574 DISCUSSION ON THE 
compartment indicates that we are already 4,000 feet above sea level, 
there are very long odds that the general trend of our journey will be 
downhill. 
It is this kind of consideration rather than exact knowledge that 
guides us in our efforts to study the evolution and final end of the universe. 
We can have no certainty and must be guided entirely by probabilities. 
Yet the odds we encounter prove always to be so immense that we may, 
for all practical purposes, treat long odds as certainties. The number of 
particles—electrons and protons—in the universe is of the order of 10”. 
As a consequence, high powers of 10” enter into all our odds, and, this 
_ being so, we need not trouble to differentiate too carefully between long 
odds and certainties. 
In our diagram of the universe there is a quantity—the entropy— 
which plays much the same part as height played in our imaginary railway 
map, except that small entropy corresponds to great height, and vice-versa. 
Thus entropy corresponds rather to depth below the top of the highest 
mountain, whose height is 4,400 feet. As most of Great Britain is only 
a little above sea-level, most of it is at a depth, in this sense of the word, 
of nearly 4,400 feet—the maximum depth possible. In the same way, 
most of our map of the universe is at the maximum entropy possible— 
all, indeed, except for tiny bits proportional to inverse powers of 10”. 
At the moment we cannot prove this statement because we have not 
yet defined ‘entropy.’ And there is no need to prove it, because the best 
definition of ‘entropy’ makes the statement true of itself and auto- 
matically. We may define maximum entropy as specifying the condition 
which is commonest in our map of the universe; we define entropy in 
general so that the more common condition is always of higher entropy 
than the less common. Entropy gives a measure of the ‘ commonness ’ of 
a given state in our map. Actually, if W is the ‘commonness,’ of a 
certain state, the mathematician defines the entropy of this state k log W, 
where & is the gas-constant. 
Just because such immense numerical factors are involved, con- 
ditions of ‘maximum’ entropy are incomparably more common than 
those whose entropy is less, and so on all down the ladder. Thus it is 
practically certain that the universe will ‘ evolve’ through a succession of 
states of ever-increasing entropy, until it ends in the final state of 
maximum entropy. Beyond this it cannot go; it must come to rest— 
not in the sense that every atom in it will have come to rest (for maximum 
entropy does not involve this) but rather in the sense that its general 
characteristics cannot change any more. 
Yet, if someone asserts that this will not happen, and that the universe 
will move to a state of lower entropy than the present, we cannot prove 
him wrong. He is entitled to his opinion, either as a speculation or as a 
pious hope. All we can say is that the odds against his dream coming 
true involve a very high power of 10°—in his disfavour. 
The question of discovering the final state of the universe is merely 
that of discovering how far its entropy can increase without violating the 
physical laws which govern the motions of its smallest parts—the physical 
properties of matter come into account as soon as we try to discover the 
state of maximum entropy. Let us take two simple instances. I pour 
—— ee Cm 
— a a ee 
oe 2) eee 
