M. F. Smoluchowski

(Marian v.-Smoluchowski).

(Obituary).

On the fifth of September, 1917, during an epidemic of dysentery in Kraków, the famous physicist Smoluchowski died at the age of 45.

He was one of those outstanding contemporary physicists—Langevin, Perrin, Smoluchowski, and Einstein—belonging to one and the same generation, who undertook a reworking of all the fundamental principles of physics and applied directly to the preceding generation of great physicists (Boltzmann, H. A. Lorentz, Planck, Poincaré, Rayleigh, and J. J. Thomson).

The nineteenth century may be called the century of the triumph of thermodynamics; the entire world-view, all explanations of natural phenomena, were being reduced to the fundamental principles of the conservation of energy and the increase of entropy; all phenomena of nature were subdivided into reversible and irreversible, and to the former were directly applied the principles of thermodynamics, which were recognized as infallible; for example, it was believed that every phenomenon occurring in nature is always accompanied by an increase in entropy and only in the limiting case by its constancy. The work of thermodynamicists at the end of the nineteenth century was directed toward the study of irreversible processes, and in this direction especially important are the investigations of the recently deceased physicist Duhem.

But thermodynamics, which had had such enormous success in the study of physical and physicochemical statics, encountered insurmountable difficulties in the study of the kinetics of natural phenomena. Moreover, a whole series of data concerning the diffusion of gases, internal friction, and the thermal conductivity of gases had, already by the end of the nineteenth century, raised general questions and yielded a mass of new facts which in no way could be explained by the principles of thermodynamics alone. For these phenomena molecular physics gave simple quantitative explanations. Smoluchowski’s first work, in 1898, also concerns these questions; he studied the temperature jump in the transition from a warm wall to a gas, a jump that is the stronger the lower the gas pressure. This temperature jump is explained by the kinetic theory of gases; precise measurements of the temperature distribution as a function of distance from the heated wall, which were made in 1910 by Lazarev, make it possible directly to calculate the mean free path of the gas molecules; thermodynamics, however, is not capable of explaining this phenomenon.

More striking, one might say a devastating blow dealt to thermodynamics, were the works on the Brownian motions of the smallest particles. Already at the end of the nineteenth century Gouy gave a general theory according to which Brownian motions are the result of molecular impacts on particles, and showed that these motions follow from the principle of equipartition of energy. This assertion seemed incredible; a whole series of physicists said that with such an explanation of Brownian motions one would obtain much weaker motions of the particles than those which are observed. Smoluchowski was the first who, in a series of works beginning in 1905, calculated the velocities of motion of microscopic particles and their mean displacement in a given interval of time; he showed that the displacements of a particle along one axis \(X\) during time \(t\) are equal to:

\[ X=\frac{R\cdot T}{N}\cdot \frac{t}{3\pi r\eta} \]

where \(R\) is the gas constant \(=8.32\cdot 10^7\) (C. G. S.), \(T\) is the absolute temperature, \(N\) is the number of molecules in one gram-molecule of a body (Avogadro’s number \(=6.1\cdot 10^{23}\)), \(r\) is the radius of the particle, and \(\eta\) is the viscosity of the liquid.

Smoluchowski also showed that the diffusion coefficient \(D\) is equal to:

\[ D=\frac{RT}{N}\cdot \frac{1}{6\pi r\eta}. \]

These formulas were then derived by other methods by Einstein in 1906 and by Langevin in 1908.

The classical experiments of Perrin and his students, and then of a whole series of other investigators, gave brilliant confirmation of these formulas, derived on the basis of kinetic theory, and laid a firm foundation for the new current of molecular physics.

Thus there was put on the agenda the question of reconsidering the basic principles of thermodynamics and in particular the second principle—the increase of entropy. Smoluchowski’s activity was entirely concentrated on these fundamental questions. He showed in 1908 that the phenomena of turbidity of liquids and mixtures of liquids near the critical temperature stand in quantitative dependence on the compressibility of these liquids and are caused by those very small changes in the concentration of molecules which constantly arise in a liquid as a result of the motion of molecules. In the same way the blue color of the sky is explained quantitatively by the constant formation of similar centers of condensation of molecules, arising from the fact that molecules are constantly moving, and by chance now in one place, now in another, there occurs an accumulation of molecules or else their rarefaction. The theory of probability makes it possible to calculate how many such centers are formed at a given moment in a definite volume of air, and from this it is clear that one can derive the diffusion of light and calculate the intensity of the blue color of the sky. Experiments made in Switzerland in the mountains by two students of Langevin and Perrin—Bauer and Moulin—fully confirmed these theoretical calculations of Smoluchowski; the very delicate and penetrating experimentalist, the young physicist Moulin, who had only just been appointed professor in Besançon, was killed in the first weeks of the war.

