166 Physical Properties [CH. vn 



were different from this minimum value, it was infinitely probable that 

 H would decrease, but on the other hand, it was infinitely probable 

 assuming the basis of probability supplied by the generalised space, that 

 the initial value of H would be equal to the minimum, in which case, as H 

 could not decrease further, the " expectation " was of a slight increase. Thus 

 the large probability of a small increase in H just balanced the small pro- 

 bability of a finite decrease in H, and on the whole the " expectation " of 

 change in H was nil. 



A precisely similar explanation holds with reference to >. Indeed it will 

 be found that $$ is very closely allied numerically, not with the simple 

 function H of Chapters II. IV. but with the more general function $ 

 of Chapter V. An increase in ^b presupposes an initial difference of 

 temperature between the two component systems ; and these initial con- 

 ditions, looked at from the point of view of abstract dynamics, and judged 

 with reference to the basis of probability supplied by a generalised space, 

 are infinitely improbable. With reference to the same basis of probability, 

 it is infinitely probable that the initial conditions will be those of equilibrium 

 of temperature, in which case the only change possible in ^ is a decrease. 

 Or, physically, the only possible alteration in the state of the system is the 

 production of inequalities of temperature. The production of such an 

 inequality, although improbable when the motion is confined to a short 

 time, is not impossible, and indeed becomes infinitely probable when the 

 motion is continued for a sufficient time. Thus the increase of entropy, 

 even granted the infinitely improbable (from the dynamical point of view) 

 initial conditions which makes such an increase possible, is only a probability 

 and not a certainty ; and when the entropy starts initially at its maximum, 

 it is infinitely probable that, granted sufficient time, the entropy will 

 decrease. 



193. Translated into physical language these results seem at first 

 sight somewhat startling. To borrow an illustration from Lord Kelvin, if 

 we have a bar of iron initially at uniform temperature, and subject neither 

 to external disturbance nor to loss of energy, it is infinitely probable that, 

 given sufficient time, the temperature of one half will at some time differ by 

 a finite amount from that of the other half. Or again, if we place a vessel 

 full of water over a fire, it is only probable, and not certain, that the water 

 will boil instead of freezing. And moreover, if we attempt to boil the water 

 a sufficient number of times, it is infinitely probable that the water will, on 

 some occasions, freeze instead of boil. The freezing of the water, in this 

 case, does not in any way imply a contravention of the laws of nature : the 

 occurrence is merely what is commonly described as a " coincidence," exactly 

 similar in kind to that which has taken place when the dealer in a game of 

 whist finds that he has all the trumps in his hand. 



