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10 The Statistical Basis of Bioenergetics
10.7 Disorder Expressed by Probabilities
Statistical averages are typically written in terms of the probabilities for the
occurrence of certain possibilities. There is a profound connection between the
entropy S for systems of distinguishable entities and the probability that a given
distribution will be found after many trials. If we make N trials, selecting unbiasedly
among a set n of entities, each labeled by ‘i’, then the entropy in the N trials is
determined by
S/k B = ln W
(10.80)
with
W =
N !
n
i=1 n i !
.
For large n, n! ≈ n ln n − n, so
S/k B ≈ N ln N − N −
n i ln n i +
n i = N ln N −
n i ln n i
= N ln N −
n i ln (n i /N ) −
n i ln N
= −N
n
i=1
P i ln P i
(10.81)
where P i = n i /N approaches the probability for the occurrence of the distribution
n i as N → ∞. The expression Eq. (10.81) was developed by Boltzmann for
statistical mechanics and later by Shannon in his definition of information (see
Sect. 12.1) in terms of the probability of the occurrence of each character in a string
making up a message.
10.8 The Partition Function for Ensembles
I would never join a group that would have me as a member.
— Groucho Marx