"On a misconception in the probability theory of
sible processes". By Mrs T. EHRENFEST~AFANASSJEWA.
Physics. -
irrever~
(Communicated at the meeting of June 27, 1925).
aften it is thought that the two following statements contradict each other:
I. A given course of a mechanical system with very many
degrees of freedom is as probable as the contrary course, and
STATEMENT
STATEMENT 11. From every state which is not the most probable, the
system tends with great probability toward the most probable state.
This impression really is false and arises from the confusion of two
statements. both equally true, as will be shown in what follows by
means of a simplified scheme. - Con si der a continuous series of discrete
points lying in a plane with all integers (positive and negative) as ab~
scissae, and with ordinates H which comply with the following conditions :
1. The ordinates are positive integers between Hmin=O arid Hmax=Hm .
2. The successive ordinates always differ by 1.
3. The various values of the ordinates have different probabilities
(i. e., appear although infinitely often yet in differing proportions), and
this probability increases from above to below. (Thus the ordinate H
0
has the greatest and the ordinate Hm the least probability).
A point whose ordinate is greater than those of' the two adjacent
ones we will call a "peak"; a point whose ordinate is less than those
of the two adjacent ones, a "valley". - The succession of two points
shall be called a "descending slope" whenever the point with the
smaller abscissa (the beginning") has a greater ordinate than that with
the greater abscissa ("the end"). In the coqtrary case we shall call it a
"rising slope". - We shall s~y that our curve "rises" at a point P
whenever it is the beginning of a rising slope; if the point P is the
beginning of a descending slope we shall say that the curve "descends"
at this point.
Now let us ask two questions :
QUESTION A. Given a certain height H. How great is the probability
W (H -+ (H - 1)) that the curve descends at this point? - This proba~
bility, as may easily be derived from property 3 1) is greater than 1/ 2
for all values of H that differ from. zero. For H
Hm it is equal to I,
for all points at a height H
H mare peaks. But for H 0 it is equal
to O. for at this height all points are valleys. - Consequently it follows,
=
=
=
=
1) Cf. PAUL und TATIANA EHRENPEST. Ueber zwei bekannte Einwände gegen das
Boltzmannsche H-Theorem . Phys. Zschr. 8, 1907, 311.
733
+
I))
that for points at every height H:;6:. 0 the probability W (H ~ (H
that the curve rises there is less than t. Thus we obtain:
THEOREM A. For all heights except H=O the probability that the
curve descends is greater than that it rises.
QUESTION B. To compare the probability of a descending and a rising
slope between the same two heights Hand H - 1. - The answer is:
THEOREM B. The probability for a descending and a rising slope
between two particular heights Hand H - 1 are equally great.
We see this by asking how a certain height H which just has been
reached at a point P can again be reached at a point P'. This can
occur if the point P is the beginning of a raising slope H ~ (H I).
and then necessarily the point P' is the end of a descending slope
(H 1) ~ H. for we consider the point P'. at which. after point P. the
height H is reached for the first time. in which case the height H - 1
is not reached at all between the points P and PI; or if point P is the
beginning of a descending slope H ~ (H - I) and P' the beg inning of
a rising slope (H - 1) -+: H. in which case neither of these two particular
slopes can occur more than on ce between Pand P' . We also notice
that point P' can not be missing from our curve. for each height between
Hm and 0 is reached infinitely of ten.
Thus in any finite part of our curve the count ZI of the slopes
H ~ H - 1 and ZIl of the slopes H - 1 ~ H must be the same or
+
+
differing by no more than one. and in the limit for ZI =
00
is ;;/ = I.
That is to say. the probabilities for the two slopes are the same.
What relation then exists between the two theorems A and B and
the two statements land II? What frequently is not perceived properly
is that statement II is a consequence of theorem A while statement I
·is not directly connected with theorem A. but is a consequence of
theorem B. - The two statements land II c1early are conflicting with
each other as little as the two theorems A and B. - However. since
the occurrence side by side of 1 and II is often considered a paradox.
we shall consider further how theorem B depends on A.
The probability of a descending slope H ~ (H-I) is the product of
two probabilities: the probability W (H) that the beg inning lies at the
height H. and the probability W (H ~ (H-I )). that the curve descends
at that point.
The probability of a rising slope (H-I) ~ H is the product of two
others probabilities. W (H-I) that the beg inning lies at the hight H-l,
and W ((H-l) ~ H) that the curve rises at that point.
Thus from (3) it follows:
W(H)
W(H-l)
and from theorem A:
<
W(H~(H-l))>t
W((H-l)~H))
< t.
48
Proceedings Royal Acad. Amsterdam. Vol. XXVIII.
734
thus
W(H~(H-I)) > W((H-I)~H).
It is thus quite easily possihle that the two products
W(H).
W(H~(H-I))
and
W(H-I).
W((H-I)~H)
are equally large. However. the topological considerations ahove must
he taken as the rigorous proof of th is equality.
Attention mayalso he called to the following: the quasi-periodic
return of the ordinates of our curve to each value assumed once occurs
only hy definition (1.2. 3) and is independent of the sort of phenomenon
of which it is taken as an image.
Leiden, May 1925.