equations Hertz seeks only after the event in agreement with experience, just as we should with atomist pictures." - Boltzmann (1897), Lectures on the Principles of Mechanics, as translated in Theoretical Physics and Philosophical Problems, pp. 225-6, D. Reidel Publishing Company, Dordrecht, Holland (1974).

"In the ideas one forms of the material points experience with extended objects are mixed up with the conceptual constructions on individual points. Who would not feel the vicious circle involved in defining a material point, for the purpose of setting up the fundamental concepts, as a very small body and then insisting that a remote consequence of the theory built on this basis will be that if we regard volume elements as simple spatial points although they are really small bodies, we are neglecting only small quantities of higher order. How much clearer to view extended bodies from the outset under the picture of many tightly packed points whose velocity always changes only very little from one to the next. Starting from the accelerations of finite bodies without first explaining the concept of the material point is again inadmissible, if we do not have prior knowledge of the theorem concerning the centre of gravity… one must choose precisely those words that will always remind us in the most appropriate way of the correct epistemological status of all concepts." - Boltzmann (1897), Lectures on the Principles of Mechanics, as translated in Theoretical Physics and Philosophical Problems, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 252-3.

"Contradictions (for example we cannot conceive of bodies being really infinitely divisible, nor yet of an extended body as arising from a finite number of points) can lie only in ways of denoting and are thus a sign that these have been inappropriately chosen." - Boltzmann, On the Question of the Objective Existence of Processes in Inanimate Nature (1897), as translated in Theoretical Physics and Philosophical Problems, D. Reidel Publishing Company, Dordrecht, Holland (1974), p. 75.

"Will man sich keiner Illusion iiber die Bedeutung einer Differentialgleichung oder überhaupt einer continuirlich ausgedehnten Grosse hingeben, so kann man nicht in Zweifel sein, dass dieses Weltbild in seinem Wesen wieder ein atomistisches sein muss, d. h. eine Vorschrift, sich die zeitlichen Veränderungen einer überaus grossen Anzahl von in einer Mannichfaltigkeit von wohl drei Dimensionen angeordneten Dingen nach bestimmten Regeln zu denken. Die Dinge konnen naturlich gleichartig oder von verschiedener Art, unveränderlich oder veränderlich sein. Das Bild konnte bei der Annahme einer grossen endlichen Zahl, oder es konnte dessen Limite bei stets wachsender Zahl alle Erscheinungen richtig darstellen." - Boltzmann (1897), Ueber die unentbehrlichkeit der atomistik in der naturwissenschaft. Annalen der Physik und Chemie 296 (2), 231-247.

"Aus den Principien dieses Aufsätzes folgt zweifellos, dass auch continuirliche geometrische Figuren, z. B. der Kreis, nur den Sinn haben, dass man sich dieselben zuerst aus einer endlichen Punktezahl bestehend zu denken hat und erst dann diese Zahl beliebig wachsen lassen muss. Die Limite, der sich der Umfang des ein und umschriebenen n-Ecks mit wachsendem n nähert, ist eben die Definition der Zahl R. Doch wird man sich den Kreis (als geometrischen Begriff) nicht aus einer grossen endlichen Atomzahl gebildet denken, da er nicht, wie der Begriff eines Grammes Wasser von 4o C. unter dem Atmosphärendruck ein Gedankensymbol für einen einzigen gleichbleibenden Complex ist, sondern wie der Zahlbegriff auf die verschiedensten Complexe mit den verschiedensten (natürlich immer sehr grossen) Atomzahlen anwendbar sein soll." - Boltzmann (1897), Ueber die unentbehrlichkeit der atomistik in der naturwissenschaft. Annalen der Physik und Chemie 296 (2), 231-247.

