Physics. — "A paradox in the theory of the Brownian movement. 

 By Prof'. P. Ehrenfest. (Communicated by Prof. H. A. Lorentz). 



(Communicated in the meeting of September 29, 1917). 



§ 1. Suppose a small sphere to be suspended in a fluid. Suppose 

 further that its Brownian movement is observed and consider a 

 moment when the sphere has a rather great velocity, e.g. upward^ 

 May we expect the surrounding fluid to move with the sphere'^ 



Prof. J. D. VAN DRR Waals Jr. and Miss A. Snethlage have 

 recently shown ^): l^t how the answer to this question is connected 

 with Eimstein's theory of the Brownian movement. 



2"^ that the statistical theory of the molecular movement demands 

 that .such a common niotion does not exist. This theory implies namely 

 that for a given place and velocity of the suspended sphere and 

 for a given configuration of the surrounding particles of the fluid, (?^w«/ 

 aiid opposite velocities of these particles are always equally pi-obable. 



The authors cited remarked already that this result is somewhat 

 paradoxical; and therefore subjected it ta a detailed discussion. 



In the following we shall make the paradox still more acute by 

 considering an analogous question for an extremely simplified model. 

 This will show us that two closely connected material points m^ 

 and m, of which our model will consist, may on one hand possess 

 mutually independent velocities, while on the other hand they still 

 accompany each other (because of the close connexion). 



§ 2. We consider two material particles with the masses m^ and 

 7??, and the following properties: 



1st }iQ{\i are constrained kinematically to move along the axis of X 

 2"*^ By a field of force their distance can never become greater 

 than Z)'), where D may be small compared with the displacements 

 of the two points in the course of time along the X-axis : 



|;..-^J<i> ........ (1) 



3'"^ Let this pair of points be placed in an infinite extension 



1) These Proc p. 1322. 



~) Let e. g. m-i be a shell in which mg remains enclosed. 



