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LIX. Relativity Theory : The Equipartition Law in a System 

 of Particles. By Richard C. Tolman, Ph.D., Assistant 

 Professor of Chemistry in the University of California *. 



Tl 1HE principle of relativity necessitates a revision of the 

 -L kinetic theory of gases. For an ideal gas containing 

 monatomic molecules, all of which have the same mass, a 

 modified form of the Maxwell distribution law for the 

 momenta of the molecules has already been derived by 

 Jiittner f . In the present article we shall consider systems 

 containing particles of different masses, and derive also for 

 this case a law for the distribution of momenta. From this 

 distribution law we shall be able to show that the average 



value of . — ° ^ is the same for particles of all different 

 masses. \ V m v 2 



This equipartition of . ° will be found true not onlv 



for a mixture of particles of different masses, but also for 

 systems of particles which are separated by a partition that 

 allows transfer of energy. Hence the equality of the average 

 values of this quantity becomes the relativity condition for 

 thermal equilibrium. Furthermore, we shall show that 



2 



— — -° — is proportional to the absolute temperature as 



V 1 — v 2 



measured on the thermodynamic scale, and hence see that in 



all such considerations this new quantity satisfies the same 



relations as the vis viva m v 2 in Newtonian mechanics, and 



indeed reduces to that quantity when the velocities are small 



compared with that of light. 



We note immediately, however, that the quantity . — 

 is neither the relativity expression for the total energy of a 



particle — j - - nor for the kinetic energy , — -- — m , 



*/l—v 2 * ^/1 — v 2 



and hence in relativity mechanics there can be no possibility of 

 an exact equipartition of energy. Finally, we shall present a 



* Communicated bv the Author. 

 t Jiittner, Ann. d. Physik, xxxiv. p. 856 (1911). 

 X Throughout the article we shall use v to express the velocity of 

 particle divided by the velocity of light. The mass of a particle will 



hence be .- — ., , its momentum .. — ^ , its total energT ,^ — =:,- , the 



VI— v- s/ L — v °" yl— v- 



(vi \ 



V 7 !— v I 