"Wenn man schon von vornherein der Ansicht ist, dass unsere Wahrnehmungen durch das Bild eines Continuums dargestellt werden, dann gehen allerdings nicht die Differentialgleichungen , wohl aber die Atomistik liber die vorgefasste Ansicht hinaus. Ganz anders, wenn man atomistiech zu denken gewohnt ist; dann kehrt sich die Sache um und die Vorstellung des Continuums scheint uber die Thatsachen hinauszugehen." - Boltzmann (1897), Ueber die unentbehrlichkeit der atomistik in der naturwissenschaft. Annalen der Physik und Chemie 296 (2), 231-247.

"Ebenso können bestimmte Integrale, welche die Lösung der Differentialgleichung darstellen, im allgemeinen nur durch mechanische Quadraturen berechnet werden, erfordern also wieder zuerst eine ZErlegung in eine endliche Anzahl von Theilen.

Man glaube doch nicht, dass man sich durch das Wort Continuum oder das Hinschreiben einer Differentialgleichung auch einen klaren Begriff des Continuums verschafft habe! Bei näherem Zusehen ist die Differentialgleichung nur der Ausdruck dafür, dass man sich zuerst eine endliche Zahl zu denken hat; dies ist die erste Vorbedingung, dann erst muss die Zahl wachsen, bis ihr weiteres Wachsthum nicht mehr von Einfluss ist. Was nützt es, die Forderung, sich eine grosse Zahl von Einzelwesen zu denken, jetzt zu verschweigen, wenn man bei Erklärung der Differentialgleichung den durch diesselbe ausgedrückten Werth durch jene Forderung definirt hat? Man verzeihe den etwas banalen Ausdruck, wenn ich sage, dass derjenige, welcher die Atomistik durch Differentialgleichungen losgeworden zu sein glaubt, den Wald vor Bäumen nicht sieht. Eine Erklärung der Differentialgleichung durch complicirtere, geometrische oder andere physikalische Begriffe würde aber erst recht die Wärmeleitungsgleichung im Lichte einer Analogie, statt einer directen Beschreibung erscheinen lassen. Wir vermögen in Wirklichkeit die benachbarten Theile nicht zu unterscheiden. Ein Bild aber, in welchem wir von allem Anfange her die benachbarten Theile nicht unterscheiden, wäre verschwommen; wir könnten daran die vorgeschriebenen Rechnungsoperationen nicht vornehmen." - Boltzmann (1897), Ueber die unentbehrlichkeit der atomistik in der naturwissenschaft. Annalen der Physik und Chemie 296 (2), 231-247.

"Wie die Wärmeleitungsgleichung, so können auch die Grundgleichungen der Elasticität allgemein nur gelöst werden, indem man sich zuerst eine endliche Zahl von Elementarkörperchen denkt, welche nach gewissen einfachen GEsetzen aufeinander wirken und dann wieder die Limite bei Vermehrung der Zahl derselben sucht. Diese Limite ist also wieder die eigentliche Definition der Grundgleichungen, und das Bild, welches von vornherein eine grosse, aber endliche Zahl annimt, erscheint abermals einfacher." - Boltzmann (1897), Ueber die unentbehrlichkeit der atomistik in der naturwissenschaft. Annalen der Physik und Chemie 296 (2), 231-247.

"Auch die Differentialquotienten nach der Zeit sprechen natürlich die Forderung aus, dass man in dem Bilde der Natur die Zeit zunächst in sehr kleine, endliche Zeittheile (Zeitatome) zerlegt denken muss." - Boltzmann (1897), Ueber die unentbehrlichkeit der atomistik in der naturwissenschaft. Annalen der Physik und Chemie 296 (2), 231-247.

"It emerges that we cannot define the infinite otherwise than as the limit of steadily growing finite quantities; at any rate no one yet has been able to construct a reasonably tangible idea of the infinite in any other way. Therefore, if we wish to get a picture of the continuum in words, we first have to imagine a large, but finite number of particles with certain properties and investigate the behavior of the ensemble of such particles. Certain properties of the ensemble may approach a definite limit as we allow the number of particles ever more to increase and their size ever more to decrease. Of these properties one can then assert that they apply to a continuum, and in my opinion this is the only uncontradictory definition of a continuum with certain properties.

