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10 The Statistical Basis of Bioenergetics
thermodynamic properties of the given system and fluctuations of the energy near
equilibrium.
The Grand Canonical Ensemble
Now let us consider a system that allows the exchange of particles with the
environment as well as the exchange of heat. A Gibbs ensemble with N such
systems can be created by allowing each system copy to exchange heat and particles
between them. The complete Gibbs ensemble is isolated. Each ensemble member
has been prepared in a quantum state with the full set of observables labeled by
‘k’. The system labeled by ‘k’ will have its own particle number N k . The energy
of the system k, called E k , is determined by its microdynamics. The E k will be a
function of the volume V of any one of the systems and of its particle number N k :
E k = E k (V , N k ). Now N E and N N are kept fixed during the evolution of the
systems, so that E and N are also fixed. All N − 1 systems outside the system k
act as a heat and particle bath for the selected system k.
The constraints on the ensemble become
k
P k = 1 .
(10.105)
k
P k E k = E .
(10.106)
k
P k N k = N .
(10.107)
At equilibrium, the entropy
S = −k B
k
P k ln P k
is maximized under the constraints, and
S A ({P k }) = S/k B + α
k
P k
− β
k
P k E k
+ γ
k
P k N k
(10.108)
becomes a maximum without any constraints. The function S A ({P k }) reaches a
maximum when
− ln P k − 1 + α − βE k + γ N k = 0 ,
or