ON OUR KNOWLEDGE OF THERMODYNAMICS. 87 



Is the Boltzmann- Maxwell Distribution unique ? 



45. We have now found the necessary and sufficient conditions for a 

 permanent or ' special ' state to be — 



{!.)/, F independent of time in the absence of collisions. 



(II.) F/=^¥'f' for all collisions between two molecules. 



Under these circumstances the principle of Conservation of Energy at 

 once establishes the Boltzmann-Maxwell Law, which asserts that 



F=Aexp-7iEp /=Bexp-/iE 



'p 



always give a solution of the problem. 



Whether the Boltzmann-Maxwell distribution is unique depends on 

 the possibility of finding other functions satisfying the above conditions, 

 and it becomes necessary to discuss particular cases separately. 



Case I. — When no forces act, the conservation of momentum parallel 

 to any line chosen as the axis of x gives for masses M, vi 



MU + 7nu = MU' + mu' 



and therefore the conditions are satisfied by 



F oc exp ^MU / oc exp kmu. 

 Combining this with the solution 



F oc exp — /iTji /oc exp —hT„^ 

 and writing k=huQ where «o is any constant, we obtain the solution 



F=Aexp-Um{(U-t6o)2 + V2+W2} ) , . 



/=Bexp-|7iM;(M-Mo)'+^-' + ^«'} I * ' ■ ^^ 



This is the law of distribution in a gas having what Burbury calls a 

 * motion of simple translation ' and Boltzmann a ' progressive motion ' of 

 velocity iig along the axis of .r ; a result agreeing with tlaose found by 

 Burbury,' Boltzmann,- and others. Here the average molecular kinetic 

 energies of the relative motion, taken with respect to a point moving with 

 velocity ?io) ^re equal, so that the quantity which represents tein2)erature 

 is, as it should be, independent of tto, the velocity of translation. 



Case II. — When the field is symmetrical about a fixed axis, the 

 constancy of angular momentum about this axis leads in a similar way 

 to the distribution of co-ordinates and velocities among the molecules of a 

 mass of rotating gas — such as that forming the atmosphere of a planet. 



This case I have worked out in detail in Appendix B, and in 

 conformity with the nomenclature of Case I. the gas may be said to have 

 a motion of simple rotation.' 



It is remarkable that in the paper last referred to Boltzmann,^ while 

 w( rking out a number of problems on the motion of gases in a field 

 of force, specially considers the case in which the gas has no initial 

 motion of rotation. Maxwell, in the second part of the paper, discussed 



' Phil. Mag., October 1890, p. 305. 



- ' Ueber Aas Wiirmegleichgewicht von Gasen auf welche aussere Kriifte wirken,' 

 Sitzber. der k. Wiener Akad., Ixxii. (ii.). Oct. 1875. 'Ueber die Aufstellung und 

 Integration der Gleichungen welche die Molecularbewegung in Gasen bestimmen,' 

 Ihid., Ixxiv. (ii.), Dec. 1876. 



» ' Ueber die Aufstellung,' &c., Wiener Sitzh., Dec. 1876. 