The question whether matter is built of atoms or is continuous therefore reduces to whether those properties approximate the observed properties of matter most exactly under the assumption of an extraordinarily large, finite number of particles, or by taking limits as the number of particles grows indefinitely. True, this does not answer the old philosophical question, but we have been cured of the endeavor to answer it in an absurd and hopeless way. The thought process by which we first examine the properties of a finite ensemble and then allow the number of members to grow extraordinarily remains the same in both cases." - Boltzmann, as quoted in Broda's Ludwig Boltzmann: Man, Physicist, Philosopher, Ox Bow Press, Woodbridge, Connecticut, USA (1983), p. 48.

"Whence comes the ancient view, that the body does not fill space continuously in the mathematical sense, but rather it consists of discrete molecules, unobservable because of their small size. For this view there are philosophical reasons. An actual continuum must consist of an infinite number of parts; but an infinite number is undefinable. Furthermore, in assuming a continuum one must take the partial differential equations for the properties themselves as initially given. However, it is desirable to distinguish the partial differential equations, which can be subjected to empirical tests, from their mechanical foundations (as Hertz emphasized in particular for the theory of electricity). Thus the mechanical foundations of the partial differential equations, when based on the coming and going of smaller particles, with restricted average values, gain greatly in plausibility; and up to now no other mechanical explanation of natural phenomena except atomism has been successful." - Boltzmann, Lectures on Gas Theory, trans. S. Brush, University of California Press (1964), p. 27.

"A remark is necessary here. The quantities previously denoted by $d\omega = d\xi d\eta d\zeta$ and now by $\omega$ are volume elements, and hence really only differentials. The number n of molecules in unit volume is indeed a very large number, but it is still finite… It may therefore be surprising that we treat the expressions $n_1 \omega, n_2 \omega$ and $f(\xi, \eta, \zeta, t) d\xi d\eta d\zeta$ as very large numbers. One could also carry out the same calculations on the assumption that these are fractions; they would then simply represent probabilities. But an actual number of objects is a more perspicuous concept than a mere probability, and the considerations just carried out would have required complicated digressions and explanations since once cannot speak of the permutation number of a fraction. Such thoughts remind us, however, that we could have chosen the volume element as large as we pleased. We could have assumed so many equivalent gases to be present in the volume element that even when $\omega$ is chosen very small, the velocity points of many molecules would still always lie in it. The order of magnitude of the volume chosen as volume element is completely independent of the order of magnitude of the volume elements $\omega$ and $d\xi d\eta d\zeta$.

Even more dubious is the assumption we shall make later, that not only the number of molecules in the volume element whose velocity points lie in a differential volume, but also the number of molecules whose centers are in such a volume element, is infinitely large. The latter assumption is no longer justified as soon as one has to deal with phenomena in which finite differences in the properties of the gas are encountered in distances that are not large compared to the mean free path… All other phenomena take place in such large spaces that one can construct a volume element for which the visible motion of the gas can be taken as a differential, yet which still contains a large number of molecules. This neglect of small terms whose order of magnitude is completely independent of the order of magnitude of the terms occurring in the final result must be carefully distinguished from the omission of terms that are of the same order of magnitude as those from which the final result is derived… While the latter omission causes an error in the result, the former is simply a necessary consequence of the atomistic conception, which characterizes the meaning of the result obtained, and is the more permissible, the smaller the dimension of the molecule compared to that of the visible bodies. In fact from the standpoint of atomistics, the differential equations of the doctrines of elasticity and hydrodynamics are not exactly valid, but rather they are themselves approximation formulae which become more nearly exact as the space in which the visible motions occur becomes large compared to the dimensions of molecules. Likewise, the distribution law for molecular velocities is not precisely correct as long as the number of molecules is not mathematically infinite. The disadvantage of giving up the supposedly exact validity of the hydrodynamic differential equations is however compensated by the advantage of greater perspicuousness." -Boltzmann, Lectures on Gas Theory, trans. S. Brush, University of California Press (1964), p. 61-62.

