26
1 Basic Background Material
8. Show that the ratio of the weight W (2N, N H ) associated with an event other
than the most probable event to the weight of the most probable event for 2N
binary outcome trials is given by
ln
W
W max
= −N H ln
N H
N
− N T ln
N T
N
when 2N is a sufficiently large number.
9. We may define a ‘deviation index’ α that provides a measure of the difference
in the numbers of heads and tails obtained upon having completed a sequence
of 2N coin tosses as α ≡ (N H − N T )/(2N). Show that expressions for N H
and N T can also be obtained in terms of N and α as N H = N(1 + α) and
N T = N(1 − α).
10. Use the results of Problems 8 and 9 to show that for |α| | 1, the ratio W/W max
may be approximated as
W
W max
e
−2Nα 2 .
11. Three tetrahedra are embedded in a host of transparent spheres in such a way
that one full face of each tetrahedron is visible. Each tetrahedron has 1 red,
1 blue, 1 white, and 1 green face. How many ways can these tetrahedra be
arranged so that two of the visible faces have the same colour?
References
1. H.S. Leff, Amer. J. Phys. 70, 792 (2002)
2. P.T. Landsberg, Thermodynamics and Statistical Mechanics, ch. 13.1 (Oxford University Press,
Oxford, 1978)
3. F. Reif, The Berkeley Physics Course: Statistical Physics, vol. 5. (McGraw Hill, New York,
1965)