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10 The Statistical Basis of Bioenergetics
In applying statistical ideas to the Gibbs ensemble, we will wait for all members
of the ensemble to relax. As the systems within evolve, they will eventually spend
most of their time in those states which maximize the entropy of the ensemble.
These systems are more numerous in the ensemble as equilibrium is established.
To find the values of the P k for the canonical ensemble at or near equilibrium,
we will maximize the entropy S under the constraints
N
k=1
P k = 1
(10.88)
and
N
k−1
P k E k = E .
(10.89)
The constrained maximization of S = −k B
k P k ln P k gives the same set of
{P k } as the unconstrained maximization of
S F ({P k }) = S/k B + α
k
P k − 1
− β
k
P k E k − E
,
(10.90)
where α and β are ‘Lagrange multipliers’, determined by the two constraints.
The maximum of S F occurs where all the slopes ∂S F /∂P k vanish. Thus, at or
near equilibrium,
− ln P k − 1 + α − βE k = 0 ,
i.e.
P k = exp (−1 + α) exp (−βE k ) .
(10.91)
The first constraint,
k P k = 1 tells us that α is fixed by
exp (−1 + α) =
1
k exp (−βE k )
,
so that
P k =
1
k exp (−βE k )
exp (−βE k ) .
(10.92)
The expression