"…the condition that the number of molecules for which the values of the variables lie within one of these regions must be very large can be satisfied only if the number of molecules in unit volume is infinite in the mathematical sense. Hence the satisfaction of the above conditions in this case remains merely an ideal; yet we still expect agreement with experience, for the following reasons.

In the molecular theory we assume that the laws of the phenomena found in nature do not essentially deviate from the limits that they would approach in the case of an infinite number of infinitesimally small molecules… It is indispensable for any application of the infinitesimal calculus to molecular theory; indeed, without it, our model which strictly deals always with a large finite number, would not be applicable to apparently continuous quantities.. This assumption will seem best justified to those who have carefully considered experiments for the direct proof of the atomic constitution of matter. Even in the smallest neighborhood of the tiniest particles suspended in a gas, the number of molecules is already so large that it seems futile to hope for any observable deviation, even in a very small time, from the limits that the phenomena would approach in the case of an infinite number of molecules.

If we accept this assumption, then we should also obtain agreement with experience by calculating the limit that the laws of the phenomena would approach in the case of an infinitely increasing number and decreasing size of the molecules. In calculating the latter limit, we again have in fact two quantities, which can independently be made arbitrarily small: the size of the volume element, and the dimensions of the molecules. For any given choice of the former, we can always choose the latter so small that each volume element still contains very many molecules, whose properties are closely defined within the given narrow limits.

If, with Kirchoff, one interprets the expressions (115) and (118) as simply statements of probabilities, then one can allow them to be fractions or even very small quantities; yet one thereby loses their perspicuousness." - Boltzmann, Lectures on Gas Theory, trans. S. Brush, University of California Press (1964), p. 318.

"the entire symbolism of the differential and integral calculus is meaningless unless one proceeds first by considering large finite numbers" - Boltzmann, Lectures on Gas Theory, trans. S. Brush, University of California Press (1964), p. 427

"Just as the differential equations [of elasticity theory and hydrodynamics] represent simply a mathematical method for calculations, whose clear meaning can only be understood by the use of models which employ a large finite number of elements, so likewise general thermodynamics… also requires the cultivations of mechanical models representing it, in order to deepen our knowledge of nature —- not in spite of, but rather precisely because these models do not always cover the same ground as general thermodynamics, but instead offer a glimpse of a new viewpoint." - Boltzmann, Lectures on Gas Theory, trans. S. Brush, University of California Press (1964), p. 445.

Secondary literature

"It has also been suggested, in view of Boltzmann's later habit of discretising continuous variables, that he somehow thought of the energy hypersurface as a discrete manifold containing only finitely many discrete cells (Gallavotti 1994). In this reading, obviously, the mathematical no-go theorems of Rozenthal and Plancherel no longer apply. Now it is definitely true that Boltzmann developed a preference towards discretizing continuous variables, and would later apply this procedure more and more (although usually adding that this was fictitious and purely for purposes of illustration and more easy understanding). However, there is no evidence in the (1868) and (1871b) papers that Boltzmann implicitly assumed a discrete structure of mechanical phase space or the energy hypersurface." - Jos Uffink

"The resort to combinatorials demands, Boltzmann states, that one begin by dividing the energy or the velocity continuum into finite cells of size ε or ε, ϛ, η. These cells must, furthermore, be large enough so that each contains many molecules. The quantities wk or wabc must, in short, be very large. While cells retain their initially given size, one may discard small quantities as well as size-dependent constants which cannot affect the form of the minimum. In this way one arrives at a sum the value of which is to be maximized, for example at:

\begin{align} \Omega = - \sum \sum^{+\infty}_{-\infty} \sum \epsilon \zeta \eta f (a\epsilon, b\zeta, c\eta) \log f(a\epsilon, b\zeta, c\eta). \end{align}

Only after reaching this point, Boltzmann says, may the quantities ε, ϛ, η be further reduced and the transition to an integral made. Furthermore, one must even then remember that the symbols dE or dudvdw, though they appear as mathematical differentials, still represent cells large enough to contain many molecules. 'This may at first glance seem strange,' Boltzmann writes, 'since the [total] number of gas molecules, though large, is still finite, while du, dv, dw are mathematical differentials. Yet on closer inspection the assumption must be regarded as self-explanatory, for all applications of the differential calculus to gas theory depend on it.'

