APPENDIX CTHE BROWNIAN-MOVEMENT EQUATION
A very simple derivation of this equation of Einstein has been given by Langevin of Paris[199]essentially as follows: From the kinetic theory of gases we havein whichis the average of the squares of the velocities of the molecules,the number of molecules in a gram molecule, andthe mass of each. Hence the mean kinetic energy of agitationof each molecule is given by.
Since in observations on Brownian movements we record only motions along one axis, we shall divide the total energy of agitation into three parts, each part corresponding to motion along one of the three axes, and, placing the velocity along the-axis equal to,we haveEvery Brownian particle is then moving about, according to Einstein’s assumption, with a mean energy of motion along each axis equal toThis motion is due to molecular bombardment, and in order to write an equation for the motion at any instant of a particle subjected to such forces we need only to know (1) the valueof the-component of all the blows struck by themolecules at that instant, and (2) the resistance offered by the medium to the motion of the particle through it. This last quantity we have set equal toand have found that in the case of the motion of oil droplets through a gashas the valueWe may then write the equation of motion of the particle at any instant under molecular bombardment in the formSince in the Brownian movements we are interested only in the absolute values of displacements without regard to their sign, it is desirable to change the form of this equation so as to involveand.This can be done by multiplying through by.We thus obtain, after substituting forits value,Langevin now considers themeanresult arising from applying this equation at a given instant to a large number of different particles all just alike.
Writing thenfor in whichdenotes the mean of all the large number of different values of,he gets after substitutingfor,and remembering that intaking the mean, since thein the last term is as likely to be positive as negative and hence that,Separating the variables this becomeswhich yields upon integration between the limitsandFor any interval of timelong enough to measure this takes the value of the first term. For when Brownian movements are at all observable,isor less, and sinceis roughly equal towe see that, taking the density of the particle equal to unity,Hence whenis taken greater than aboutseconds,rapidly approaches zero, so that for any measurable time intervalsorand, lettingrepresent the change inin the timeThis equation means that if we could observe a large numberof exactly similar particles through a time,square the displacement which each undergoes along the-axis in that time, and average all these squared displacements, we should get the quantity.But we must obviously obtain the same result if we observe the same identical particle through-intervals each of lengthand average these-displacements. The latter procedure is evidently the more reliable, since the former must assume the exact identity of the particles.