28 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



once distinguishing between a state of a considered system and 

 the complexion of the considered system. A state of the system is 

 determined by the law of locus and velocity distribution, i.e., by 

 a statement of the number of particles which lie in each ele- 

 mentary district of space and the number of particles which lie in 

 each elementary velocity realm, assuming that among themselves 

 these districts and realms are alike and each such infinitesimal 

 element still harbors very many particles. Accordingly a particu- 

 lar state of the system embraces a very large number of com- 

 plexions. For if any two particles belonging to different regions 

 swap their co-ordinates and velocities, we get thereby a new 

 complexion, but still the same state. Now BOLTZMANN assumes all 

 complexions to be equally probable and therefore the number of 

 complexions included in a particular state furnishes at the same 

 time the numerical value for the Probability of the state in 

 question. Illustration taken from the simultaneous throwing of 

 two, ordinary, cubical dice. Suppose that the sum is to be 4 for 

 each throw, then this can be realized by the following three com- 

 plexions: 



First cube shows i, the second cube shows 3; 

 First cube shows 2, the second cube shows 2; 

 First cube shows 3, the second cube shows i. 



The requirement that the sum on the two cubes shall be 2, how- 

 ever, involves but one complexion. Under the circumstances 

 therefore the probability of throwing the sum 4 is three times 

 as great as throwing the sum 2. 



In closing this part of our presentation, we may make what is 

 now an almost obvious remark. The long-lasting difficulty in 

 giving a physical meaning to entropy and the Second Law is due 

 to the fact of its intimate dependence on considerations of prob- 

 ability. It is only quite recently that such considerations have 

 attained the dignity of a great working principle in the domain of 

 Physics. 