The last of those sentences is, of course, a non sequitur, and it suggests that there is something about the argument which Boltzmann is not seeing. The transition to an integral form depends for its legitimacy, not on some mathematical limiting procedure but on the validity of a physical hypothesis, and the latter is by no means self-evident. For plausible values of gas density and thus of n, it must be possible to choose cell size large enough so that each cell contains many molecules, yet small enough so that the variation of f is small as one moves from the center of one cell to that of the next. Boltzmann sees the first of these conditions as merely mathematical and never alludes to the second one at all, an oversight presumably facilitated by his little studied views about the relation between the continuous and the discrete. Planck's subsequent insensitivity to an important difference between his distribution of energy over resonators and Boltzmann's distribution over gas molecules may be traceable in part to this aspect of Boltzmann's thought. Until sometime after 1906, Planck did not notice, or at least did not note the consequences of, the fact that under quite usual physical circumstances his distribution function varies markedly from cell to cell." - Thomas Kuhn, Black-Body Theory and the Quantum Discontinuity, pp. 59-60, Oxford University Press, 1978.

"In a development of his reply to the reversibility paradox, Boltzmann considered the formula $S = -\int f \ln f d^3r d^3v$, which gives the entropy of the distribution f(r,v) of the molecules of a perfect gas in configuration space. He had already reached this logarithmic expression through various methods, but now wanted to relate it directly to something like the probability of the distribution f. Intuitively, this probability had to be proportional to the number of microscopic configurations of the set of molecules compatible with the given distribution f.

In order to give this number a definite meaning, Boltzmann invented a 'fiction' in which the microstate of a molecule λ is given by a discrete energy value iλε (iλ is a positive integer), the microstate of the whole gas corresponds to the list λ —> iλ (called complexion), and the distribution f is replaced with a discrete distribution (N0, N1, … Ni, … ), where Ni is the number of molecules carrying the energy . The probability of this distribution is assumed to be proportional to its permutability P, i.e. the corresponding number of complexions. According to the combinatorics W3 this number is:

\begin{align} P = n! / \Pi_i n_i ! \end{align}

To turn from the discrete fiction to the continuous reality, Boltzmann just had to re-intrepret (N0, N1, … Ni, … ) as a coarse characterization of a continuous energy-distribution f(E). More specifically, the energy axis is divided into intervals of uniform length ε (later called cells by Planck); a complexion tells in which cell each molecule is; and Ni represents the number of molecules in the i-th interval. To be quite realistic (that is to say, to lead to Maxwell's distribution), the cell divisions must be uniform in the (r,v)-configuration space of the molecule. Then, if the cells are small enough to make (N0, N1, … Ni, … ) a discrete approximation of f(r,v) (but still large enough to allow the Stirling approximation), one has:

\begin{align} \ln P \sim -\sum_i N_i \ln N_i + N \ln N \sim -\int f \ln f d^3r d^3v + constant \end{align}

This number reaches its maximum (for a given value of the total energy and the total number of molecules) if $N_i = \alpha e^{-\beta i \epsilon}$(α and β are the Lagrange parameters), or $f = \alpha' e^{-\beta m v^2 /2}$, which is Maxwell's distribution; and its corresponding value is equal (up to a function of N) to the classical entropy of a perfect gas (temperatures being measured in energy units)." - Olivier Darrigol, "Statistics and combinatorics in early quantum theory, II: Early symptoms of indistinguishability and holism", Historical Studies in the Physical Sciences, 21:237-298.