The formation of such centers of condensation and rarefaction of molecules,

which so fully explained the blue color of the sky and the clouding of liquids near the critical state, could be observed directly.

Two kinds of phenomena made it possible to carry out such observations: on the one hand, observations, under the ultramicroscope, of the number of colloidal particles which, in a series of equal short intervals of time, are found in a definite volume; on the other hand, observations of the decay of radioactive bodies, in which one can follow and record how individual molecules, one after another, disintegrate, emitting particles \(\alpha\), i.e. helium atoms charged positively. The first method was used by Th. Svedberg on colloidal gold, the second method by Mme Curie together with Debierne and their students.

Smoluchowski showed that if in a given volume the average number of molecules under uniform distribution should be equal to the value \(\nu\), then the probability that in this volume there will be obtained a number of molecules equal to \(n\) is equal to:

\[ W(n)=\frac{\nu^n e^{-\nu}}{n!}, \]

and if we take the relative condensation \(\delta=\frac{n-\nu}{\nu}\), then the mean square of this condensation will be equal to

\[ \overline{\delta^2}=\frac{1}{\nu}. \]

In Svedberg’s experiments, the numbers of particles of colloidal gold found every \(\frac{1}{39}\) minute in a given volume of liquid are distributed as follows:

From this the mean square of the condensation is calculated to be \(\overline{\delta^2}=0.637\), while Smoluchowski’s theory gives \(\delta^2=0.645\), a confirmation, one may say, brilliant.

These considerations lead, obviously, to broad generalizations. It becomes clear that all the phenomena which we observe are ordinarily only average data; in reality, constant fluctuations about these averages occur in nature; these fluctuations occur both in time and in space; for example, the temperature of a given volume of gas or liquid constantly fluctuates, now rising, now falling; the temperature changes from one place to another, and these fluctuations will be the more noticeable the smaller the volume under consideration that we take. Likewise the pressure of a given volume of gas is only an apparently constant איז?

quantity; if pressure is observed in sufficiently small partial volumes, it will constantly fluctuate, now rising, now falling.

In general, if the quantity \(E\) denotes a deviation from the mean state, then the probability of obtaining a state lying between \(E\) and \(E+dE\) is given by the formula:

\[ W(E)dE=A.e^{-\frac{N}{RT}\cdot \chi(E)}.dE. \]

In this formula, obtained by Smoluchowski, \(A\) is a constant, \(N\) is Avogadro’s number, and \(\chi(E)\) expresses the work required for the transition from the mean state to a position with deviation \(E\).

Thus, we see that if we consider the world from the point of view of the microcosm, entropy may equally well increase or decrease, heat may equally well pass from a warm body to a cold one and back again, molecules may equally well move from more concentrated parts to less concentrated ones and back again—in general, all phenomena of nature appear to us reversible; one need only be able to wait a sufficiently long time and to observe sufficiently small volumes. Thus, for example, Smoluchowski calculates how long one would have to wait for there to accumulate, in a volume equal to one cubic centimeter and containing a mixture of oxygen and nitrogen, one half with 1% more oxygen than in the other, as a result of the disordered motion of the molecules, and finds that this time is equal to \(10^{10^{11}}\) seconds. If, however, one takes a volume similar to those observed in an ultramicroscope, i.e. \((0.2\mu)^3\), then such a deviation in the distribution of oxygen molecules would be observed once every \(10^{-9}\) seconds.

The second principle of thermodynamics thus appears only as a convenient factual rule, suitable only for our macrocosm and not corresponding to the phenomena of nature under their fine analysis.

We see what great significance were the works of the physicist Smoluchowski, who died so prematurely, and who, together with his contemporaries, contributed to the transformation of our entire philosophical worldview.

Victor Henri.