# On the theory of brownian motion

On the theory of brownian motion

PHYSICS--On the Theory of Brownian Motion

A note from M. Paul Langevin, presented by M. Eleuthere Mascart.

I. The very great theoretical importance presented by the phenomena of Brownian motion has been brought to our attention by M. Gouy. (Footnote 1) We are indebted to this physicist for having clearly formulated the hypothesis which sees in this continual movement of particles suspended in a fluid an echo of molecular-thermal agitation, and for having demonstrated this experimentally, at least in a qualitative manner, by showing the perfect permanence of Brownian motion, and its indifference to external forces when the latter do not modify the temperature of the environment. A quantitative verification of this theory has been made possible by M. Einstein (Footnote 2), who has recently given a formula that allows one to predict, at the end of a given time tau, the mean square (delta x)^2 of displacement delta x of a spherical particle in a given direction x as the result of Brownian motion in a liquid as a function of the radius a of the particle, of the viscosity mu of the liquid, and of the absolute temperature T. This formula is:

${\displaystyle \scriptstyle {\overline {(delta.x)^{2}}}={\frac {R*T}{N}}*{\frac {1}{3*Pi*mu*a}}*tau}$

(1)

where R is the perfect gas constant relative to one grammolecule and N the number of molecules in one grammolecule, a number well known today and around 8.10^23. M. Smoluchowski (Footnote 3) has attempted to approach the same problem with a method that is more direct than those used by M. Einstein in the two successive demonstrations he has given of his formula, and he has obtained for mean square (delta x)^2 an expression of the same form as (1) but which differs from it by the coefficient 64/27

II. I have been able to determine, first of all, that a correct application of the method of M. Smoluchowski leads one to recover the formula of M. Einstein precisely, and, furthermore, that it is easy to give a demonstration that is infinitely more simple by means of a method that is entirely different. The point of departure is still the same: The theorem of the equipartition of the kinetic energy between the various degrees of freedom of a system in thermal equilibrium requires that a particle suspended in any kind of liquid possesses, in the direction x, an average kinetic energy RT/2N equal to that of a gas molecule of any sort, in a given direction, at the same temperature. If ksi=dx/dt is the speed, at a given instant, of the particle in the direction that is considered, one therefore has for the average extended to a large number of identical particles of mass m

${\displaystyle \scriptstyle m.{\overline {ksi^{2}}}={\frac {R*T}{N}}}$

(2)

A particle such as the one we are considering, large relative to the average distance between the molecules of the liquid, and moving with respect to the latter at the speed ksi, experiences a viscous resistance equal to -6*Pi*m*a*ksi according to Stokes’ formula. In actual fact, this value is only a mean, and by reason of the irregularity of the impacts of the surrounding molecules, the action of the fluid on the particle oscillates around the preceding value, to the effect that the equation of the motion in the direction x is

${\displaystyle \scriptstyle m.{\frac {d^{2}x}{dt^{2}}}=(-6*Pi*mu*a)*{\frac {dx}{dt}}+X}$

(3)

About the complementary force X, we know that it is indifferently positive and negative and that its magnitude is such that it maintains the agitation of the particle, which the viscous resistance would stop without it. Equation (3), multiplied by x, may be written as:

${\displaystyle \scriptstyle {\frac {m}{2}}*{\frac {d^{2}x^{2}}{dt^{2}}}-m*ksi^{2}=(-3*Pi*mu*a)*{\frac {dx^{2}}{dt}}+X*x}$

(4)

If we consider a large number of identical particles, and take the mean of the equations (4) written for each one of them, the average value of the term X*x is evidently null by reason of the irregularity of the complementary forces X. It turns out that, by setting z= mean square (dx^2)/dt,

${\displaystyle \scriptstyle {\frac {m}{2}}*{\frac {dz}{dt}}+3*Pi*mu*a*z={\frac {R*T}{N}}}$

(5)

The general solution

${\displaystyle \scriptstyle z={\frac {R*T}{N}}*{\frac {1}{3*Pi*mu*a}}+C*e^{({\frac {-6*Pi*mu*a}{m}})*t}}$

(6)

enters a constant regime in which it assumes the constant value of the first term at the end of a time of order m/6*Pi*mu*a or approximately 10^(-8) seconds for the particles for which Brownian motion is observable. One therefore has, at a constant rate of agitation,

${\displaystyle \scriptstyle {\frac {d{\overline {x^{2}}}}{dt}}={\frac {R*T}{N}}*{\frac {1}{3*Pi*mu*a}}}$

(7)

hence, for a time interval tau,

${\displaystyle \scriptstyle {\overline {x^{2}}}-{\overline {(x_{0})^{2}}}={\frac {R*T}{N}}*{\frac {1}{3*Pi*mu*a}}*tau}$

(8)

The displacement delta x of a particle is given by

${\displaystyle \scriptstyle x=x_{0}+delta.x}$

(9)

and, since these displacements are indifferently positive and negative,

${\displaystyle \scriptstyle {\overline {(delta.x)^{2}}}={\overline {x^{2}}}-{\overline {(x_{0})^{2}}}={\frac {R*T}{N}}*{\frac {1}{3*Pi*mu*a}}*tau}$

(10)

thence the formula (1).

III. A first attempt at experimental verification has just been made by M. T. Svedberg (Footnote 4), the results of which differ from those given by formula (1) only by about the ratio 1 to 4 and are closer to the ones calculated with M. Smoluchowski’sformula. The two new demonstrations of M. Einstein’s formula, one of which I obtained by following the direction begun by M. Smoluchowski, definitely rule out, it seems to me, the modification suggested by the latter. Furthermore, the fact that M. Svedberg does not actually measure the quantity mean square (delta x^2) that appears in the formula and the uncertainty of the real diameter of the ultramicroscopic granules he observed call for new measurements. These, preferably, should be made on microscopic granules whose dimensions are easier to measure precisely and for which the application of the Stokes formula, which neglects the effects of the inertia of the liquid, is certainly more legitimate.

FOOTNOTES

[translators note: In the original, footnote numbering started anew on each page; here, in order to avoid confusion, numbering is sequential throughout the paper.]

1. Gouy, Journ. de Phys., 2e serie, t. VII, 1888, p. 561; Comptes rendus, t. CIX, 1889, p. 102.
2. A. Einstein, Ann. d. Physik, 4e serie, t. XVII, 1905, p. 549; Ann. d. Physik, 4e serie, t. XIX, 1906, p. 371.
3. M. von Smoluchowski, Ann. d. Physik, 4e serie, t. XXI, 1906, p. 756.
4. T. Svedberg, Studien zer Lehre von den kolloïden Lösungen. Upsala, 1907.