212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



statistical information about a system by following an individual member 

 through a large number of random processes. The result of such a pro- 

 cedure is knowledge about one member of the assembly for a long period 

 of time. Time averages of various kinds can be obtained from such data ; 

 these time averages are then set equal to instantaneous averages over the 

 assembly, in accordance with ergodic theory. In our case, an ion was 

 followed through 10,000 collisions. On an average, the collisions were 

 isotropic in the center of mass system (11 (x) = 1) and obeyed a mean 

 free time condition r = const. Actually, both the free time and the scat- 

 tering angles varied from collision to collision; the angles varied in a 

 random fashion over a unit sphere and r was random within an ex- 

 ponential distribution. 



A Monte Carlo calculation of this type consists of three parts. In the 

 first part the random numbers having the required distributions are 

 obtained and recorded. In the present problem there were three such 

 random numbers required for each collision, namely a time and two 

 angles. These numbers were placed on 10,000 IBM cards, along with 

 suitable identification. In the second part a calculating machine simulates 

 the successive collisions and keeps a record of the initial and final veloci- 

 ties for each one. The third part consists in analyzing statistically the 

 numerical material accumulated in the second. For the first part of the 

 calculation particular values must be chosen for the acceleration a and 

 the mean free time t. These values were 



a = 1 



r = logio e = 0.43429 



However, the dimensional analysis of Section ID shows us at this point 

 that these two constants enter into the problem only through their 

 product ar which scales all velocities. It is therefore convenient at the 

 statistical stage to remove these factors and to analyze the results in 

 terms of a dimensionless variable which by (26c) we take in the form 



w = -^ (75) 



ar 



In view of the a priori information for mean free time problems which 

 is gathered in Section I IB we can use the statistical data from the 

 Monte Carlo calculation in two ways. We may (a) check the numerical 

 computation itself or (b) gain new information not available otherwise. 



(a) The check of the numerical calculation by statistical analysis 

 proceeds as follows. From deductive reasoning we have obtained the 



